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Prove that S = 8S1 − 4S2 - [IMO LongList 1971]

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January 1, 2011
geometrygeometric transformationreflectiongeometry proposed

Problem Statement

Let squares be constructed on the sides BC,CA,ABBC,CA,AB of a triangle ABCABC, all to the outside of the triangle, and let A1,B1,C1A_1,B_1, C_1 be their centers. Starting from the triangle A1B1C1A_1B_1C_1 one analogously obtains a triangle A2B2C2A_2B_2C_2. If S,S1,S2S, S_1, S_2 denote the areas of trianglesABC,A1B1C1,A2B2C2 ABC,A_1B_1C_1,A_2B_2C_2, respectively, prove that S=8S1−4S2.S = 8S_1 - 4S_2.
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