# Compass & Straightedge Construction and the Impossible Constructions

### From Math Images

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| -                                      | {{Image Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +                                                    | {{Image Description Ready                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        | |ImageName=Creating a regular hexagon with a ruler and compass                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |                                                      | |ImageName=Creating a regular hexagon with a ruler and compass                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        | |Image=HexagonConstructionAni.gif                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                      | |Image=HexagonConstructionAni.gif                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        | |ImageIntro=This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      | |ImageIntro=This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| -                                      | |ImageDescElem=Many have learned about compass and straightedge construction in a very simplified manner. Nonetheless, it is one of the most interesting aspect of geometry a very hands-on experience. It makes us to seriously appreciate geometry in its purest and simplest form. With a compass and a unmarked straightedge, the possible constructions are almost unlimited. However, it is always the case that many geometry textbooks only touche on the subject in an introductory and perfunctory manner, not reaching the deeper issues surrounding the subject. On the other hand, information on the subject from the internet has great breadth but little depth. Hence, this page is in intended to introduce compass and straightedge construction to those to whom the subject is new and also give a fair amount of in-depth analysis on the subject. Hopefully as a result, this page can serve as a gateway that leads the abler and more interested to places where they can find more information on the subject.                                                                                                                                                                                                                                            | +                                                    | |ImageDescElem=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | Let's assume we only have a compass and a unmarked straightedge. What can we construct and how can we construct them? That were the problems that Euclid pondered not only because those were probably the only instruments that he had at his time but also he wanted to build his theorems with as few assumptions, or {{EasyBalloon|Link=axioms|Balloon=In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.}}, as possible.<ref>Peterson, 2003</ref> With these two simple tools, he managed to build myriad of theorems in both plane and solid geometry .<ref>Hartshorne, 2000</ref> These theorem are as good as they were some 2000 years ago and it is this enduring quality of Euclid's work that inspired this page. In the main image of this page, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon. It seems to be a fairly simple construction but you should be prompted to ask two questions: Are other polygons constructible  and is every polygon constructible, that is able to be constructed using only compass and straightedge? To extend the question, what are constructible and what are not? That is the problem that is resolved in this page. |
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| -                                      | Let's assume we only have a compass and a unmarked straightedge. How could we construct certain geometric shapes and prove certain theorems? That was the problems that Euclid pondered not only because those were probably the only instrument that he had at his time but also he wanted to construct his theorems with as few assumptions, or axioms, as possible. In this picture, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon. It seems to be a fairly simple construction but it should prompt you to ask a question: is every polygon constructible, that is able to be constructed using only compass and straightedge? To extend the question, what are constructible and what is not? The issue will be addressed in a later subsection.                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                    | |ImageDesc===What is Compass & Straightedge Constructions==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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| -                                      | |ImageDesc===The Compass and straightedge==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:Compass.jpg|border|right|150px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | A Compass is a tool which can be used for drawing circles. It has two legs, one end of which is fixed on the plane of construction and the other end is of desirable distance away and maintains the distance throughout the construction. A normal form of compass is shown on the right but there are many other variants, crude or precise. For example, a crude form will be pinning a thread of string on the plane and fixing a pencil/pen at a certain distance away from the pin. We usually see this kind of crude compass in military field operations. Look out for such a compass in HBO's TV series '''''Band of Brothers''''' second episode when the paratroopers were trying to figure out where they would be dropped on D-Day based on the distances of their preparatory flights.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | A straightedge is a tool which can be used for drawing straight lines, or segments thereof. It should be noted that the difference between a straightedge and a ruler is that the former has no graduations on it (and does not allow any markings as well) while the latter has divisions on them according to certain unit system, be it the metric or the British customary system.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | ==What is Compass&Straightedge Constructions==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        | ==='''Introduction'''===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                      | ==='''Introduction'''===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | We start by familiarizing ourselves with Euclid's three Postulates in his books '''''Elements'''''.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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| -                                      | Compass & Straightedge Construction is the construction of points, lengths, angles, and other geometric figures using only an idealized straightedge and compass. The straightedge is infinite in length, has no markings on it and only one edge. The compass collapses when lifted from the page, so may not be directly used to transfer distances. However, it turns out that this restriction makes no difference due to the Compass Equivalence Theorem which was stated as Proposition II of Book I of Euclid's Elements. It stated that from a given point, it was possible to construct a straight line equal to a given straight line using collapsible compass. Euclid's proof for the Compass Equivalence Theorem will be presented after the section of Basic Construction.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                    | <blockquote>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{{!}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}'''''Let it be granted                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}'''''1. that a straight line may be drawn from any one point to any other point;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}-                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}'''''3. that given any straight line segment, a circle may be described having the segment as radius and one endpoint as center.<ref>Taylor, 1895, p. 14</ref>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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| -                                      | We start by familiarizing ourselves with Euclid' three Postulates in his book Elements                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                    | {{{!}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | It should be carefully noted that Euclid started with two given points and produced a line segment, from where he could extend into a straight line if he pleased. Then ONLY from the original two points, he could use one point as center and ONLY spread the legs of compass the distance of the line segment to produce a circle. Even though many translations say that a circle can be described at any center with any radius, we must take it with a pinch of salt. He could not specify any points and lengths other than what was already given, that is to say he could not claim "I wanted to spread the legs of the compass <math>\pi</math> centimeters apart (or any specified denominations) with the center of the circle half way between the two given points". In all, Compass and Straightedge Constructions only allow us to start with points (and hence lengths) we have been given(or constructed from given points), and create the ones we don't.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | Thus, we define Compass & Straightedge Construction as the construction of points, lengths, angles, and circles using only ideal straightedge and compass. A straightedge is infinite in length, has no markings on it and only one edge. A compass has two legs, one end of which is fixed on the plane of construction and the other end is of given distance away and maintains the distance throughout the construction. It collapses when lifted from the page, so may not be '''directly''' used to transfer distances. However, it turns out that this restriction makes no difference due to the Compass Equivalence Theorem which was stated as Proposition II of Book I of Euclid's Elements. It stated that from a given point, it was possible to construct a line segment equal to a given line segment using collapsible compass in any desirable direction. Euclid's proof for the Compass Equivalence Theorem will be presented after the section of Basic Construction. Since Euclid has proven this using only the three postulates, then he did not have to use a collapsible compass any more.<ref>Peterson, 2003</ref>                                                                                                                                                                                                                                                                                                                                                       |
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| -                                      | <blockquote> "Let it be granted                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | ===Some Basic Constructions===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| -                                      | # that a straight line may be drawn from any one point to any other point;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | The constructions below are some basic ones from where many more constructions are possible and they are by no means exhaustive. In the figures below, what we are given are in blue; intermediate steps are in dotted black; the resulting products are in red. The proofs for these constructions are relatively simple and only require the knowledge of [[congruent triangles]]. Euclid derived the theories on congruency and congruent triangles directly from  his Postulates. Try proving the theorems yourself!                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | # that a line segment may be extended into a straight line;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # that given any straight line segment, a circle may be described having the segment as radius and one endpoint as center. " </blockquote>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | ===='''Line Segment Bisection'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | [[Image:CS1.png|border|550px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{SwitchPreview|ShowMessage=Click here to show construction|HideMessage=Click here to hide construction|PreviewText=Given points <math>{\color{Gray}A}</math> and <math>{\color{Gray}B}</math> and the straight line passing through it. Construct a line that bisects line segment <math>AB</math>.|FullText=Given points <math>A</math> and <math>B</math> and the straight line passing through it. Construct a line that bisects line segment <math>AB</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | It should be carefully noted that Euclid started with two given points and produced a line segment, from where he could extend into a straight line if he pleases. Then from the original two points, he could use one point as center and spread the legs of compass the distance of the line segment to produce a circle. He could not (so couldn't we)however, choose any two points as he please. The three postulates was stated in ancient Greek and both its old and modern translations have not been consistent with each other. What is stated above is an assortment of different translations that give the least confusions about what could and could not be done with a compass and a straightedge. Throughout his book, Euclid used only these three operations to do all his plane geometry: the drawing of straight line through two given points, and circle with a given center to pass through a given point. The terms ''Euclidean construction'', ''construction'', ''construct'' and ''constructible'' all refer to Euclid's three operations repeated any finite number of times.                                                                                                                                                                          | +                                                    | # Draw a circle centered at point <math>A</math> with radius equals <math>AB</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | ===Some Basic Construction===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | The constructions below are some basic ones from where many more constructions and operations are possible and they are by no means exhaustive. The resulting products are in red. The proofs for these constructions are relatively simple and only require the knowledge of congruent triangles. Try proving the theorems yourself!                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | ===='''Line Bisection'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:CS1.png|border|700px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | Given points <math>A</math> and <math>B</math> and the straight line passing through it. Construct a line that bisects line segment <math>AB</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Draw a circle centered at point <math>A</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | # Next, draw a circle centered at point <math>B</math> with the same radius.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      | # Next, draw a circle centered at point <math>B</math> with the same radius.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| -                                      | # The two circles intersect at points <math>C</math> and <math>D</math>(Of course, you have to make sure they intersect in the first place. One way to do this is to let the radius be equal to <math>AB</math>).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                    | # Where the two circles intersect, call those points <math>C</math> and <math>D</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        | # Draw a line through points <math>C</math> and <math>D</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      | # Draw a line through points <math>C</math> and <math>D</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # <math>CD</math> intersects <math>AB</math> at the midpoint.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | It should be noted that <math>CD \perp AB</math> as well. Line <math>CD</math> is the perpendicular bisector of line segment <math>AB</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                    | Line <math>CD</math> intersects line segment <math>AB</math> at the midpoint. It should be noted that <math>CD \perp AB</math> as well. Line <math>CD</math> is the perpendicular bisector of line segment <math>AB</math>.}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        | ===='''Angle Bisection'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                      | ===='''Angle Bisection'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| -                                      | [[Image:CS2.png|border|700px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    | [[Image:CS2.png|border|550px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
| -                                      | Given angle <math>\angle AOB</math>, construct a line that bisects the angle.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                    | {{SwitchPreview|ShowMessage=Click here to show construction|HideMessage=Click here to hide construction|PreviewText=Given angle <math>{\color{Gray}\angle AOB}</math>, construct a line that bisects the angle.|FullText=Given angle <math>\angle AOB</math>, construct a line that bisects the angle.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Construct a circle centered at point <math>O</math> with radius <math>OA</math>. This circle should intersect at points <math>A</math> and <math>B</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                    | # Construct a circle centered at point <math>O</math> with radius <math>OA</math>. This circle intersects <math>OA</math> and <math>OB</math> at point <math>A</math> and <math>B</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -                                      | # Keeping the same radius, draw a circle at points <math>A</math> and <math>B</math>. They must intersect each other at <math>C</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                    | # Keeping the same radius, draw a circle at point <math>A</math> and <math>B</math> respectively. Where they intersect each other, call it point <math>C</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
| -                                      | # Draw a line through points <math>C</math> and <math>O</math>. This line bisects  <math>\angle AOB</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    | # Draw a line through point <math>C</math> and <math>O</math>. This line bisects  <math>\angle AOB</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | ===='''Perpendicular Through a Point'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | [[Image:Perp.png|border|450px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{SwitchPreview|ShowMessage=Click here to show construction|HideMessage=Click here to hide construction|PreviewText=Given a point <math>{\color{Gray}M}</math> on a line <math>{\color{Gray}AB}</math>, construct a line that is perpendicular to the given line through <math>M</math>.|FullText=Given a point <math>M</math> on a line <math>AB</math>, construct a line that is perpendicular to the given line through <math>M</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | ===='''Perpendicular through a point'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    | # Draw a circle centered at <math>M</math> with radius <math>MB</math> (or <math>MA</math>. It is your choice).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:CS3.png|border|600px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    | # Where the circle intersects the original line, construct a perpendicular bisector (see the line segment bisector construction above).}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -                                      | Given a point on a line, construct a line that is perpendicular to the given line through the given point.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Draw a circle centered at <math>M</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Where the circle intersects the original line, construct a perpendicular bisector.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | ===='''Parallel Line'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | [[Image:CS4.png|border|550px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{SwitchPreview|ShowMessage=Click here to show construction|HideMessage=Click here to hide construction|PreviewText=Given two points, <math>{\color{Gray}A}</math> and <math>{\color{Gray}B}</math> and the straight line passing through them, construct a line that is parallel to the given line through another given point <math>C</math>.|FullText=Given two points, <math>A</math> and <math>B</math> and the straight line passing through them, construct a line that is parallel to the given line through another given point <math>C</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | ===='''Parallel'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                    | # Draw a circle at <math>A</math>, crossing <math>C</math>. Where the circle <math>A</math> intersects <math>AB</math>, call the point <math>D</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | [[Image:CS4.png|border|750px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    | # Centered at points <math>C</math> and <math>D</math>, draw circles crossing point <math>A</math>. Where these two circles intersect each other, call it <math>E</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -                                      | Given two points and the straight line passing through, construct a line that is parallel to the given line through another given point.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                    | # Draw a line through point <math>C</math> and <math>E</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Draw a circle at <math>A</math>, crossing <math>C</math>. Circle <math>A</math> intersects <math>AB</math> at <math>D</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Centered at points <math>C</math> and <math>D</math>, draw circles crossing point <math>A</math>. These two circles intersect each other at <math>E</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Connect points <math>C</math> and <math>E</math> with a line. <math>CE</math> is parallel to <math>AB</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | <math>CE</math> is parallel to <math>AB</math>}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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|                                        | ===='''Tangent Line to a Circle'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                      | ===='''Tangent Line to a Circle'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | [[Image:CS5.png|border|750px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    | [[Image:CS5.png|border|550px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
| -                                      | Given a circle and another given point, construct a line that is tangent to the circle.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             | +                                                    | {{SwitchPreview|ShowMessage=Click here to show construction|HideMessage=Click here to hide construction|PreviewText=Given a circle centered at <math>{\color{Gray}O}</math> and another given point, <math>{\color{Gray}A}</math>, construct a line that is tangent to the circle.|FullText=Given a circle centered at <math>O</math> and another given point, <math>A</math>, construct a line that is tangent to the circle.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Connect the point <math>A</math>, with the center of the circle <math>O</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                    | # Draw a line through point <math>A</math> and the center of the circle <math>O</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        | # Let <math>M</math> be the midpoint of <math>OA</math>(construction omitted since we know how to construction mid point). Draw the circle centered at <math>M</math> going through <math>A</math> and <math>O</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                      | # Let <math>M</math> be the midpoint of <math>OA</math>(construction omitted since we know how to construction mid point). Draw the circle centered at <math>M</math> going through <math>A</math> and <math>O</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        | # Let the point where the two circles meet be <math>C</math>. Draw line segment <math>AC</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                      | # Let the point where the two circles meet be <math>C</math>. Draw line segment <math>AC</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
| -                                      | # To see that <math>AC</math> is tangent to the circle, connect <math>OC</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # <math>\angle ACO</math> is an inscribed angle in the circle about <math>M</math>, so the inscribed <math>\angle ACO</math> is <math>90^\circ</math>. So <math>OC</math> is perpendicular to <math>AC</math>. Since the tangent is perpendicular to the radius to the point of tangency, by the uniqueness of the line through <math>C</math> perpendicular to <math>OC</math>, <math>AC</math> is tangent to the circle.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | To see that <math>AC</math> is tangent to the circle, connect <math>OC</math>. <math>\angle ACO</math> is an inscribed angle in the circle about <math>M</math>, so the inscribed <math>\angle ACO</math> is <math>90^\circ</math>. So <math>OC</math> is perpendicular to <math>AC</math>. Since the tangent is perpendicular to the radius to the point of tangency, by the uniqueness of the line through <math>C</math> perpendicular to <math>OC</math>, <math>AC</math> is tangent to the circle.}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | ===='''Euclid's Proof of Compass Equivalence Theorem'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                      | ===='''Euclid's Proof of Compass Equivalence Theorem'''====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | This part reconnects with the previous section about the issue of compass being collapsible. Euclid's proof is presented in its originality. My comments are contained in the parenthesis.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | This part refers back to the previous section about the issue of compass being collapsible. Euclid's original proof is presented. Additional comments are contained in the parenthesis.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:CETpic.png|border|center|600px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | From a given point to draw a line segment equal to a given line segment.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | <u>Let <math>A</math> be the given point, and <math>BC</math> the given straight line : it is required to draw from the point <math>A</math> a straight line equal to <math>BC</math>.</u>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | [[Image:CETpic.png|border|center|450px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{SwitchPreview|ShowMessage=Click here to show construction|HideMessage=Click here to hide construction|PreviewText=From a given point to draw a line segment equal to a given line segment.|FullText=From a given point to draw a line segment equal to a given line segment. <u>Let <math>A</math> be the given point, and <math>BC</math> the given straight line : it is required to draw from the point <math>A</math> a straight line equal to <math>BC</math>.</u>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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|                                        | # From the point <math>A</math> to <math>B</math> draw the straight line <math>AB</math> and on it describe the equilateral triangle <math>DAB</math>(this is done the same way as bisecting a line segment), and produce the straight lines <math>DA</math>, <math>DB</math> to <math>E</math> and <math>F</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                      | # From the point <math>A</math> to <math>B</math> draw the straight line <math>AB</math> and on it describe the equilateral triangle <math>DAB</math>(this is done the same way as bisecting a line segment), and produce the straight lines <math>DA</math>, <math>DB</math> to <math>E</math> and <math>F</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | # From the center <math>B</math>, at the distance <math>BC</math>, describe the circle <math>CGH</math>, meeting <math>DF</math> at <math>G</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +                                                    | # From the center <math>B</math>, at the distance <math>BC</math>(with the radius equals to <math>BC</math>), describe the circle <math>CGH</math>, meeting <math>DF</math> at <math>G</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| -                                      | # From the center <math>D</math>, at the distance <math>DG</math>, describe the circle <math>GKL</math>, meeting <math>DE</math> at <math>L</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 | +                                                    | # From the center <math>D</math>, at the distance <math>DG</math>(with the radius equals to <math>DG</math>), describe the circle <math>GKL</math>, meeting <math>DE</math> at <math>L</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        | # <math>AL</math> shall be equal to <math>BC</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                      | # <math>AL</math> shall be equal to <math>BC</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| Line 109:                              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | Line 104:                                            |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | Wherefore from the given point <math>A</math> a straight line <math>AL</math> has been drawn equal to the given straight line <math>BC</math>.∎                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | Wherefore from the given point <math>A</math> a straight line <math>AL</math> has been drawn equal to the given straight line <math>BC</math>.∎<ref>Taylor, 1895, p. 18</ref>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | It should be noted that from <math>AL</math>, we could "duplicate" <math>AL</math> in all directions by construct a circle centered at <math>A</math> with radius <math>AL</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | It should be noted that from <math>AL</math>, we could "duplicate" <math>AL</math> in all directions by construct a circle centered at <math>A</math> with radius <math>AL</math>.}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
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|                                        | ==Algebraicization of Compass & Straightedge Constructions==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      | ==Algebraicization of Compass & Straightedge Constructions==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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| -                                      | ==='''A simple derivation'''===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | ==='''What is Algebraicization?'''===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
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| -                                      | From the few basic constructions, you would have probably realized that the different possibilities seems infinite. However, by intuition, we know that the possibility could not be infinite. Hence, mathematician are curious to find out what are constructible and what aren't and for this purpose, the language of pure geometry seems to have "limited vocabulary". Back in ancient times, mathematicians had limited algebraic knowledge and were more familiar with geometry. But in modern times, the reverse is true. Hence, today's mathematicians go back to their familiar realm of Algebra and try to find the link between geometry and algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | ''Algebraicization''? This is certainly a strange word to you. You probably cannot even find it in a dictionary or encyclopedia entry. However show this to any mathematician or math majors, they will immediately tell you what that is all about. ''Algebraicization'' is the translation of any problem statements into algebraic problems. In the case of Compass & Straightedge construction, we algebraicize each step of a straightedge and compass construction, and consequently obtaining general results about the nature of constructibility. Hilda P. Hudson put it aptly in his lecture ''Ruler & Compasses'',                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | <blockquote>"each step of a ruler and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics. For this reason, problems that can be solved with ruler only are called linear problems, and those that can be solved with ruler and compasses are called quadratic problems. Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider that these algebraic processes can lead to , when combined in every possible way, and that enables us to answer the question before us and say that those problems and those problems alone can be solved by ruler only, which can be made to depend on a linear equation, whose root can be calculated by carrying out rational operations only; and that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only."</blockquote> | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | In order to algebraicize straightedge and compass construction, we begin by choosing a specific length to be considered one unit.  Then, every time we construct a figure, we think of it instead as constructing a set of ''numbers'' representing the lengths of the constructed line segments.Compass & straightedge construction only allow us to construct points that are at intersections of lines and/or circles.  This means that every time we do a construction, we can use equations of circles and lines from coordinate geometry to figure out what numbers are being constructed.  In this way, a geometric process is translated into an algebraic process.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | Firstly, we define "1" on a straight line by specifying the length between any two point. But some will say, hey, in Euclid's postulates, he did not say you can choose any points; a point is the intersection of two lines, two circles or one line and one circle. That is a good observation. To have a line, we have to have given points. Then we draw the straight line. Choose one of the two given points, and use that as the center for our compass. Now, spread the legs of the compass by a distance between the two given points. The intersection is a new point. With the new point, we now have one arbitrarily chosen points.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | From the few basic constructions, you would have probably realized that the different possibilities seems infinite. Hence, mathematician are curious to find out what are constructible and what aren't and for this purpose, the language of pure geometry seems to have "limited vocabulary"<ref>Hudson, 1916, p. 3</ref>. Back in ancient times, mathematicians had limited algebraic knowledge and were more familiar with geometry. But in modern times, the reverse is true. Hence, today's mathematicians go back to their familiar realm of Algebra and try to find the link between geometry and algebra.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
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| -                                      | Note, the "1" does not necessarily have to measure meter nor foot since we have chosen two random points. It could measure a new length of your choosing. Then, once you have chosen that length to be "1", you have to stick to this specification throughout your construction.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                    | ''Algebraicization'' is the translation of any problem statements into algebraic problems. In the case of Compass & Straightedge construction, we algebraicize each step of a straightedge and compass construction, and consequently obtaining general results about the nature of constructibility. Hilda P. Hudson put it aptly in his lecture '''''Ruler & Compasses''''',                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | <blockquote>''"each step of a ruler and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics...... Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider that these algebraic processes can lead to , when combined in every possible way, and that enables us to answer the question before us......"''<ref>Hudson, 1916, p. 3</ref></blockquote>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
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| -                                      | Next, it is very obvious that we could construct all the integers, that is <math>\cdots -3,-2,-1,0,1,2,3,\cdots</math> (or <math>x</math> = <math>\{x|- \infty < x < \infty,x \in \mathbb{Z}\}</math>). How so? Well, once we have the "1", all we have to do is to use the Compass Equivalence Theorem finite number of times to duplicate the length "1" that we previously defined. Now, that means that we could have any two random integers, <math>a</math> and <math>b</math>, and for the sake of this discussion and clarity, we are talking about positive integers here. Next, I will show that from <math>a</math> and <math>b</math>, we could construct <math>a \pm b</math>, <math>a \times b</math> and <math>\frac {a}{b}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                    | Hudson lectured on this in the early 20th century and certain phrases of his could potentially cause confusion. The take-away from this paragraph is that in order to algebraicize straightedge and compass construction, we begin by designating a given point as the origin and the coordinates of another given point (we are given two points at least) as <math>(1,0)</math> or <math>(0,1)</math>. Thus we have established the Cartesian Coordinates. Then, every time we construct a straight line or a circle, we think of it instead as adding a new equation into a system of equations. These equations represent the coordinates of all the points on the line or circle, but that is easy since we all know the expression for a line and a circle as <math>y = ax + b</math> and <math>(x-m)^2 + (y-n)^2 = r^2</math>. However, the only times we can pinpoint a point (and find its coordinates as a result) is when a line intersects with a line, or a circle, or a circle intersects with another circle in which case we can pinpoint 2 points. We then conclude that only those coordinates of the points of intersections are constructible. In this way, a geometric process is translated into an algebraic process.                                                                                                                                                                                                                                                      |
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| -                                      | To construct <math>a \pm b</math>, we will use <math>a</math> as the center and use <math>b</math> as radius. The two points of intersection with the line will be <math>a+b</math> and <math>a-b</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                    | ==='''A Simple Derivation'''===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:A+-b.png|center|border|700px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | Firstly, we define "1" on a straight line as stated previously. Then, once you have chosen that point to be <math>(1,0)</math>, you have to stick to this specification throughout your construction. Next, it is very obvious that we could construct all the integers, that is <math>\cdots -3,-2,-1,0,1,2,3,\cdots</math> (or <math>x</math> = <math>\{x|- \infty < x < \infty,x \in \mathbb{Z}\}</math>). How so? Well, once we have the "1", all we have to do is to use the Compass Equivalence Theorem finite number of times to duplicate the length "1" that we previously defined. Now, that means that we could have any two random integers, <math>a</math> and <math>b</math>, and for the sake of this discussion and clarity, we are talking about positive integers here. Next, it is shown that from <math>a</math> and <math>b</math>, we could construct <math>a \pm b</math>, <math>a \times b</math>, <math>\frac {a}{b}</math> and <math>\sqrt {a}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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| -                                      | To construct <math>a \times b</math>, we have <math>0</math>, <math>1</math>, <math>a</math> and <math>b</math> on the straight line.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    | {{{!}}border="1"                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}align="center"{{!}}<math>a \pm b</math>{{!}}{{!}}align="center"{{!}}<math>a \times b</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}-                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}align="center"{{!}}[[Image:A+-b2.png|center|border|300px]]{{!}}{{!}}align="center"{{!}}[[Image:Achub.png|center|border|400px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}-                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}To construct <math>a \pm b</math>, we will use <math>a</math> as the center and use <math>b</math> as radius. The two points of intersection with the line will be <math>a+b</math> and <math>a-b</math>.{{!}}{{!}}To construct <math>a \times b</math>, we have <math>0</math>, <math>1</math>, <math>a</math> and <math>b</math> on the straight line <math>l_0</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
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| -                                      | # Draw any straight line through <math>0</math>, call it <math>l_1</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | # Draw a straight line through <math>0</math>, call it <math>l_1</math>. <math>l_1</math> could be constructed in many ways. For example, it could be the tangent line to circle centered at <math>a</math> with radius equals to the line segment connecting <math>b</math> to <math>a</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | # Construct circle centered at <math>0</math> with radius <math>b</math>, intersecting <math>l_1</math> at <math>B</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      | # Construct circle centered at <math>0</math> with radius <math>b</math>, intersecting <math>l_1</math> at <math>B</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        | # Connect <math>B</math> and <math>1</math>, call it <math>l_2</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                      | # Connect <math>B</math> and <math>1</math>, call it <math>l_2</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| Line 149:                              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | Line 139:                                            |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | # Construct circle centered at <math>0</math> with radius <math>0A</math>, intersecting <math>l_0</math> at a point.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                      | # Construct circle centered at <math>0</math> with radius <math>0A</math>, intersecting <math>l_0</math> at a point.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
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| -                                      | The distance between <math>0</math> and that point is <math>ab</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                    | The distance between <math>0</math> and that point is <math>a \times b</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}-                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}align="center"{{!}}<math>\frac {a}{b}</math>{{!}}{{!}}align="center"{{!}}<math>\sqrt {a}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}-                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}align="center"{{!}}[[Image:Axb.png|center|border|400px]]{{!}}{{!}}align="center"{{!}}[[Image:Sqrta2.png|center|border|350px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}-                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}To construct <math>\frac {a}{b}</math>, we have <math>0</math>, <math>1</math>, <math>a</math> and <math>b</math> on the straight line.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:Atimesb3.png|center|border|600px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | # Draw straight line through <math>0</math>, call it <math>l_1</math>. Construction of <math>l_1</math> is the same as above.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | Similarly, we could construct <math>\frac {a}{b}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Draw any straight line through <math>0</math>, call it <math>l_1</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | # Construct circle centered at <math>0</math> with radius <math>b</math>, intersecting <math>l_1</math> at <math>B</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      | # Construct circle centered at <math>0</math> with radius <math>b</math>, intersecting <math>l_1</math> at <math>B</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        | # Connect <math>B</math> and <math>1</math>, call it <math>l_2</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                      | # Connect <math>B</math> and <math>1</math>, call it <math>l_2</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| Line 163:                              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | Line 154:                                            |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | The distance between <math>0</math> and that point is <math>\frac {a}{b}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |                                                      | The distance between <math>0</math> and that point is <math>\frac {a}{b}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}{{!}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | Therefore, it has been proven that we could construction all the rational numbers since <math>a</math> and <math>b</math> are any arbitrary integers. The natural question to ask right now is that what else is possible to construct? It is not hard to think of numbers that are not rational. For example, <math>\sqrt{2}</math> is constructible. Construct a unit square and the diagonal is of length <math>\sqrt{2}</math>. So is it possible to construct <math>\sqrt{a}</math> given any constructible number <math>a</math>? It turns out that we could. See below for method.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:Ab.png|center|border|750px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                    | To construct <math>\sqrt {a}</math>, we start with <math>0</math>, <math>1</math>, <math>a</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | I will leave the proofs to you since they are very simple using similar triangles.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                    | # Construct <math>a+1</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | # Construct <math>\frac {1}{2}(a+1)</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | # Construct circle at <math>\frac {1}{2}(a+1)</math> with radius <math>\frac {1}{2}(a+1)</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | # Draw perpendicular through <math>a</math>, intersecting the circle at point <math>A</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | <math>aA=\sqrt {a}</math>.Again, I will leave the proof to you as well using similar triangles.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | {{!}}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | Therefore, it has been proven that we could construction all the rational numbers since <math>a</math> and <math>b</math> are any arbitrary integers.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | The natural question to ask right now is that what else is possible to construct? It is not hard to think of numbers that are not rational. For example, <math>\sqrt{2}</math> is constructible. Construct a unit square and the diagonal is of length <math>\sqrt{2}</math>. So is it possible to construct <math>\sqrt{a}</math> given any constructible number <math>a</math>? It turns out that we could. See below for method.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:Sqrta.png|center|border|650px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Construct <math>a+1</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Construct <math>\frac {1}{2}(a+1)</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Construct circle at <math>\frac {1}{2}(a+1)</math> with radius <math>\frac {1}{2}(a+1)</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # Draw perpendicular through <math>a</math>, intersecting the circle at point <math>A</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | # <math>aA=\sqrt {a}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | Again, I will leave the proof to you as well using similar triangles.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | I will leave the proofs to you since they are very simple using similar triangles.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | Next, we moved to the general solution of the problem.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                      | Next, we moved to the general solution of the problem.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| Line 198:                              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | Line 183:                                            |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        | Since <math>x_1</math>, <math>x_2</math>, <math>y_1</math> and <math>y_2</math> are constant we can express this as <math>ax+by=p</math> which is the general expression of a straight line.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      | Since <math>x_1</math>, <math>x_2</math>, <math>y_1</math> and <math>y_2</math> are constant we can express this as <math>ax+by=p</math> which is the general expression of a straight line.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | [[Image:Arbitrarypoint.png|center|border|650px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                    | [[Image:Arbitrarypoint.png|center|border|500px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        | \end{cases}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |                                                      | \end{cases}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
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| -                                      | To solve for the points of intersection, we only need the operations of addition, subtraction, multiplication and division along with the <u>extraction of square roots</u>. Therefore, from this analysis, we have turned geometric problem into algebraic problem and come to the conclusion that '''a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots'''.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                    | Similarly, if you there are two circles intersecting, then the two points of intersections have to satisfy the two equations of the circles. To solve for the points of intersection, we only need the operations of addition, subtraction, multiplication and division along with the <u>extraction of square roots</u>. Therefore, from this analysis, we have turned geometric problem into algebraic problem and come to the conclusion that '''a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots'''.<ref>Bryant, & Sangwin, 2008, p. 77</ref>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
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|                                        | What I have presented above is a simplified version of the derivation towards the theorem. To see a rigorous proof of this theorem at a college level, refer to the text below which is mainly taken from I. N. Herstein's ''Topics in Algebra, Second Edition''. You need some knowledge in Linear Algebra and/or Abstract Algebra. Also see [http://en.wikipedia.org/wiki/Constructible_number Constructible Numbers]. You should not be discouraged should you find it hard to understand. Instead, you should be marveled by the simplicity and elegance of the algebraic proof.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                      | What I have presented above is a simplified version of the derivation towards the theorem. To see a rigorous proof of this theorem at a college level, refer to the text below which is mainly taken from I. N. Herstein's ''Topics in Algebra, Second Edition''. You need some knowledge in Linear Algebra and/or Abstract Algebra. Also see [http://en.wikipedia.org/wiki/Constructible_number Constructible Numbers]. You should not be discouraged should you find it hard to understand. Instead, you should be marveled by the simplicity and elegance of the algebraic proof.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
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| -                                      | {{HideShow|1=We have proven that if <math>a</math> and <math>b</math> are constructible numbers, then so are <math>a \pm b</math>, <math>ab</math>, and when <math>b \ne 0</math>, <math>\frac {a}{b}</math>. Therefore, the set of constructible numbers form a subfield, <math>W</math>, of the [http://en.wikipedia.org/wiki/Field_(mathematics)#Constructible_numbers field] of real numbers.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   | +                                                    | {{SwitchPreview|ShowMessage=Click here to show proof|HideMessage=Click here to hide proof|PreviewText=We have proven that if <math>{\color{Gray}a}</math> and <math>{\color{Gray}b}</math> are constructible numbers,|FullText=<blockquote>We have proven that if <math>a</math> and <math>b</math> are constructible numbers, then so are <math>a \pm b</math>, <math>ab</math>, and when <math>b \ne 0</math>, <math>\frac {a}{b}</math>. Therefore, the set of constructible numbers form a subfield, <math>W</math>, of the [http://en.wikipedia.org/wiki/Field_(mathematics)#Constructible_numbers field] of real numbers.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        | # '''If <math>a</math> is constructible then <math>a</math> lies in some extension of the rationals of degree a power of 2.'''                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |                                                      | # '''If <math>a</math> is constructible then <math>a</math> lies in some extension of the rationals of degree a power of 2.'''                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| -                                      | # '''If the real number <math>a</math> satisfies an irreducible polynomial over the field of rational numbers of degree <math>k</math>, and if <math>k</math> is not a power of 2, then <math>a</math> is not constructible.''' }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                    | # '''If the real number <math>a</math> satisfies an irreducible polynomial over the field of rational numbers of degree <math>k</math>, and if <math>k</math> is not a power of 2, then <math>a</math> is not constructible.'''<ref>Herstein, 1975, p. 229</ref></blockquote>}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | |AuthorName=Wikipedia                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | |SiteURL=http://en.wikipedia.org/wiki/Compass_and_straightedge_constructions                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | |Field=Geometry                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | ==What is Impossible to Construct (of course, using compass and straightedge alone)?==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | Below is the brief introduction of a few of the impossible constructions. Remember that a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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| -                                      | ==='''What is Impossible to Construct (of course, using compass and straightedge alone)?'''===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                    | <ol>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | <li><math>\pi</math> cannot be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots. In fact, <math>\pi</math> belongs to a special class of numbers called the transcendental number that does not satisfy any rational polynomials. In other words, <math>\pi</math> is not a solution of any polynomials with rational coefficients. Too see complete proof that <math>\pi</math> is {{EasyBalloon|Link=transcendental|Balloon=a number (possibly a complex number) which is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients.<ref>Wikipedia (Transcendental number)</ref>}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | Now, having derived what is possible to construct, I will briefly introduce a few of the impossible constructions.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                    | , see [http://en.wikipedia.org/wiki/Transcendental_number Transcendental number] and [http://sprott.physics.wisc.edu/pickover/trans.html The 15 Most Famous Transcendental Numbers].</li>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | <li>From the above impossible construction, it follows that it is impossible to "square the circle (that is to construct a square that has the same area as a given circle)" because given a circle with radius 1, which is constructible, the area of the circle will be <math>\pi</math> and we have to construct square with sides equal to <math>\sqrt \pi</math> which is not constructible. Due to this exception, there is no general method to square the circle.</li>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
| -                                      | # <math>\pi</math> is transcendental since it does not satisfy any rational polynomials. Too see complete proof that <math>\pi</math> is transcendental, see [http://en.wikipedia.org/wiki/Transcendental_number Transcendental number] and [http://sprott.physics.wisc.edu/pickover/trans.html The 15 Most Famous Transcendental Numbers].                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    | <li>We could not double the volume of a given cube. Say we start with cube of volume 1, which is constructible. Then we have to construct cube of volume 2, which means we have to construct sides of <math>\sqrt [3]{2}</math> which is impossible to construct. So we can't double the cube.</li>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | # From the above impossible construction, it follows that it is impossible to "square the circle(that is to construct a square that has the same area as a given circle" because given a circle with radius <math>r</math> and hence area <math>\pi r^2</math>, we can not construct <math>\sqrt {\pi} r</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                    | <li>We generally can not trisect any given angle because the process involves taking cube root. For example, it is impossible to trisect <math>60^\circ</math>. See below for proof. For more, refer to [http://www.jimloy.com/geometry/trisect.htm#curves Trisection of an Angle]for explanation in great detail. Proof of <math>60^\circ</math> is impossible to trisect. {{HideShowThis|ShowMessage=Click here to show proof|HideMessage=Click here to hide proof|HiddenText=If we could trisect <math>60^\circ</math> by compass and straightedge, then the length <math>a = \cos 20^\circ</math> would be constructible. Since <math>\cos 3\theta = 4\cos^3\theta - 3 \cos \theta</math>. Substituting <math>\theta = 20^\circ</math> and <math>\cos60^\circ=\frac {1}{2}</math>, we obtain <math>4a^3-3a=\frac {1}{2}</math>. Thus <math>a</math> is a root of a cubic polynomial over the rational field. Since this polynomial is irreducible over the rational field  and its degree is 3, <math>a</math> is not constructible. Thus <math>60^\circ</math> cannot be trisected.<ref>Herstein, 1975, p. 230</ref>}}                                                                                                                                                                                                                                                                                                                                                                       |
| -                                      | # We could not double the volume of a given cube because we could not construct <math>\sqrt[3]{2}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                    | </li>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
| -                                      | # We generally can not trisect any given angle because the process involves taking cube root. For example, it is impossible to trisect <math>60^\circ</math>. See below for proof. For more, refer to [http://www.jimloy.com/geometry/trisect.htm#curves Trisection of an Angle]for explanation in great detail.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                    | <li>There are certain polygons that are impossible to construct. See [http://en.wikipedia.org/wiki/Constructible_polygon Constructible polygon] for more detail. </li>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                      | # There are certain polygons that are impossible to construct. See [http://en.wikipedia.org/wiki/Constructible_polygon Constructible polygon] for more detail.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                    | </ol>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
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|                                        | Number 2, 3 and 4 are the so-called [http://mathworld.wolfram.com/GeometricProblemsofAntiquity.html Geometric Problems of Antiquity]. Though they have been proven impossible to construct with straightedge and compass, it does not deter amateur mathematicians to come up with false proofs even today.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                      | Number 2, 3 and 4 are the so-called [http://mathworld.wolfram.com/GeometricProblemsofAntiquity.html Geometric Problems of Antiquity]. Though they have been proven impossible to construct with straightedge and compass, it does not deter amateur mathematicians to come up with false proofs even today.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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| -                                      | Proof that <math>60^\circ</math> is impossible to trisect.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | If we could trisect <math>60^\circ</math> by compass and straightedge, then the length <math>a = \cos 20^\circ</math> would be constructible. Since <math>\cos 3\theta = 4\cos^3\theta - 3 \cos \theta</math>. Substituting <math>\theta = 20^\circ</math> and <math>\cos60^\circ=\frac {1}{2}</math>, we obtain <math>4a^3-3a=\frac {1}{2}</math>. Thus <math>a</math> is a root of a cubic polynomial over the rational field. Since this polynomial is irreducible over the rational field  and its degree is 3, <math>a</math> is not constructible. Thus <math>60^\circ</math> cannot be trisected.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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| -                                      | |SiteURL=http://en.wikipedia.org/wiki/Compass_and_straightedge_constructions                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        | *):http://hptgn.tripod.com/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                      | *):http://hptgn.tripod.com/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        | *):http://en.wikipedia.org/wiki/Compass_and_straightedge_constructions                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                      | *):http://en.wikipedia.org/wiki/Compass_and_straightedge_constructions                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | *):http://www.mathopenref.com/tocs/constructionstoc.html                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | *):http://mathforum.org/library/drmath/view/66052.html                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | *):http://mathforum.org/library/drmath/view/52601.html                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | *):http://www.ams.org/notices/200004/fea-hartshorne.pdf                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | =Notes=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | <references/>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                        | |References=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      | |References=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
| -                                      | |ToDo=zzz                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                    | #Peterson, D. (2003, November 21). Collapsible Compass. Retrieved from The Math Forum: http://mathforum.org/library/drmath/view/66052.html                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| -                                      | |InProgress=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | #Hartshorne, Robin . (2000). Teaching geometry according to euclid. NOTICES OF THE AMS, 47(4), 460-465.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | #Taylor, H. M. (1895). Euclid's elements of geometry. Cambridge: Cabridge University Press.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | #Hudson, H. P. (1916). Ruler & compass. London: Longmans Green & Company, Inc..                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | #Bryant, John, & Sangwin, Christopher. (2008). How Round is your circle?. Princeton & Oxford: Princeton Univ Pr.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | #Herstein, I. N. (1975). Topics in algebra. John Wiley & Sons Inc.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | #Wikipedia (Transcendental number). (n.d.). Transcendental number. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Transcendental_number                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                    | |InProgress=No                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
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## Current revision

Creating a regular hexagon with a ruler and compass
Field: Geometry
Image Created By: Wikipedia
Website: Compass and Straightedge Constructions

Creating a regular hexagon with a ruler and compass

This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.

# Basic Description

Let's assume we only have a compass and a unmarked straightedge. What can we construct and how can we construct them? That were the problems that Euclid pondered not only because those were probably the only instruments that he had at his time but also he wanted to build his theorems with as few assumptions, or axiomsIn traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths., as possible.[1] With these two simple tools, he managed to build myriad of theorems in both plane and solid geometry .[2] These theorem are as good as they were some 2000 years ago and it is this enduring quality of Euclid's work that inspired this page. In the main image of this page, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon. It seems to be a fairly simple construction but you should be prompted to ask two questions: Are other polygons constructible and is every polygon constructible, that is able to be constructed using only compass and straightedge? To extend the question, what are constructible and what are not? That is the problem that is resolved in this page.

# A More Mathematical Explanation

Note: understanding of this explanation requires: *A little Geometry and Some Abstract Algebra
[Click to view A More Mathematical Explanation]

## What is Compass & Straightedge Constructions

### Introduction

We start by familiarizing [...]

[Click to hide A More Mathematical Explanation]

## What is Compass & Straightedge Constructions

### Introduction

We start by familiarizing ourselves with Euclid's three Postulates in his books Elements.

Let it be granted
1. that a straight line may be drawn from any one point to any other point;
2. that a line segment may be extended into a straight line;
3. that given any straight line segment, a circle may be described having the segment as radius and one endpoint as center.[3]

It should be carefully noted that Euclid started with two given points and produced a line segment, from where he could extend into a straight line if he pleased. Then ONLY from the original two points, he could use one point as center and ONLY spread the legs of compass the distance of the line segment to produce a circle. Even though many translations say that a circle can be described at any center with any radius, we must take it with a pinch of salt. He could not specify any points and lengths other than what was already given, that is to say he could not claim "I wanted to spread the legs of the compass $\pi$ centimeters apart (or any specified denominations) with the center of the circle half way between the two given points". In all, Compass and Straightedge Constructions only allow us to start with points (and hence lengths) we have been given(or constructed from given points), and create the ones we don't.
Thus, we define Compass & Straightedge Construction as the construction of points, lengths, angles, and circles using only ideal straightedge and compass. A straightedge is infinite in length, has no markings on it and only one edge. A compass has two legs, one end of which is fixed on the plane of construction and the other end is of given distance away and maintains the distance throughout the construction. It collapses when lifted from the page, so may not be directly used to transfer distances. However, it turns out that this restriction makes no difference due to the Compass Equivalence Theorem which was stated as Proposition II of Book I of Euclid's Elements. It stated that from a given point, it was possible to construct a line segment equal to a given line segment using collapsible compass in any desirable direction. Euclid's proof for the Compass Equivalence Theorem will be presented after the section of Basic Construction. Since Euclid has proven this using only the three postulates, then he did not have to use a collapsible compass any more.[4]

### Some Basic Constructions

The constructions below are some basic ones from where many more constructions are possible and they are by no means exhaustive. In the figures below, what we are given are in blue; intermediate steps are in dotted black; the resulting products are in red. The proofs for these constructions are relatively simple and only require the knowledge of congruent triangles. Euclid derived the theories on congruency and congruent triangles directly from his Postulates. Try proving the theorems yourself!

#### Line Segment Bisection

Given points ${\color{Gray}A}$ and ${\color{Gray}B}$ and the straight line passing through it. Construct [...]

Given points $A$ and $B$ and the straight line passing through it. Construct a line that bisects line segment $AB$.

1. Draw a circle centered at point $A$ with radius equals $AB$.
2. Next, draw a circle centered at point $B$ with the same radius.
3. Where the two circles intersect, call those points $C$ and $D$.
4. Draw a line through points $C$ and $D$.

Line $CD$ intersects line segment $AB$ at the midpoint. It should be noted that $CD \perp AB$ as well. Line $CD$ is the perpendicular bisector of line segment $AB$.

#### Angle Bisection

Given angle ${\color{Gray}\angle AOB}$, construct a line that bisects the angle.

Given angle $\angle AOB$, construct a line that bisects the angle.

1. Construct a circle centered at point $O$ with radius $OA$. This circle intersects $OA$ and $OB$ at point $A$ and $B$.
2. Keeping the same radius, draw a circle at point $A$ and $B$ respectively. Where they intersect each other, call it point $C$.
3. Draw a line through point $C$ and $O$. This line bisects $\angle AOB$.

#### Perpendicular Through a Point

Given a point ${\color{Gray}M}$ on a line ${\color{Gray}AB}$, construct a line that is perpendicular to th [...]

Given a point $M$ on a line $AB$, construct a line that is perpendicular to the given line through $M$.

1. Draw a circle centered at $M$ with radius $MB$ (or $MA$. It is your choice).
2. Where the circle intersects the original line, construct a perpendicular bisector (see the line segment bisector construction above).

#### Parallel Line

Given two points, ${\color{Gray}A}$ and ${\color{Gray}B}$ and the straight line passing through them, con [...]

Given two points, $A$ and $B$ and the straight line passing through them, construct a line that is parallel to the given line through another given point $C$.

1. Draw a circle at $A$, crossing $C$. Where the circle $A$ intersects $AB$, call the point $D$.
2. Centered at points $C$ and $D$, draw circles crossing point $A$. Where these two circles intersect each other, call it $E$.
3. Draw a line through point $C$ and $E$.

$CE$ is parallel to $AB$

#### Tangent Line to a Circle

Given a circle centered at ${\color{Gray}O}$ and another given point, ${\color{Gray}A}$, construct a line [...]

Given a circle centered at $O$ and another given point, $A$, construct a line that is tangent to the circle.

1. Draw a line through point $A$ and the center of the circle $O$
2. Let $M$ be the midpoint of $OA$(construction omitted since we know how to construction mid point). Draw the circle centered at $M$ going through $A$ and $O$.
3. Let the point where the two circles meet be $C$. Draw line segment $AC$.

To see that $AC$ is tangent to the circle, connect $OC$. $\angle ACO$ is an inscribed angle in the circle about $M$, so the inscribed $\angle ACO$ is $90^\circ$. So $OC$ is perpendicular to $AC$. Since the tangent is perpendicular to the radius to the point of tangency, by the uniqueness of the line through $C$ perpendicular to $OC$, $AC$ is tangent to the circle.

#### Euclid's Proof of Compass Equivalence Theorem

This part refers back to the previous section about the issue of compass being collapsible. Euclid's original proof is presented. Additional comments are contained in the parenthesis.

From a given point to draw a line segment equal to a given line segment.

From a given point to draw a line segment equal to a given line segment. Let $A$ be the given point, and $BC$ the given straight line : it is required to draw from the point $A$ a straight line equal to $BC$.

1. From the point $A$ to $B$ draw the straight line $AB$ and on it describe the equilateral triangle $DAB$(this is done the same way as bisecting a line segment), and produce the straight lines $DA$, $DB$ to $E$ and $F$.
2. From the center $B$, at the distance $BC$(with the radius equals to $BC$), describe the circle $CGH$, meeting $DF$ at $G$.
3. From the center $D$, at the distance $DG$(with the radius equals to $DG$), describe the circle $GKL$, meeting $DE$ at $L$.
4. $AL$ shall be equal to $BC$.

Because the point $B$ is the center of the circle $CGH$, $BC$ is equal to $BG$. And because the point $D$ is the center of the circle $GKL$, $DL$ is equal to $DG$ and $DA$, $DB$ parts of them are equal therefore the remainder $AL$ is equal to the remainder $BG$.

But it has been shewn that $BC$ is equal to $BG$ ; therefore $AL$ and $BC$ are each of them equal to $BG$. But things which are equal to the same thing are equal to one another. Therefore $AL$ is equal to $BC$.

Wherefore from the given point $A$ a straight line $AL$ has been drawn equal to the given straight line $BC$.∎[5]

It should be noted that from $AL$, we could "duplicate" $AL$ in all directions by construct a circle centered at $A$ with radius $AL$.

## Algebraicization of Compass & Straightedge Constructions

### What is Algebraicization?

From the few basic constructions, you would have probably realized that the different possibilities seems infinite. Hence, mathematician are curious to find out what are constructible and what aren't and for this purpose, the language of pure geometry seems to have "limited vocabulary"[6]. Back in ancient times, mathematicians had limited algebraic knowledge and were more familiar with geometry. But in modern times, the reverse is true. Hence, today's mathematicians go back to their familiar realm of Algebra and try to find the link between geometry and algebra.

Algebraicization is the translation of any problem statements into algebraic problems. In the case of Compass & Straightedge construction, we algebraicize each step of a straightedge and compass construction, and consequently obtaining general results about the nature of constructibility. Hilda P. Hudson put it aptly in his lecture Ruler & Compasses,

"each step of a ruler and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics...... Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider that these algebraic processes can lead to , when combined in every possible way, and that enables us to answer the question before us......"[7]

Hudson lectured on this in the early 20th century and certain phrases of his could potentially cause confusion. The take-away from this paragraph is that in order to algebraicize straightedge and compass construction, we begin by designating a given point as the origin and the coordinates of another given point (we are given two points at least) as $(1,0)$ or $(0,1)$. Thus we have established the Cartesian Coordinates. Then, every time we construct a straight line or a circle, we think of it instead as adding a new equation into a system of equations. These equations represent the coordinates of all the points on the line or circle, but that is easy since we all know the expression for a line and a circle as $y = ax + b$ and $(x-m)^2 + (y-n)^2 = r^2$. However, the only times we can pinpoint a point (and find its coordinates as a result) is when a line intersects with a line, or a circle, or a circle intersects with another circle in which case we can pinpoint 2 points. We then conclude that only those coordinates of the points of intersections are constructible. In this way, a geometric process is translated into an algebraic process.

### A Simple Derivation

Firstly, we define "1" on a straight line as stated previously. Then, once you have chosen that point to be $(1,0)$, you have to stick to this specification throughout your construction. Next, it is very obvious that we could construct all the integers, that is $\cdots -3,-2,-1,0,1,2,3,\cdots$ (or $x$ = $\{x|- \infty < x < \infty,x \in \mathbb{Z}\}$). How so? Well, once we have the "1", all we have to do is to use the Compass Equivalence Theorem finite number of times to duplicate the length "1" that we previously defined. Now, that means that we could have any two random integers, $a$ and $b$, and for the sake of this discussion and clarity, we are talking about positive integers here. Next, it is shown that from $a$ and $b$, we could construct $a \pm b$, $a \times b$, $\frac {a}{b}$ and $\sqrt {a}$.

$a \pm b$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             $a \times b$

To construct $a \pm b$, we will use $a$ as the center and use $b$ as radius. The two points of intersection with the line will be $a+b$ and $a-b$.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    To construct $a \times b$, we have $0$, $1$, $a$ and $b$ on the straight line $l_0$.  Draw a straight line through $0$, call it $l_1$. $l_1$ could be constructed in many ways. For example, it could be the tangent line to circle centered at $a$ with radius equals to the line segment connecting $b$ to $a$.  Construct circle centered at $0$ with radius $b$, intersecting $l_1$ at $B$.  Connect $B$ and $1$, call it $l_2$.  Construct $l_3$ that is parallel to $l_2$, intersecting $l_1$ at $A$.  Construct circle centered at $0$ with radius $0A$, intersecting $l_0$ at a point.  The distance between $0$ and that point is $a \times b$.
$\frac {a}{b}$                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             $\sqrt {a}$

To construct $\frac {a}{b}$, we have $0$, $1$, $a$ and $b$ on the straight line.  Draw straight line through $0$, call it $l_1$. Construction of $l_1$ is the same as above.  Construct circle centered at $0$ with radius $b$, intersecting $l_1$ at $B$.  Connect $B$ and $1$, call it $l_2$.  Construct circle centered at $0$ with radius $a$, intersecting $l_1$ at $A$.   Construct $l_3$ that is parallel to $l_2$, intersecting $l_0$ at a point.  The distance between $0$ and that point is $\frac {a}{b}$.  Therefore, it has been proven that we could construction all the rational numbers since $a$ and $b$ are any arbitrary integers. The natural question to ask right now is that what else is possible to construct? It is not hard to think of numbers that are not rational. For example, $\sqrt{2}$ is constructible. Construct a unit square and the diagonal is of length $\sqrt{2}$. So is it possible to construct $\sqrt{a}$ given any constructible number $a$? It turns out that we could. See below for method.  To construct $\sqrt {a}$, we start with $0$, $1$, $a$.   Construct $a+1$.  Construct $\frac {1}{2}(a+1)$.  Construct circle at $\frac {1}{2}(a+1)$ with radius $\frac {1}{2}(a+1)$.  Draw perpendicular through $a$, intersecting the circle at point $A$.  $aA=\sqrt {a}$.Again, I will leave the proof to you as well using similar triangles.

I will leave the proofs to you since they are very simple using similar triangles.

Next, we moved to the general solution of the problem.

Assume we have two points $P_1$ and $P_2$ with coordinates $(x_1, y_1)$ and $(x_2, y_2)$. Take an arbitrary point $X (x,y)$ on the line.

By similar triangle, $\frac {x_1-x}{x-x_2} = \frac {y-y_1}{y_2-y}$.

Rearranging the above we have

$x(y_1-y_2)+y(x_2-x_1)=x_2y_1-x_1y_2$.

Since $x_1$, $x_2$, $y_1$ and $y_2$ are constant we can express this as $ax+by=p$ which is the general expression of a straight line.

Now, if we have two lines specified by four given points, $P_1$,....,$P_4$ with coordinates $(x_i,y_i)$. The intersection of the two lines, $P (x,y)$ will satisfy two equations

$\begin{cases} x(y_1-y_2)+y(x_2-x_1)=x_2y_1-x_1y_2\\ x(y_3-y_4)+y(x_4-x_3)=x_4y_3-x_3y_4\\ \end{cases}$

You may say that the there might not be a solution. True the two lines do not have to intersect. But if they do, we only need the operations of addition, subtraction, multiplication and division to find the point.

Now, we move onto circle. Say we have circle centered at some point $P_5$ with coordinates $(x_5, y_5)$ and radius $r$. We know that the explicit expression for a circle is $(x-x_5)^2+(y-y_5)^2=r^2$. Hence, if that circle intersects with one of the straight lines, then the points of intersection will satisfy

$\begin{cases} x(y_1-y_2)+y(x_2-x_1)=x_2y_1-x_1y_2\\ (x-x_5)^2+(y-y_5)^2=r^2\\ \end{cases}$

Similarly, if you there are two circles intersecting, then the two points of intersections have to satisfy the two equations of the circles. To solve for the points of intersection, we only need the operations of addition, subtraction, multiplication and division along with the extraction of square roots. Therefore, from this analysis, we have turned geometric problem into algebraic problem and come to the conclusion that a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots.[8]

### A Rigorous Proof

What I have presented above is a simplified version of the derivation towards the theorem. To see a rigorous proof of this theorem at a college level, refer to the text below which is mainly taken from I. N. Herstein's Topics in Algebra, Second Edition. You need some knowledge in Linear Algebra and/or Abstract Algebra. Also see Constructible Numbers. You should not be discouraged should you find it hard to understand. Instead, you should be marveled by the simplicity and elegance of the algebraic proof.

We have proven that if ${\color{Gray}a}$ and ${\color{Gray}b}$ are constructible numbers,

We have proven that if $a$ and $b$ are constructible numbers, then so are $a \pm b$, $ab$, and when $b \ne 0$, $\frac {a}{b}$. Therefore, the set of constructible numbers form a subfield, $W$, of the field of real numbers. In particular, since $1 \in W$, $W$ must contain $F_0$, the field of rational numbers. If $w \in W$, we can reach $w$ from the rational field by a finite number of constructions. Let $F$ be any subfield of the field of the field of real numbers. Consider all the points $(x,y)$ in the real Euclidean plane both of whose coordinates $x$ and $y$ are in $F$; we call the set of these points the plane of $F$. Any straight line joining two points in the plane of $F$ has an equation of the form $ax+by+c=0$ where $a$,$b$,$c$ are all in $F$. Moreover,any circle having as center a point in the plane of $F$ and having as radius an element of $F$ has equation of the form $x^2+y^2+ax+by+c=0$, where all of $a$, $b$, $c$ are in $F$. Given two lines in $F$ which intersect in the real plane, then their intersection point is a point in the plane of $F$. On the other hand, the intersection of a line in $F$ and a circle in $F$ need not yield a point in the plane of $F$. But, using the fact that the equation of a line in $F$ is of the form $ax+by+c=0$ and that of a circle in F is of the form $x^2+y^2+dx+ey+f=0$, where $a$, $b$, $c$, $d$, $e$, $f$ are all in $F$, we can show that when a line and circle of $F$ intersect in the real plane, they intersect either in a point in the plane of $F$ or in the plane of $F(\sqrt {r})$ for some positive $r$ in $F$. Finally, the intersection of two circles in $F$ can be realized as that of a line in $F$ and a circle in $F$, for if these two circles are $x^2+y^2+a_1x+b_1y+c_1=0$ and $x^2+y^2+a_2x+b_2y+c_2=0$, then their intersection is the intersection of either of these with the line $(a_1-a_2)x+(b_1-b_2)y+(c_1-c_2)=0$, so also yields a point either in the plane of $F$ or of $F(\sqrt{r})$ for some positive $r$ in $F$. Thus lines and circles of $F$ lead us to points either in $F$ or in quadratic extensions of $F$. If we now are in $F(\sqrt{r_1})$ intersect in in points in the plane of $F(\sqrt{r_1},\sqrt{r_2})$ where $r_2$ is a positive number in $F(\sqrt{r_1})$. A point is constructible from $F$ is we can find real numbers $r_1,...,r_n$, such that $r_1^2 \in F$, $r_2^2 \in F(r_1)$, $r_3^2 \in F(r_1,r_2),...,$$r_n^2 \in F(r_1,...,r_{n-1})$, such that the point is in the plane of $F(r_1,...,r_n)$. Conversely, if $r \in F$ is such that $\sqrt{r}$ is real then we can realize $r$ as an intersection of lines and circles in $F$. Thus a point is constructible from $F$ is and only if we can find a finite number of real numbers $r_1,...,r_n$, such that
1. $[F(r_1:F)] = 1$ or $2$;
2. $[F(r_1,...,r_i):F(r_1,...,r_{i-1}]=1$ or $2$ for $i=1,2,...,n$;
and such that our point lies in the plane of $F(r_1,...,r_n)$. We have defined a real number $a$ to be constructible if by use of straightedge and compass we can construct a line segment of length $a$. But this translates, in terms of the discussion above, into: $a$ is constructible if starting from the plane of the rational numbers, $F_0$, we can imbed a in a field obtained from $F_0$ by a finite number of quadratic extensions. And therefore,
1. If $a$ is constructible then $a$ lies in some extension of the rationals of degree a power of 2.
2. If the real number $a$ satisfies an irreducible polynomial over the field of rational numbers of degree $k$, and if $k$ is not a power of 2, then $a$ is not constructible.[9]

# Why It's Interesting

## What is Impossible to Construct (of course, using compass and straightedge alone)?

Below is the brief introduction of a few of the impossible constructions. Remember that a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots.

1. $\pi$ cannot be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots. In fact, $\pi$ belongs to a special class of numbers called the transcendental number that does not satisfy any rational polynomials. In other words, $\pi$ is not a solution of any polynomials with rational coefficients. Too see complete proof that $\pi$ is transcendentala number (possibly a complex number) which is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. , see Transcendental number and The 15 Most Famous Transcendental Numbers.
2. From the above impossible construction, it follows that it is impossible to "square the circle (that is to construct a square that has the same area as a given circle)" because given a circle with radius 1, which is constructible, the area of the circle will be $\pi$ and we have to construct square with sides equal to $\sqrt \pi$ which is not constructible. Due to this exception, there is no general method to square the circle.
3. We could not double the volume of a given cube. Say we start with cube of volume 1, which is constructible. Then we have to construct cube of volume 2, which means we have to construct sides of $\sqrt [3]{2}$ which is impossible to construct. So we can't double the cube.
4. We generally can not trisect any given angle because the process involves taking cube root. For example, it is impossible to trisect $60^\circ$. See below for proof. For more, refer to Trisection of an Anglefor explanation in great detail. Proof of $60^\circ$ is impossible to trisect.

If we could trisect $60^\circ$ by compass and straightedge, then the length $a = \cos 20^\circ$ wou [...]

If we could trisect $60^\circ$ by compass and straightedge, then the length $a = \cos 20^\circ$ would be constructible. Since $\cos 3\theta = 4\cos^3\theta - 3 \cos \theta$. Substituting $\theta = 20^\circ$ and $\cos60^\circ=\frac {1}{2}$, we obtain $4a^3-3a=\frac {1}{2}$. Thus $a$ is a root of a cubic polynomial over the rational field. Since this polynomial is irreducible over the rational field and its degree is 3, $a$ is not constructible. Thus $60^\circ$ cannot be trisected.[11]
5. There are certain polygons that are impossible to construct. See Constructible polygon for more detail.

Number 2, 3 and 4 are the so-called Geometric Problems of Antiquity. Though they have been proven impossible to construct with straightedge and compass, it does not deter amateur mathematicians to come up with false proofs even today.

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# About the Creator of this Image

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# Notes

1. ↑ Peterson, 2003
2. ↑ Hartshorne, 2000
3. ↑ Taylor, 1895, p. 14
4. ↑ Peterson, 2003
5. ↑ Taylor, 1895, p. 18
6. ↑ Hudson, 1916, p. 3
7. ↑ Hudson, 1916, p. 3
8. ↑ Bryant, & Sangwin, 2008, p. 77
9. ↑ Herstein, 1975, p. 229
10. ↑ Wikipedia (Transcendental number)
11. ↑ Herstein, 1975, p. 230

# References

1. Peterson, D. (2003, November 21). Collapsible Compass. Retrieved from The Math Forum: http://mathforum.org/library/drmath/view/66052.html
2. Hartshorne, Robin . (2000). Teaching geometry according to euclid. NOTICES OF THE AMS, 47(4), 460-465.
3. Taylor, H. M. (1895). Euclid's elements of geometry. Cambridge: Cabridge University Press.
4. Hudson, H. P. (1916). Ruler & compass. London: Longmans Green & Company, Inc..
5. Bryant, John, & Sangwin, Christopher. (2008). How Round is your circle?. Princeton & Oxford: Princeton Univ Pr.
6. Herstein, I. N. (1975). Topics in algebra. John Wiley & Sons Inc.

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