The Japanese theorem for nonconvex polygons
By David Richeson
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.
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