IMO Shortlist 1992 problem 5


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2. travnja 2012.
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A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular.

Alternative formulation. Given a convex quadrilateral ABCD with congruent diagonals AC = BD. Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other.

Original formulation: Let ABCD be a convex quadrilateral such that AC = BD. Equilateral triangles are constructed on the sides of the quadrilateral. Let O_1,O_2,O_3,O_4 be the centers of the triangles constructed on AB,BC,CD,DA respectively. Show that O_1O_3 is perpendicular to O_2O_4.
Izvor: Međunarodna matematička olimpijada, shortlist 1992