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r^2+r_a^2+r_b^2+ r_c^2 >= 2S

Source: VI Soros Olympiad 1990-00 R1 10.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 28, 2024
geometrygeometric inequalityGeometric Inequalitiesexradiusinequalities

Problem Statement

Prove that the inequality r2+ra2+rb2+rc2≥2S r^2+r_a^2+r_b^2+ r_c^2 \ge 2S holds for an arbitrary triangle, where rr is the radius of the circle inscribed in the triangle, rar_a, rbr_b, rcr_c are the radii of its three excribed circles, SS is the area of the triangle.
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