Japanese Triangulation Theorem

Let a convex Cyclic Polygon be Triangulated in any manner, and draw the Incircle to each Triangle so constructed. Then the sum of the Inradii is a constant independent of the Triangulation chosen. This theorem can be proved using Carnot's Theorem. It is also true that if the sum of Inradii does not depend on the Triangulation of a Polygon, then the Polygon is Cyclic.

See also Carnot's Theorem, Cyclic Polygon, Incircle, Inradius, Triangulation

References

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 24-26, 1985.

Lambert, T. ``The Delaunay Triangulation Maximizes the Mean Inradius.'' Proc. Sixth Canadian Conf. Comput. Geometry. Saskatoon, Saskatchewan, Canada, pp. 201-206, August 1994.