Trisection of a Line

It always seemed odd to me that one can trisect a line but not an angle. It's one of those things that makes sense arithmetically—trisecting an angle requires a cubic equation, and circles are only quadratic—but I don't have a good visual intuition for why that's so.

I set out to play around with trisecting a line in an attempt to get a better feel for the problem, but it didn't help: the numbers fell out very easily and I feel like I have no better an understanding than when I started. Oh well.

Given points A and B, draw

1. circle AB
2. circle BA intersecting circle AB at points C and D,
3. circle CD intersecting circle AB at point E≠D and circle BA at point F≠D,
4. circle EB,
5. circle FC intersecting circle EB at points G and H,
6. circle GB,
7. circle HB intersecting GB at I≠B;

then I trisects AB. This one's straightforward to analyze using trigonometry. Let's first observe that EABF are all collinear, so we'll just worry about distances along the line EF. If we define AB=1, then EA=BF=1 as well, making EF=EA+AB+BF=3. FC=CD=√3. Suppose GH intersects EF at X: using the formula for the intersection of two circles, EX=(3²-√3²+2²)/(2·3)=5/3, therefore AX=EX-EA=2/3 and XB=AB-AX=1/3. I is the reflection of B about GH, therefore IX=XB=1/3 and AI=AX-IX=1/3.