Given my blog title, how have I gone this long without
discussing triangle geometry? I will rectify this gross
negligence in the next few weeks. Let’s begin with a seemingly
impossible fact due to Frank Morley. Start with a triangle ABC,
with any shape you wish. Now cut its angles into three equal
parts, and extend these angle trisectors until they meet in
pairs as illustrated below, forming triangle PQR. Morley’s
amazing theorem says that this Morley triangle, PQR, will always
be equilateral! Why would this be true? Well, Morley’s theorem
tells us that this diagram has three nice 60-degree angles in
the middle, but we may suspect that, in fact, all of the angles
are nice! This key insight lets us piece together the following
argument, where we build up the diagram backwards from its
constituent pieces. Draw seven separate “puzzle piece” triangles
with angles and side-lengths as shown below, where the original
triangle ABC has angles \(3\alpha,3\beta,3\gamma\) respectively.
(Be sure to check that puzzle piece