Something Neat About Triangles

I was reading a biography of Donald Coxeter, one of the most important geometers of the 20th century, and it mentioned in passing something Coxeter referred to as Morley’s Miracle Theorem. The theorem was proved in 1899 by Frank Morley, who taught at Haverford College (if that sounds vaguely familiar that’s because you remember it’s where Dave Barry went) and then Johns Hopkins (which may be familiar on the strength of its lacrosse team), and published this in the first issue of the Transactions of the American Mathematical Society. And, yes, perhaps it isn’t actually important, but the result is so unexpected and surprising that I wanted to share it with you. The biography also includes a proof Coxeter wrote for the theorem, one that’s admirably straightforward, but let me show the result without the proof so you can wonder about it.

First, start by drawing a triangle. It doesn’t have to have any particular interesting properties other than existing. I’ve drawn an example one.

The next step is to cut into three equal pieces each of the interior angles of the triangle, and draw those lines. I’m doing that in separate diagrams for each of the triangle’s three original angles because I want to better suggest the process.

I should point out, this trisection of the angles can be done however you like, which is probably going to be by measuring the angles with a protractor and dividing the angle by three. I made these diagrams just by sketching them out, so they aren’t perfect in their measure, but if you were doing the diagram yourself on a sheet of scratch paper you wouldn’t bother getting the protractor out either. (And, famously, you can’t trisect an angle if you’re using just compass and straightedge to draw things, but you don’t have to restrict yourself to compass and straightedge for this.)

Now the next bit is to take the points where adjacent angle trisectors intersect — that is, for example, where the lower red line crosses the lower green line; where the upper red line crosses the left blue line; and where the right blue line crosses the upper green line. Draw lines connecting these points together and …

This new triangle, drawn in purple on my sketch, is an equilateral triangle!

(It may look a little off, but that’s because I didn’t measure the trisectors when I drew them in and just eyeballed it. If I had measured the angles and drawn the new ones in carefully, it would have been perfect.)

I’ve been thinking back on this and grinning ever since reading it. I certainly didn’t see that punch line coming.