A cyclic polygon is a polygon all of whose vertices lie on a circle (for now we assume that the polygon is convex; that is, the edges do not cross). The inradius and circumradius of a triangle are the radii of the inscribed and circumscribed circles, respectively. The theorem stated and proved by Honsberger is the following.
The Japanese theorem for polygons. Triangulate a convex cyclic polygon using nonintersecting diagonals. The sum of the inradii of the triangles is independent of the choice of triangulation.
Let \(P\)
The following applet gives three different triangulations of a cyclic polygon. Move the vertices to see that the sums remain the same.
Remark: if the polygon is not cyclic then the total inradius is not independent of triangulation. In 1994 Lambert ([L]) proved that in general the largest total inradius is achieved by the Delaunay triangulation, which is the planar dual to the well-known Voronoi diagram.