Question 126
Suppose the convex quadrilateral $ABCD$ has an inscribed circle. The circle touches $AB$, $BC$, $CD$, $DA$ at $A_1$, $B_1$, $C_1$, $D_1$ respectively. Let points $E$, $F$, $G$, $H$ be the midpoints of $A_1B_1$, $B_1C_1$, $C_1D_1$, $D_1A_1$ respectively. Prove that quadrilateral $EFGH$ is a rectangle if and only if $ABCD$ is a cyclic quadrilateral. (posed by Feng Zhigang)