The Japanese Theorem for Nonconvex Polygons - The Japanese Theorem for Quadrilaterals

David Richeson

The Japanese Theorem for Quadrilaterals

The art of Maruyama's sangaku problem implies that in any such configuration \(d_N = d_E + d_W - d_S\), where \(d_N, d_E, d_W\), and \(d_S\) are the diameters of the northern, eastern, western, and southern circles, respectively. Equivalently, using radii,

\[r_N + r_S = r_E + r_W.\]

This surprising result is now known as the Japanese theorem.

The Japanese theorem for quadrilaterals. Add a diagonal to a convex quadrilateral whose vertices lie on a circle. Inscribe circles in the resulting triangles. The sum of the radii of the inscribed circles does not depend on the choice of diagonal.

Below we have an applet illustrating this invariance. The circle has radius 1. Move the points around the circle and observe that the sum of the radii of the red circles equals the sum of the radii of the blue circles. There is an additional fascinating fact about this construction that we will not pursue: the centers of the four circles form a rectangle ([p. 255, Jo1]).



In this article we examine this extremely beautiful theorem. We not only prove the theorem, but we also endeavor to illustrate why the theorem is true. We explore some consequences of the theorem, and we generalize it further to polygons with intersecting sides.