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Geometry inequality about distances From M to AB,BC,CA

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September 23, 2010
geometryinequalitiesgeometric inequalityarea of a triangleCentroidIMO Shortlist

Problem Statement

Let ABCABC be an arbitrary triangle and MM a point inside it. Let da,db,dcd_a, d_b, d_c be the distances from MM to sides BC,CA,ABBC,CA,AB; a,b,ca, b, c the lengths of the sides respectively, and SS the area of the triangle ABCABC. Prove the inequality abdadb+bcdbdc+cadcda≤4S23.abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}. Prove that the left-hand side attains its maximum when MM is the centroid of the triangle.
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