A couple of years ago I wrote blog posts about two beautiful theorems from geometry: the so-called Japanese theorem and Carnot’s theorem. Today I finished a draft of a web article that looks at both of these theorems in more detail. It contains all that you could want—connections between these theorems, generalizations of them, and consequence of them. It even has a little topology (the n-dimensional torus makes an appearance). The mathematics is supplemented by some Geogebra applets so that you can see the theorems in action.
You can find the draft of the article here: The Japanese theorem for nonconvex polygons [The article is now online at the MAA's Loci.]