Miquel’s pivot theorem

Euclidean geometry dates back at least to Euclid (circa 300 BC), and so you might think it’s been pretty well picked over by now. And yet people still occasionally discover new plane geometry theorems. Some of these new theorems are complicated, asking question that the ancients would not have asked. But once in a while someone discovers a gem that the ancients could have appreciated but didn’t find. One example is Miquel’s pivot theorem [1]. The theorem was discovered in 1838, which relative to the timeline of Euclidean geometry makes it a recent discovery. Choose a point on each side of a triangle. Then for each vertex draw a circle through it and the chosen points on the adjacent sides. Miquel’s theorem says the three circles meet in one point. Here’s an example. For a trangle ABC, choose points D, E, and F on each side. The three triangles described in the theorem intersect at M. Now the three points D, E, and F don’t have to be limited to the sides of the triangle; they can be on…