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:

1. Construct two circles, each with radius AB, one centered around A and one around B.
2. Label the two intersections of the circles "C" and "D."
3. Label the intersections of these new segments "E" and "F."
4. 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.)
5. Label the intersection of AD and BE "G". Label the intersection of BD and AF "H."
6. Construct line segments connecting C to G and H.
7. 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 I've withdrawn this, and turned in my mathematician's badge and gun.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.