An idea for trisecting an angle. Take an arbritrary angle BOC. Let OA be the bisector
of the angle. Draw a circle of arbitrary radius with O as center. Here the circle passes
through A, B and C. Mark off three equal chords on that circle: AD, DE, and
EF. Draw FG parallel to AO, and let G be that point where that line
intersects OB. Draw the circle with center O and radius OG.. Now mark off
three equal chords on that circle, each equal to the previous chords: HI, IJ, and JK.
Then, for small angles, it looks like K = G. If K did equal G, then
OI and OJ would trisect angle AOB, hence OJ would be one of the
trisectors of BOC. Unfortunately, K is never equal to G. So this proposed
method for trisecting angles doesn't work.
It was proved in the 19th century that angles cannot be trisected with the Euclidean tools of
straightedge and compass.
David E. Joyce
Department of Mathematics and Computer Science
Worcester, MA 01610