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Geometric Inequality on the length sides - ISL 1978

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September 20, 2010
geometrytriangle inequalitygeometric inequalityarea of a triangleIMO Shortlist

Problem Statement

Let T1T_1 be a triangle having a,b,ca, b, c as lengths of its sides and let T2T_2 be another triangle having u,v,wu, v,w as lengths of its sides. If P,QP,Q are the areas of the two triangles, prove that 16PQ≤a2(−u2+v2+w2)+b2(u2−v2+w2)+c2(u2+v2−w2).16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2). When does equality hold?
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