An old Japanese theorem

Abstract

Let a convex polygon (a shape is convex if with every pair of points that belong to the shape, the shape contains the whole straight line segment connecting the two points) which is inscribed in a circle, be triangulated by drawing all the diagonals from one of the vertices and let the inscribed circle be drawn in each of the triangles. Then the sum of the radii of all these circles is a constant which is independent of which vertex is used to form the triangulation. R is the radius of the outer circle while T is the radius of the inner circle inscribed in the triangle. To prove the theorem, we must first prove the following two lemmas.