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

David Richeson

The Japanese Theorem for Polygons

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 \(P\)   be any cyclic polygon inscribed in a circle that is triangulated by diagonals. Define the total inradius of \(P\) , denoted \(r_P\), to be the sum of the inradii of the triangles. The Japanese theorem states that \(r_P\) is independent of triangulation.

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.