BALKAN 1997 Problem 1

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

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