David Richeson's Extension
of an Old Japanese Theorem
A 1800 sangaku known as the Old Japanese Theorem has been recently extended by David Richeson of Dickinson College. The original theorem reads:
Let a convex polygon, 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.
David's extension applies to more general polygons - they have to be cyclic but not necessarily convex. A triangulation of a polygon - even of a non-convex one - is obtained by introducing not intersecting diagonals, i.e., that is the line segments that join not adjacent vertices of the polygon. The essential difference with the convex case is that now not all triangles in a triangulation have the same orientation. Accordingly, David introduced the notion of a signed inradius. The inradius of a triangle with negative orientation is also considered negative. David proved the following theorem:
Triangulate a cyclic polygon using diagonals. The sum of the signed inradii of the triangles is independent of the choice of triangulation.
The applet below serves to illustrate the new development (and the old case as well.) The incircles of the positively oriented triangles are painted blue, those of the negatively oriented - red.
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Copyright © 1996-2012 Alexander Bogomolny
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