CWMO 2003 Problem 6

A circle can be inscribed in the convex quadrilateral $ABCD$ . The incircle touches the sides $AB, BC, CD, DA$ at $A_1, B_1, C_1, D_1$ respectively. The points $E, F, G, H$ are the midpoints of $A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $EFGH$ is a rectangle if and only if $A, B, C, D$ are concyclic.