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inequalities with distances of G to sides of triangle

Source: 1999 Spanish Mathematical Olympiad P5

July 21, 2018
inequalitiesdistancegeometrygeometric inequalityCentroid

Problem Statement

The distances from the centroid GG of a triangle ABCABC to its sides a,b,ca,b,c are denoted ga,gb,gcg_a,g_b,g_c respectively. Let rr be the inradius of the triangle. Prove that: a) ga,gb,gc≥23rg_a,g_b,g_c \ge \frac{2}{3}r b) ga+gb+gc≥3rg_a+g_b+g_c \ge 3r
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