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.
Read more about the Japanese Theorem here, here, and here. For more about Sangaku problems see Hidetoshi and Rothman’s new book Sacred Mathematics: Japanese Temple Geometry.
Other readers
That is neat. I think I will add proving it to my summer todo list. Thank you.
By: Kate on June 2, 2009
at 9:30 pm
I love it — thank you — from a math tutor always interested in new (to me) constructions
By: jedward706 on June 3, 2009
at 9:42 am
It is a cool problem. Be warned, Kate, that in Hidetoshi and Rothman’s book they have chapters titled Easier Temple Geometry Problems, Harder Temple Geometry Problems, and Still Harder Temple Geometry Problems. This problem is in that third chapter.
It is tricky to prove from basic geometric results, but if you happen to know this geometric theorem http://cli.gs/UBZX06 (I’ve shortened the URL in case you didn’t want the proof “spoiled”), it has a quick proof.
By: Dave Richeson on June 3, 2009
at 9:59 pm
[...] case you’re wondering, I’ll be speaking about generalizations and consequences of the Japanese Theorem. Or as much as I can in only 10 [...]
By: Short talks at conferences « Division by Zero on June 4, 2009
at 3:37 pm
[...] but not least, Dave Richeson at Division by Zero has some beautiful GeoGebra applets for playing with the Japanese Theorem, which relates the radii of circles inscribed in triangulations of cyclic [...]
By: Carnival of Mathematics #53 « The Math Less Traveled on June 5, 2009
at 6:35 pm
[...] I wrote about the Japanese Theorem. If you were unsuccessful in proving this beautiful theorem, try again using Carnot’s [...]
By: Carnot’s Theorem « Division by Zero on June 22, 2009
at 5:06 pm
[...] 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, [...]
By: I need to learn how to say no « Division by Zero on August 3, 2009
at 9:17 am