[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. Sangaku problems were geometric problems posted in Buddhist temples and Shinto shrines in Japan as offerings to the gods. They were created during the Edo period when the nation was closed to the outside world.
Essentially the problem amounted to the following. Inscribe a quadrilateral in a circle. Draw one of the diagonals and inscribe a red circle in each of the two resulting triangular regions. Then repeat with the other diagonal, creating two blue circles (see image below). The amazing fact is that the sum of the radii of the red circles is the same as the sum of the radii of the blue circles. Here’s a GeoGebra applet illustrating this behavior. (Incidentally, there is another cool fact about this construction: the centers of the four circles form a rectangle!)
Once the quadrilateral version of the Japanese theorem has been established it is not difficult to extend it to general cyclic polygons. Take any cyclic polygon and triangulate it using nonintersecting diagonals. Inscribe circles in each of the triangles. Then the sum of the radii is independent of the choice of triangulation. For example, the sum of the radii of the blue circles below is equal to the sum of the radii of the red circles. Here’s a GeoGebra applet illustrating this behavior.