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Convex quadrilateral and O inside becomes square and center

Source: Balkan MO 1997, Problem 1

April 24, 2006
geometrytrigonometryinequalitiesgeometry proposed

Problem Statement

Suppose that OO is a point inside a convex quadrilateral ABCDABCD such that OA2+OB2+OC2+OD2=2A[ABCD], OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] , where by A[ABCD]\mathcal A[ABCD] we have denoted the area of ABCDABCD. Prove that ABCDABCD is a square and OO is its center. Yugoslavia
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