# Problem of Apollonius

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|                                        | |Image=Apollonian.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                     | |Image=Apollonian.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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|                                        | |ImageIntro=This an example of a fractal that can be created by repeatedly solving the Problem of Apollonius.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                     | |ImageIntro=This an example of a fractal that can be created by repeatedly solving the Problem of Apollonius.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        | |ImageDescElem=<div style="width:auto; position:relative;">[[Image:Apollonius.png|thumb|150px|Image by: [http://commons.wikimedia.org/wiki/File:Apollonius_solution_3B.png Wikipedia]|left]]<div style="position:relative;">The problem of Apollonius involves trying to find a circle that is tangent to three objects: points, lines, or circles in a plane. The most famous of these is the case involving three different circles in a plane, as seen in the picture to the left. The given three circles are in red, green, and blue, while the solution circle is in black.</div></div> |                                                     | |ImageDescElem=<div style="width:auto; position:relative;">[[Image:Apollonius.png|thumb|150px|Image by: [http://commons.wikimedia.org/wiki/File:Apollonius_solution_3B.png Wikipedia]|left]]<div style="position:relative;">The problem of Apollonius involves trying to find a circle that is tangent to three objects: points, lines, or circles in a plane. The most famous of these is the case involving three different circles in a plane, as seen in the picture to the left. The given three circles are in red, green, and blue, while the solution circle is in black.</div></div> |
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|                                        | Given three points, the problem only has one solution. In the cases of one line and two points; two lines and one point; and one circle and two points, the problem has two solutions. Four solutions exist for the cases of three lines; one circle, one line, and one point; and two circles and one point. There are eight solutions for the cases of two circles and one line; and one circle and two lines, in addition to the three circle problem.                                                                                                                                     |                                                     | Given three points, the problem only has one solution. In the cases of one line and two points; two lines and one point; and one circle and two points, the problem has two solutions. Four solutions exist for the cases of three lines; one circle, one line, and one point; and two circles and one point. There are eight solutions for the cases of two circles and one line; and one circle and two lines, in addition to the three circle problem.                                                                                                                                     |
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| -                                      | <h1> A More Mathematical Explanation</h1>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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|                                        | There are many different ways of solving the problem of Apollonius. The few that are easiest to understand include using an algebraic method or an [[Inversion|inverse geometry]] method.                                                                                                                                                                                                                                                                                                                                                                                                     |                                                     | There are many different ways of solving the problem of Apollonius. The few that are easiest to understand include using an algebraic method or an [[Inversion|inverse geometry]] method.                                                                                                                                                                                                                                                                                                                                                                                                     |
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|                                        | === Algebraic Method ===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |                                                     | === Algebraic Method ===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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| -                                      | {{hide|1=This method only uses math up to the level of understanding quadratic equations.  We will proceed by setting up a system of quadratic equations and solving for the radius, r, of the unknown circle.                                                                                                                                                                                                                                                                                                                                                                                | +                                                   | This method only uses math up to the level of understanding quadratic equations.  We will proceed by setting up a system of quadratic equations and solving for the radius, r, of the unknown circle.                                                                                                                                                                                                                                                                                                                                                                                         |
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|                                        | We start by labeling the center of each of the given circles <math>(x</math><sub>1</sub><math>,y</math><sub>1</sub><math>)</math>, <math>(x</math><sub>2</sub><math>,y</math><sub>2</sub><math>)</math>, and <math>(x</math><sub>3</sub><math>,y</math><sub>3</sub><math>)</math>.  We will call the center of the unknown circle <math>(x,y)</math>. <math>r</math><sub>1</sub>,<math>r</math><sub>2</sub>, and <math>r</math><sub>3</sub> are the different radii of each of the given circles.                                                                                             |                                                     | We start by labeling the center of each of the given circles <math>(x</math><sub>1</sub><math>,y</math><sub>1</sub><math>)</math>, <math>(x</math><sub>2</sub><math>,y</math><sub>2</sub><math>)</math>, and <math>(x</math><sub>3</sub><math>,y</math><sub>3</sub><math>)</math>.  We will call the center of the unknown circle <math>(x,y)</math>. <math>r</math><sub>1</sub>,<math>r</math><sub>2</sub>, and <math>r</math><sub>3</sub> are the different radii of each of the given circles.                                                                                             |
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|                                        | Second minus first gives us:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |                                                     | Second minus first gives us:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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| -                                      | :*<math>2(x</math><sub>1</sub><math>-x</math><sub>2</sub><math>)x+2(y</math><sub>1</sub><math>-y</math><sub>2</sub>)<math>+2(\pm r</math><sub>1</sub><math>\pm r</math><sub>2</sub><math>)r=(x</math><sub>1</sub><math>^2+y</math><sub>1</sub><math>^2-r</math><sub>1</sub><math>^2)-(x</math><sub>2</sub><math>^2+y</math><sub>2</sub><math>^2-r</math><sub>2</sub><math>^2)</math>}}                                                                                                                                                                                                        | +                                                   | :*<math> 2(x_1-x_2)x+2(y_1-y_2)y+2(\pm r_1 \pm r_2)r=(x_1^2+y_1^2-r_1^2)-(x_2^2+y_2^2-r_2^2) </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Third minus first gives us:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | :*<math> 2(x_1-x_3)x+2(y_1-y_3)y+2(\pm r_1 \pm r_3)r=(x_1^2+y_1^2-r_1^2)-(x_3^2+y_3^2-r_3^2) </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
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| -                                      | <h1> Apollonian Gasket </h1>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                   | For the sake of simplicity, we'll define some new variables. Let                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| -                                      | {{Switch|link1=Show|link2=Hide|1=<div style="float:left; margin-right:25px"><pausegif id="2" wiki="no" border="no">Problemapollonius3.gif</pausegif></div>|2=The Apollonian gasket is an example of one of the earliest studied fractals and was first constructed by Gottfried Leibniz. It can be constructed by solving the problem of Apollonius iteratively. It was a precursor to [[Sierpinski's Triangle]],  and in a special case, it forms [[Ford Circles]].                                                                                                                          | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_2=2(x_1-x_2) </math>; <math> b_2=2(y_1-y_2)</math>  ; <math> c_2=2(\pm r_1 \pm r_2) </math> ; <math>d_2=(x_1^2+y_1^2-r_1^2)-(x_2^2+y_2^2-r_2^2) </math>                                                                                                                                                                                                                                                                                                                                                                                                                              |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_3=2(x_1-x_3) </math>; <math> b_3=2(y_1-y_3)</math>  ; <math> c_3=2(\pm r_1 \pm r_3) </math> ; <math>d_2=(x_1^2+y_1^2-r_1^2)-(x_3^2+y_3^2-r_3^2) </math>                                                                                                                                                                                                                                                                                                                                                                                                                              |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Now our two equations can be written as                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | :<math> a_2 x+b_2 y +c_2 r=d_2 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | :<math> a_3 x+b_3 y +c_3 r=d_3 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Since this is a simple linear system of equations, we can solve it for x and y in terms of r.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | {{Switch|link1=click to show algebra |link2=click to hide algebra|1=|2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Solving the first equation for x:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_2x=d_2-c_2r-b_2y \rightarrow x=\frac{d_2-c_2 r -b_2 y}{a_2} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Substituting that in to the second equation allows us to find y in terms of known values and r:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_3\left(\frac{d_2-c_2 r -b_2 y}{a_2} \right)+b_3 y +c_3 r =d_3 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_3 d_2-a_3c_2 r -a_3 b_2 y+a_2 b_3 y +a_2 c_3 r =d_3 a_2 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> y(a_2 b_3-a_3 b_2)=a_2 d_3-a_3 d_2 +(a_3 c_2  -a_2 c_3)r </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | *<math> y=\frac{a_2 d_3-a_3 d_2 +(a_3 c_2  -a_2 c_3)r}{a_2 b_3-a_3 b_2} </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Rather than substituting back in to find x, it is actually simpler to go through the same process we used to find y to get x in terms of known values and r.                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | First, we solve the first equation for y.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> b_2y=d_2-c_2r-a_2x \rightarrow y=\frac{d_2-c_2 r -a_2 x}{b_2} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Plugging into the second equation gives us                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_3 x+b_3\left(\frac{d_2-c_2 r -a_2 x}{b_2}\right) +c_3 r =d_3 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_3 b_2 x+b_3 d_2 -b_3 c_2 r-a_2 b_3 x +b_2 c_3 r=b_2 d_3 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> x(a_3 b_2-a_2 b_3)=b_2 d_3-b_3 d_2 +(b_3 c_2-b_2 c_3) r </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | *<math> x=\frac{b_2 d_3-b_3 d_2 +(b_3 c_2-b_2 c_3) r}{a_3 b_2-a_2 b_3} </math>.}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | With <math> x=\frac{b_2 d_3-b_3 d_2 +(b_2 c_3-b_3 c_2) r}{a_3 b_2-a_2 b_3} </math> and <math> y=\frac{a_2 d_3-a_3 d_2 +(a_3 c_2  -a_2 c_3)r}{a_2 b_3-a_3 b_2} </math>, we plug in values for the a's, b's and c's (which we get from the original information about the centers and radii of the three circles) to calculate x and y. Using our very first equations for the circles, we can then solve for r.                                                                                                                                                                                |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | {{Switch|link1=click to show example|link2=click to hide example|1=|2=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Let's pick three circles and find the mutually tangent ones. Let's choose some coordinates and radii for our three circles.                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Take <math> (x_1,y_1,r_1)=(0,3,2) </math>, <math> (x_2,y_2,r_2)=(-2,-2,1) </math> and <math> (x_3,y_3,r_3)=(3,-3,3)</math>. These three circles are shown below.                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | [[Image:Circleex1.png|300]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Now we want to calculate the a's, b's, and d's first.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_2=2(x_1-x_2)=2(0--2)=4</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> a_3=2(x_1-x_3)=2(0-3)=-6</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> b_2=2(y_1-y_2)=2(3--2)=10</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> b_3=2(y_1-y_3)=2(3--3)=12</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> d_2=(x_1^2+y_1^2-r_1^2)-(x_2^2+y_2^2-r_2^2) =(0^2+3^2-2^2)-(2^2+2^2-1^2)=(9-4)-(4+4-1)=-2 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> d_3=(x_1^2+y_1^2-r_1^2)-(x_3^2+y_3^2-r_3^2) =(0^2+3^2-2^2)-(3^2+3^2-3^2)=(9-4)-(9+9-9)=-4 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Calculating the c terms requires a bit more thought since there are the <math> \pm </math> signs. The choice of these signs determines which circle we are solving for. We simply must be consistent in all of our applications of signs for a given r. For the first example, let's simply take all of the plus signs. Then                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> c_2=2(r_1+r_2)=2(2+1)=6</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> c_3=2(r_1+r_3)=2(2+3)=10 </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Now we can calculate x and y for this first circle.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math>x=\frac{b_2 d_3-b_3 d_2 +(b_3 c_2-b_2 c_3) r}{a_3 b_2-a_2 b_3}=\frac{(10)(-4)-(12)(-2)+(12(6)-10(10))r}{-6(10)-4(12)}=\frac{-16-(28)r}{-108} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> x=\frac{1}{27}(4+7r) </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math>y=\frac{a_2 d_3-a_3 d_2 +(a_3 c_2  -a_2 c_3)r}{a_2 b_3-a_3 b_2}=\frac{(4)(-4)-(-6)(2)+ (-6(6)-4(10))r}{12(4)-(10)(-6)}=\frac{-28-(76)r}{108} </math>                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> y=\frac{-7-19r}{27} </math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Now we can return to one of our first equations to find r.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> (x-x_1)^2+(y-y_1)^2=(r+r_1)^2 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Note that here the sign of <math> r_1 </math> is positive. That is because we took the positive sign when we solved for the c values. Plugging in values, we get                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <math> \left(\frac{1}{27}(4+7r)\right)^2+\left(\frac{-7-19r}{27}-3 \right)^2=(r+2)^2 </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | This equation is quadratic in r, so it can be solved using the quadratic formula (or a graphing calculator, if you prefer). When the dust clears, we get <math> r\approx 4.729 </math> (the other value that comes out of the quadratic formula does not work when plotted).                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Now we can plot this circle with center <math> \left(\frac{1}{27}(4+7(4.729)),\frac{-7-19(4.729)}{27} \right) </math>. It is shown below in red, with the original circles in black.                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | [[Image:Circleex2.png|300px|center]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | We can see that it is indeed tangent to the three original circles!                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | This process can be repeated choosing different signs for the different r values in the c coefficients to find the other seven circles.                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | == Apollonian Gasket ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | The Apollonian gasket is an example of one of the earliest studied fractals and was first constructed by Gottfried Leibniz. It can be constructed by solving the problem of Apollonius iteratively. It was a precursor to [[Sierpinski's Triangle]],  and in a special case, it forms [[Ford Circles]].                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        | <div style="float:left"><pausegif id="2" wiki="no" border="no">Problemapollonius3.gif</pausegif></div>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                     | <div style="float:left"><pausegif id="2" wiki="no" border="no">Problemapollonius3.gif</pausegif></div>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
| Line 62:                               |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | Line 173:                                           |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        | Repeating this process over and over again with each set of three mutually tangent circles will create the Apollonian gasket.}}                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     | Repeating this process over and over again with each set of three mutually tangent circles will create the Apollonian gasket.}}                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | {{-}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                   | == Interactive Applet ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | {{#iframe:https://www.cs.drexel.edu/~rlw82/Canvas/ResizeCircles/|632|632|border=0}}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | <BR>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Drag the circles around for solutions to be drawn                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Drag the red circles to resize that circle                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | == References ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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| -                                      | <h1>References</h1>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        | Math Pages, [http://www.mathpages.com/HOME/kmath113.htm Apollonius' Tangency Problem]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                     | Math Pages, [http://www.mathpages.com/HOME/kmath113.htm Apollonius' Tangency Problem]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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|                                        | MathWorld, [http://mathworld.wolfram.com/ApolloniusProblem.html Apollonius' Problem]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                     | MathWorld, [http://mathworld.wolfram.com/ApolloniusProblem.html Apollonius' Problem]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | Wikibooks , [http://en.wikibooks.org/wiki/Fractals/Apollonian_fractals Apollonian fractals]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | == Mathematica Programs ==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   | [[User:AnnaP|Anna]] created several programs that solve the problem of Apollonius in the case of three non-tangent, non-intersecting circles that the user inputs. The programs were written in Mathematica 7, but are likely compatible with Mathematica 5 and/or 6.                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | |AuthorName=Paul Nylander                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                   | The first program automatically plots all eight solutions. [http://www.sccs.swarthmore.edu/users/09/anna09/mathematica/apolloniussolution.nb Click here to download this program]                                                                                                                                                                                                                                                                                                                                                                                                             |
| -                                      | |SiteName=Fractals                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | |SiteURL=http://bugman123.com/Fractals/Fractals.html                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                   | The second program allows the user to choose which solutions to plot in groups of two. [http://www.sccs.swarthmore.edu/users/09/anna09/mathematica/apolloniussolutionoptions.nb Click here to download this program]                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |Field=Geometry                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | |Field2=Fractals                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | |InProgress=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |

## Current revision

Fields: Geometry and Fractals
Image Created By: Paul Nylander
Website: Fractals

This an example of a fractal that can be created by repeatedly solving the Problem of Apollonius.

# Basic Description

Image by: Wikipedia
The problem of Apollonius involves trying to find a circle that is tangent to three objects: points, lines, or circles in a plane. The most famous of these is the case involving three different circles in a plane, as seen in the picture to the left. The given three circles are in red, green, and blue, while the solution circle is in black.

Apollonius of Perga posed and solved this problem in his work called Tangencies. Sadly, Tangencies has been lost, and only a report of his work by Pappus of Alexandria is left. Since then, other mathematicians, such as Isaac Newton and Descartes, have been able to recreate his results and discover new ways of solving this interesting problem.

Click to stop animation.
The problem usually has eight different solution circles that exist that are tangent to the given three circles in a plane. The given circles must not be tangent to each other, overlapping, or contained within one another for all eight solutions to exist.

Given three points, the problem only has one solution. In the cases of one line and two points; two lines and one point; and one circle and two points, the problem has two solutions. Four solutions exist for the cases of three lines; one circle, one line, and one point; and two circles and one point. There are eight solutions for the cases of two circles and one line; and one circle and two lines, in addition to the three circle problem.

# A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

There are many different ways of solving the problem of Apollonius. The few that are easiest to under [...]

[Click to hide A More Mathematical Explanation]

There are many different ways of solving the problem of Apollonius. The few that are easiest to understand include using an algebraic method or an inverse geometry method.

### Algebraic Method

This method only uses math up to the level of understanding quadratic equations. We will proceed by setting up a system of quadratic equations and solving for the radius, r, of the unknown circle.

We start by labeling the center of each of the given circles $(x$1$,y$1$)$, $(x$2$,y$2$)$, and $(x$3$,y$3$)$. We will call the center of the unknown circle $(x,y)$. $r$1,$r$2, and $r$3 are the different radii of each of the given circles.

From this we are able to write our equations:

• $(x - x$1$)^2 + (y - y$1$)^2 = (r \pm r$1$)^2$
• $(x - x$2$)^2 + (y - y$2$)^2 = (r \pm r$2$)^2$
• $(x - x$3$)^2 + (y - y$3$)^2 = (r \pm r$3$)^2$

Next we are able to expand each of the equations to see better how they can relate to each other.

Expanding gives us:

• $(x^2+y^2-r^2)-2xx$1$-2yy$1$\pm2rr$1$+(x$1$^2+y$1$^2-r$1$^2)=0$
• $(x^2+y^2-r^2)-2xx$2$-2yy$2$\pm2rr$2$+(x$2$^2+y$2$^2-r$2$^2)=0$
• $(x^2+y^2-r^2)-2xx$3$-2yy$3$\pm2rr$3$+(x$3$^2+y$3$^2-r$3$^2)=0$

We can now look at the equations and see how we can subtract them from each other. So we will take the second and third equation minus the first equation.

Second minus first gives us:

• $2(x_1-x_2)x+2(y_1-y_2)y+2(\pm r_1 \pm r_2)r=(x_1^2+y_1^2-r_1^2)-(x_2^2+y_2^2-r_2^2)$

Third minus first gives us:

• $2(x_1-x_3)x+2(y_1-y_3)y+2(\pm r_1 \pm r_3)r=(x_1^2+y_1^2-r_1^2)-(x_3^2+y_3^2-r_3^2)$

For the sake of simplicity, we'll define some new variables. Let

$a_2=2(x_1-x_2)$; $b_2=2(y_1-y_2)$  ; $c_2=2(\pm r_1 \pm r_2)$ ; $d_2=(x_1^2+y_1^2-r_1^2)-(x_2^2+y_2^2-r_2^2)$

$a_3=2(x_1-x_3)$; $b_3=2(y_1-y_3)$  ; $c_3=2(\pm r_1 \pm r_3)$ ; $d_2=(x_1^2+y_1^2-r_1^2)-(x_3^2+y_3^2-r_3^2)$

Now our two equations can be written as

$a_2 x+b_2 y +c_2 r=d_2$
$a_3 x+b_3 y +c_3 r=d_3$

Since this is a simple linear system of equations, we can solve it for x and y in terms of r.

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Solving the first equation for x:

$a_2x=d_2-c_2r-b_2y \rightarrow x=\frac{d_2-c_2 r -b_2 y}{a_2}$

Substituting that in to the second equation allows us to find y in terms of known values and r:

$a_3\left(\frac{d_2-c_2 r -b_2 y}{a_2} \right)+b_3 y +c_3 r =d_3$

$a_3 d_2-a_3c_2 r -a_3 b_2 y+a_2 b_3 y +a_2 c_3 r =d_3 a_2$

$y(a_2 b_3-a_3 b_2)=a_2 d_3-a_3 d_2 +(a_3 c_2 -a_2 c_3)r$

• $y=\frac{a_2 d_3-a_3 d_2 +(a_3 c_2 -a_2 c_3)r}{a_2 b_3-a_3 b_2}$.

Rather than substituting back in to find x, it is actually simpler to go through the same process we used to find y to get x in terms of known values and r.

First, we solve the first equation for y.

$b_2y=d_2-c_2r-a_2x \rightarrow y=\frac{d_2-c_2 r -a_2 x}{b_2}$

Plugging into the second equation gives us

$a_3 x+b_3\left(\frac{d_2-c_2 r -a_2 x}{b_2}\right) +c_3 r =d_3$

$a_3 b_2 x+b_3 d_2 -b_3 c_2 r-a_2 b_3 x +b_2 c_3 r=b_2 d_3$

$x(a_3 b_2-a_2 b_3)=b_2 d_3-b_3 d_2 +(b_3 c_2-b_2 c_3) r$

• $x=\frac{b_2 d_3-b_3 d_2 +(b_3 c_2-b_2 c_3) r}{a_3 b_2-a_2 b_3}$.

With $x=\frac{b_2 d_3-b_3 d_2 +(b_2 c_3-b_3 c_2) r}{a_3 b_2-a_2 b_3}$ and $y=\frac{a_2 d_3-a_3 d_2 +(a_3 c_2 -a_2 c_3)r}{a_2 b_3-a_3 b_2}$, we plug in values for the a's, b's and c's (which we get from the original information about the centers and radii of the three circles) to calculate x and y. Using our very first equations for the circles, we can then solve for r.

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Let's pick three circles and find the mutually tangent ones. Let's choose some coordinates and radii for our three circles.

Take $(x_1,y_1,r_1)=(0,3,2)$, $(x_2,y_2,r_2)=(-2,-2,1)$ and $(x_3,y_3,r_3)=(3,-3,3)$. These three circles are shown below.

Now we want to calculate the a's, b's, and d's first.

$a_2=2(x_1-x_2)=2(0--2)=4$

$a_3=2(x_1-x_3)=2(0-3)=-6$

$b_2=2(y_1-y_2)=2(3--2)=10$

$b_3=2(y_1-y_3)=2(3--3)=12$

$d_2=(x_1^2+y_1^2-r_1^2)-(x_2^2+y_2^2-r_2^2) =(0^2+3^2-2^2)-(2^2+2^2-1^2)=(9-4)-(4+4-1)=-2$

$d_3=(x_1^2+y_1^2-r_1^2)-(x_3^2+y_3^2-r_3^2) =(0^2+3^2-2^2)-(3^2+3^2-3^2)=(9-4)-(9+9-9)=-4$

Calculating the c terms requires a bit more thought since there are the $\pm$ signs. The choice of these signs determines which circle we are solving for. We simply must be consistent in all of our applications of signs for a given r. For the first example, let's simply take all of the plus signs. Then

$c_2=2(r_1+r_2)=2(2+1)=6$

$c_3=2(r_1+r_3)=2(2+3)=10$.

Now we can calculate x and y for this first circle.

$x=\frac{b_2 d_3-b_3 d_2 +(b_3 c_2-b_2 c_3) r}{a_3 b_2-a_2 b_3}=\frac{(10)(-4)-(12)(-2)+(12(6)-10(10))r}{-6(10)-4(12)}=\frac{-16-(28)r}{-108}$

$x=\frac{1}{27}(4+7r)$.

$y=\frac{a_2 d_3-a_3 d_2 +(a_3 c_2 -a_2 c_3)r}{a_2 b_3-a_3 b_2}=\frac{(4)(-4)-(-6)(2)+ (-6(6)-4(10))r}{12(4)-(10)(-6)}=\frac{-28-(76)r}{108}$

$y=\frac{-7-19r}{27}$.

Now we can return to one of our first equations to find r.

$(x-x_1)^2+(y-y_1)^2=(r+r_1)^2$

Note that here the sign of $r_1$ is positive. That is because we took the positive sign when we solved for the c values. Plugging in values, we get

$\left(\frac{1}{27}(4+7r)\right)^2+\left(\frac{-7-19r}{27}-3 \right)^2=(r+2)^2$

This equation is quadratic in r, so it can be solved using the quadratic formula (or a graphing calculator, if you prefer). When the dust clears, we get $r\approx 4.729$ (the other value that comes out of the quadratic formula does not work when plotted).

Now we can plot this circle with center $\left(\frac{1}{27}(4+7(4.729)),\frac{-7-19(4.729)}{27} \right)$. It is shown below in red, with the original circles in black.

We can see that it is indeed tangent to the three original circles!

This process can be repeated choosing different signs for the different r values in the c coefficients to find the other seven circles.

## Apollonian Gasket

The Apollonian gasket is an example of one of the earliest studied fractals and was first constructed by Gottfried Leibniz. It can be constructed by solving the problem of Apollonius iteratively. It was a precursor to Sierpinski's Triangle, and in a special case, it forms Ford Circles.

Click to stop animation.

Constructing the gasket begins with three mutually tangent circles. By solving this case of the problem of Apollonius we know that there are two other circles that are tangent to the three given circles. We now have five circles from which to start again.

Repeat the process with two of the original circles and one of the newly generated circles. Again, by solving Apollonius' problem we can find two circles that are tangent to this new set of three circles. Although, we already know one of the two solutions for this set of three circles; it is the other of the three circles that we started with.

Repeating this process over and over again with each set of three mutually tangent circles will create the Apollonian gasket.

# Teaching Materials

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## Interactive Applet

Drag the circles around for solutions to be drawn Drag the red circles to resize that circle

## References

Math Pages, Apollonius' Tangency Problem

MathWorld, Apollonius' Problem

Wikibooks , Apollonian fractals

## Mathematica Programs

Anna created several programs that solve the problem of Apollonius in the case of three non-tangent, non-intersecting circles that the user inputs. The programs were written in Mathematica 7, but are likely compatible with Mathematica 5 and/or 6.

The second program allows the user to choose which solutions to plot in groups of two. Click here to download this program
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