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 degrees.
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.
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.