#### Title

The Japanese Theorem for Nonconvex Polygons

#### Document Type

Article

#### Publication Date

12-2013

#### Department

Mathematics

#### Language

English

#### Publication Title

Covergence

#### 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.

"The Japanese Theorem for Nonconvex Polygons" includes the following subsections:

- The Japanese Theorem for Nonconvex Polygons - A Japanese Temple Problem
- The Japanese Theorem for Nonconvex Polygons - The Japanese Theorem for Quadrilaterals
- The Japanese Theorem for Nonconvex Polygons - First Generalizations
- The Japanese Theorem for Nonconvex Polygons - The Japanese Theorem for Polygons
- The Japanese Theorem for Nonconvex Polygons - Carnot's Theorem
- The Japanese Theorem for Nonconvex Polygons - Carnot's Theorem for Cyclic Polygons
- The Japanese Theorem for Nonconvex Polygons - The Space of Cyclic Polygons
- The Japanese Theorem for Nonconvex Polygons - The Total Inradius Function
- The Japanese Theorem for Nonconvex Polygons - Extreme Values for the Radial Sum Function
- The Japanese Theorem for Nonconvex Polygons - Regular Polygons
- The Japanese Theorem for Nonconvex Polygons - Limiting Behavior
- The Japanese Theorem for Nonconvex Polygons - Irrational Rotations of the Circle
- The Japanese Theorem for Nonconvex Polygons - The Generalized Japanese Theorem
- The Japanese Theorem for Nonconvex Polygons - A Further Generalization of Carnot's Theorem
- The Japanese Theorem for Nonconvex Polygons - A Proof of the Generalized Japanese Theorem
- The Japanese Theorem for Nonconvex Polygons - Extreme Values for Cyclic Polygons
- The Japanese Theorem for Nonconvex Polygons - The Bibliography

#### Recommended Citation

Richeson, David. "The Japanese Theorem for Nonconvex Polygons." *Convergence* (December 2013). https://www.maa.org/press/periodicals/loci/the-japanese-theorem-for-nonconvex-polygons

## Comments

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