Go Geometry Problem Sangaku Problem (An Old Japanese Theorem)

Let a convex inscribed 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. Hints: Carnot's Theorem.
 

 

References: Fukagawa Hidetoshi, Tony Rothman, "Sacred Mathematics: Japanese Temple Geometry" (Princenton University Press, 2008).

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