IR-MO 1993 Problem 5

In a convex quadrilateral $ABCD$ , diagonals $AC$ and $BD$ are equal. We construct four equilateral triangles with centers $O_1,O_2,O_3,O_4$ on the sides sides $AB, BC, CD, DA$ outside of this quadrilateral, respectively. Show that $O_1O_3 \perp O_2O_4$ .