Napoleon's Theorem

Napoleon's Theorem is a theorem some attribute (naturally enough) to Napoleon.

It says that if you take any triangle and draw equilateral triangles on each side, then the incenters of those triangles form a fourth equilateral triangle.

There is a variant in which the new triangles are drawn as reflections of the other ones, that is, inside the original triangle.

There is a terrific vector proof that I diagram here. (I think I got the idea for the proof from Alexander Bogomolny, but I can't find it at the moment. Wonderful site).

Define vectors for paths to and from the incenters based on the following.

Then apply a simple test for the adjacent sides of an equilateral triangle:

The details depend on the definition of the direction of rotation, and the path taken around the putative equilateral triangle. Details in the links below. Here is a variant of the problem:

My write-up is here. Probably the neatest thing is we get the variant basically for free, once the setup is done. I also (finally) got a proof on ProofWiki here as well as the variant (here)