Trisecting an Angle
Given any angle, we already know that it can be bisected by Euclidean construction. Then how about trisecting an angle? For some special angles like , we can certainly trisect it because we known how to construct a angle. However, what we really want is a way to trisect ANY angle by Euclidean construction. In this section, we will show that this is IMPOSSIBLE - there exists an angle that cannot be trisected!
To prove this, we first need to define what is a "constructible angle" and its relation to constructible numbers. Then we use some trigonometric identities to derive a cubic equation so as to prove the impossibility using the Main Theorem.
Constructible angles
An angle is said to be constructible if it can be constructed by straightedge and compass only i.e the angle is formed by three constructible points.
It is not hard to deduce the following simple but important result:
An angle is constructible if and only if is a constructible number.