Japanese theorem for cyclic polygons

The sum of the radii of the green circles is equal to the sum of the radii of the red circles

In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.

Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic. The Japanese theorem follows from Carnot's theorem; it is a Sangaku problem.

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