@jmw150 The proof of Monsky's theorem has a similar style but is much shorter and simpler, prove that a square can't be subdivivded into an odd number of triangles with equal area, and to do so you need sperner's lemma and p-adic numbers and a really goofy chain of reasoning that shows it can't be done. Completely useless but all the more interesting.

I think my actual favorite theorem is Euler's formula for convex polyhedra because there's a proof with a beautiful visualization. Imagine the polyhedron as a wire frame and place it within a concentric sphere, at the center of the sphere place a light that casts shadows of the wireframe onto the surface of the sphere, using spherical geometry you can double count all the great circle arcs and intersections and faces, and a little algebra gives you Euler's formula.