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sum R^4/P_i^2 >= 16, geometric inequality with areas P_i, by incenter I

Source: JBMO Shortlist 2018 G4

July 22, 2019
geometric inequalityinequalitiesgeometryincentercircumradius

Problem Statement

Let ABCABC be a triangle with side-lengths a,b,ca, b, c, inscribed in a circle with radius RR and let II be ir's incenter. Let P1,P2P_1, P_2 and P3P_3 be the areas of the triangles ABI,BCIABI, BCI and CAICAI, respectively. Prove that R4P12+R4P22+R4P32≥16\frac{R^4}{P_1^2}+\frac{R^4}{P_2^2}+\frac{R^4}{P_3^2}\ge 16
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