Question 1
A convex quadrilateral $ABCD$ has perpendicular diagonals, and the opposite sides $AB$ and $CD$ are not parallel. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.