Chapter 1

Trisecting Angles

One of the most famous questions in mathematics is, “Can every angle be

trisected?” A more specific statement of the problem is as follows: Can

every angle be trisected by compass and straightedge alone? Thus you are

not allowed to use a protractor or any other device for measuring angles,

only a compass and straightedge. Still this is not the precise statement of

the problem, which must wait until we progress a bit.

Besides being one of the most famous, the problem is also one of the

oldest in mathematics, dating from the time of the ancient Greeks. More-

over, the problem seems to be more famous than its solution. Many in the

non-mathematical world know of this question, but only a few seem to know

that the problem has been solved. The result of this is a steady outpouring

of “proofs” that every angle can be trisected. Not only do these so-called

proofs contain errors, but the conclusion is also false; there are angles (in

particular, the

60◦

angle) that cannot be trisected by straightedge and com-

pass alone. This will be proved in this chapter.

Many of the false proofs of the trisection of angles come from not un-

derstanding the nature of the problem. This will be our first undertaking

after we review a few things about ruler and compass constructions. Indeed

there are variations on this problem that, while interesting and challenging,

differ from the classical problem.

Why should you bother to study ruler and compass constructions and

learn the answer to the question, “Can every angle be trisected?” If one

wishes to be called an educated mathematician, there are some things that

have to be mastered, though it is debatable whether this fits that category.

Certainly the trisection problem is part of mathematical history and is wor-

thy of study from that point of view. But perhaps the best reason to study

1

http://dx.doi.org/10.1090/mbk/075/01