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 \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 be a convex quadrilateral such that AC \equal{} BD. Equilateral triangles are constructed on the sides of the quadrilateral. Let O1,O2,O3,O4 be the centers of the triangles constructed on AB,BC,CD,DA respectively. Show that O1O3 is perpendicular to O2O4. geometryconvex quadrilateraldiagonalsperpendicular linesIMO Shortlist