Inequality with sum of inradii in polygon
Source: Vietnam TST 1985 Problem 4
February 5, 2010
inequalitiestrigonometrygeometry proposedgeometry
Problem Statement
A convex polygon is inscribed in a circle with center and radius so that lies inside the polygon. Let the inradii of the triangles A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n \minus{} 1}A_n be denoted by r_1,r_2,\cdots ,r_{n \minus{} 2}. Prove that r_1 \plus{} r_2 \plus{} ... \plus{} r_{n \minus{} 2}\leq R(n\cos \frac {\pi}{n} \minus{} n \plus{} 2).