The Japanese theorem for nonconvex polygons
By David Richeson
Dickinson College

Abstract. The so-called "Japanese theorem" dates back over 200 years; in its original form it states that given a quadrilateral inscribed in a circle, the sum of the inradii of the two triangles formed by the addition of a diagonal does not depend on the choice of diagonal. Later it was shown that this invariance holds for any cyclic polygon that is triangulated by diagonals. In this article we examine this theorem closely, discuss some of its consequences, and generalize it further. In particular, we explore its relationship with Carnot's classical theorem on triangles, we look for extreme values for this sum of inradii, we look at the limit of this value as the number of sides goes to infinity, and we generalize the theorem to nonconvex cyclic polygons. We include interactive applets throughout the article to give the theorems a tangible credibility.

Table of contents:

  1. A Japanese temple problem
  2. The Japanese theorem for quadrilaterals
  3. First generalizations
  4. The Japanese theorem for polygons
  5. Carnot's theorem
  6. Carnot's theorem for cyclic polygons and a proof of the Japanese theorem
  7. The space of convex polygons
  8. The total inradius function
  9. Extreme values for the radial sum function
  10. Regular polygons
  11. Limiting behavior
  12. Irrational rotations of the circle
  13. The generalized Japanese theorem
  14. A further generalization of Carnot's theorem
  15. Proof of the generalized Japanese theorem
  16. Extreme values for cyclic polygons
  17. Acknowledgments and bibliography

Pages that contain applets:

 

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The Japanese Theorem