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quadrilateral ABCD without parallel sides is circumscribed

Source: ARO 2005 - problem 11.7

April 30, 2005
geometryincenterparallelogramtrigonometrycomplex numbersRussia

Problem Statement

A quadrilateral ABCDABCD without parallel sides is circumscribed around a circle with centre OO. Prove that OO is a point of intersection of middle lines of quadrilateral ABCDABCD (i.e. barycentre of points A, B, C, DA,\,B,\,C,\,D) iff OA⋅OC=OB⋅ODOA\cdot OC=OB\cdot OD.
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