## It is Impossible to trisect an angle of 60 degrees. So What?

The book New First Course in the Theory of Equations by L.E. Dickson has a proof that it is impossible to trisect a 60° angle with just a compass and unmarked straight edge.  From this fact Dickson uses this cool reasoning pattern.

Constructing a 20° angle is impossible else we could have trisected 60°.  Therefore constructing a 40° angle is impossible since  we could bisect a 40° angle and get a 20° angle.  Therefore it is impossible to construct a nonagon since the central angle of the nine triangles composing it is $360/9 = 40$ degrees.

Nonagon

Actually the reasoning process should start by stating that we can construct a 60° angle.  Just make an equilateral triangle.

A further implication is that a 10° angle can’t be constructed since any angle can be doubled by constructing two right triangles as in the diagram and constructing a 20° is impossible.

Doubling an Angle

Extending the reasoning, no angle that can be formed by successive doubling or halving 10° can be constructed.

Even more, an angle of 70° can not be constructed since otherwise an isosceles triangle with two 70° angles could be constructed and the other angle would be 40°.

All sorts of impossible results come from the one fact that trisecting a 60° is impossible.  I thought that was pretty cool.