The impossibility of angle bisection

In a typical introductory course in abstract algebra, after you have proven the impossibility of trisecting an arbitrary angle using just straightedge and compasses, you sum up the argument as follows: “We have just shown that cos 20° is not constructible, and so we cannot construct a 20° angle either; thus we cannot trisect a 60° angle, and so we cannot trisect an arbitrary angle.”

You can often create some consternation by continuing: “Now the fact that we cannot construct a 20° angle also shows that we cannot bisect a 40° angle and so you cannot bisect an arbitrary angle with compasses and straightedge.” ♣

Since an angle bisection is possible with straightedge and compasses, all that has been shown is that an angle of 40° is not so constructible. If a 40° angle was given, it would have had to have been determined by some measuring device. A 60° angle is constructible, so if a trisection were possible, we would be able to obtain a 60° angle and then trisect it to obtain a 20°angle.

Contributed by Eric Chandler of Randolph-Macon Woman's College in Lynchburg, VA.

Trisecting an angle with ruler and compasses

Construction. Let the angle to be trisected be BAC. With center A and respective radii of two, three and four units, draw arcs PU, QV and RW to intersect the arms of the angle. Determine D, E, F and G, the respective midpoints of arcs PU, RW, RE and EW.