MathNet 2000 — Math Olympiad Vault Question 74

A convex quadrilateral $ABCD$ is inscribed in a circle. Prove that the chord joining the midpoints of arcs $\widehat{AB}$ and $\widehat{CD}$ is perpendicular to the chord joining the midpoints of arcs $\widehat{BC}$ and $\widehat{DA}$.