Question 34
Let $ABCD$ be a quadrilateral, and let $O$ be the intersection of $AC$ and $BD$. Quadrilateral $A'B'C'D'$ is obtained by rotating $ABCD$ about $O$ by some angle. Let $A_1, B_1, C_1, D_1$ be the intersection points of the lines $A'B'$ and $AB$, $B'C'$ and $BC$, $C'D'$ and $CD$, $D'A'$ and $DA$, respectively. Prove that quadrilateral $A_1B_1C_1D_1$ is cyclic if and only if $AC$ is perpendicular to $BD$.