[UPDATE: we have a proof! I included it at the end of the blog post.]
Yesterday I was looking at a few methods of angle trisection.
For instance, I made this applet showing how to use the “cycloid of Ceva” to trisect an angle. (It is based on Archimedes’s neusis [marked straightedge] construction.)
I also found David Alan Brook’s College Mathematics Journal article “A new method of trisection.” He shows how you can use the squared-off end of a straightedge (or equivalently, a carpenter’s square) to trisect an angle. (This is a different carpenter’s square construction than the one I wrote about recently.)
To perform the trisection of (see figure below), bisect the segment at Then draw the segment perpendicular to . Draw a circle with center and radius Next, arrange the carpenter’s square so that one edge goes through , one edge is tangent to the circle, and the vertex, , sits on . Then .
I made an applet to illustrate this trisection.
Brooks’s proof used trigonometry. My question is: Is there a geometric proof that I spent a little while working on it yesterday and couldn’t find one. If you can, let me know!
UPDATE: We have a proof! Thank you Marius Buliga!
In the proof we are referring to the figure below. Let and Then it suffices to show that Let be the point of tangency. Draw segment and extend to on Then draw segments and Because lines and are parallel, and hence is a parallelogram. This implies that is the midpoint of the diagonal so Moreover, Because is the hypotenuse of the right triangle and is the median, We see that so Finally, because is a chord of the circle, that is,
Update 2: Andrew Stacey just sent me an another proof:
In the proof we are referring to the figure below. Let and Then it suffices to show that Let be the point of tangency of the carpenter’s square and the circle. Because Construct Because is a tangent line and is a radius of the circle, Because and are parallel. Thus Construct a point so that is a parallelogram. Notice that and are collinear. Let be the midpoint of . Construct the line segment and the circle with center and radius . Note that the circle passes through and —the first three because and the fourth because and is a diameter of the circle. Because is parallel to Finally, is a central angle and is an inscribed angle and both cut off the same arc of the circle; thus