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Lines joining centers of equilateral triangles perpendicular

Source: IMO Shortlist 1992, Problem 5

October 26, 2003
geometryconvex quadrilateraldiagonalsperpendicular linesIMO Shortlist

Problem Statement

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 ABCD with congruent diagonals AC \equal{} 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 ABCD be a convex quadrilateral such that AC \equal{} BD. Equilateral triangles are constructed on the sides of the quadrilateral. Let O1,O2,O3,O4 O_1,O_2,O_3,O_4 be the centers of the triangles constructed on AB,BC,CD,DA AB,BC,CD,DA respectively. Show that O1O3 O_1O_3 is perpendicular to O2O4. O_2O_4.
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