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Equilateral triangles on the exterior of a quadrilater ABCD

Source: Mediterranean MO 2000

November 23, 2010
geometry proposedgeometry

Problem Statement

Suppose that in the exterior of a convex quadrilateral ABCDABCD equilateral triangles XAB,YBC,ZCD,WDAXAB,YBC,ZCD,WDA with centroids S1,S2,S3,S4S_1,S_2,S_3,S_4 respectively are constructed. Prove that S1S3⊥S2S4S_1S_3\perp S_2S_4 if and only if AC=BDAC=BD.
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