The Japanese theorem for nonconvex polygons

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…

I need to learn how to say no

I’m heading to MathFest in a few days. I’m giving a talk on some generalizations of the Japanese Theorem (which I hope to blog about at some point), I am a panelist on the AWM panel called Family Matters, and, since I’m on the MAA Committee for Minicourses, I will be monitoring two minicourses. I also hope…

Carnot’s Theorem

Here’s a neat theorem from geometry. Begin with any triangle. Let R be the radius of its circumscribed circle and r be the radius of its inscribed circle. Let a, b, and c be the signed distances from the center of the circumscribed circle to the three sides. The sign of a, b, and c…

Short talks at conferences

I just received an email from the MAA regarding MathFest 2009 that made me laugh. I’m scheduled to give a ten minute (10 minute!) talk in one of the sessions. Those of you who’ve given a 10 minute talk know that it is impossible to say anything in that amount of time. (This reminds me…

The Japanese Theorem

[Update: I’ve written quite a bit more about this theorem since 2009. See this page for more details.] I’ve been playing with GeoGebra for the last few days. As an exercise I decided to create applets to demonstrate the extremely beautiful Japanese Theorem. The first appearance of the Japanese theorem was as a Sangaku problem….