Once the magical forthcoming Everything2 image embedding capability is working, I'll revise this, but for now, I've uploaded a diagram of how this all works to my website; here's the direct link.
To trisect line segment AB:
- Construct two circles, each with radius AB, one centered around A and one around B.
- Label the two intersections of the circles "C" and "D."
- Label the intersections of these new segments "E" and "F."
- Construct line segments connecting E and F to B and A. (They're already connected to A and B, so you just need to connect them up the other way.)
- Label the intersection of AD and BE "G". Label the intersection of BD and AF "H."
- Construct line segments connecting C to G and H.
- Label the intersections of CG and CH with AB "I" and "J." These points trisect AB.
An edit: I, in my infinite wisdom, assumed I had determined a general method for trisecting an angle based on looks alone. I've withdrawn this, and turned in my mathematician's badge and gun. I'm sorry.