Posted by: Dave Richeson | June 2, 2009

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. 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!)

Picture 9

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.

Picture 10

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.



  1. That is neat. I think I will add proving it to my summer todo list. Thank you.

  2. I love it — thank you — from a math tutor always interested in new (to me) constructions

  3. 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 (I’ve shortened the URL in case you didn’t want the proof “spoiled”), it has a quick proof.

  4. […] 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 […]

  5. […] 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 […]

  6. […] I wrote about the Japanese Theorem. If you were unsuccessful in proving this beautiful theorem, try again using Carnot’s […]

  7. […] 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, […]

  8. […] couple of years ago I wrote a 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 […]

  9. Very cool! How do you insert text into GeoGebra that includes variables from the construction – like your “Sum of radii …” text?

    • You can find instructions for that here under “mixed text.”

      • Thank you!

  10. WOw.. it’s great… I have studied Circumscribable Quadrilateral just last summer and now I meet inscribable quadrilateral with a great characteristic.. Has this been proven?i’ll try to make some theorems and proofs for this..:)

  11. […] Sacred Mathematics: Japanese Temple Geometry (Sangaku problem) on divisbyzero with GeoGebra Applet […]

  12. […] Sacred Mathematics: Japanese Temple Geometry (Sangaku problem) on divisbyzero with GeoGebra Applet […]


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