Japanese theorem for cyclic polygons

For a cyclic polygon, the sum of the inradii of the triangles in a triangulation from one vertex is independent of the chosen vertex.

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Inradius: The radius of a circle inscribed in a polygon, tangent to every side. Each triangle contributes its inradius to the vertex-independent sum.
Cyclic polygon: A polygon whose vertices all lie on a common circle. The theorem applies specifically when all polygon vertices are concyclic.
Japanese mathematics: Mathematical work developed in Japan, including the distinctive early-modern sangaku tradition. The theorem’s name connects it to Japanese geometry.
Cyclic pentagon: A five-sided polygon whose vertices lie on one circle. Its three-triangle fans provide a direct instance of the invariant sum.
Semiperimeter: Half the perimeter of a polygon. For a triangle, its area equals its inradius times its semiperimeter.
Circumcircle: The circle passing through the vertices of a polygon or the three vertices of a triangle. It is the common circle containing the polygon’s vertices.
Sangaku: Japanese wooden votive tablets displaying geometric problems, especially from the Edo period. Sangaku problems helped preserve and popularize Japanese geometry.
Cyclic hexagon: A six-sided polygon whose vertices lie on one circle. Its larger collection of fan triangulations illustrates the same independence.
Heron's formula: A formula giving a triangle’s area from its three side lengths. Together with the inradius-area identity, it relates each contribution to side lengths.
Triangle: A polygon with three sides and three vertices. Every region in the theorem’s vertex-based triangulation is a triangle.