Question 9
Let $ABCD$ be a convex quadrilateral with $AD$ not parallel to $BC$. The circles with diameters $AB$ and $CD$ meet at two points $E,F$ inside $ABCD$. Let $Q,R,S$ be the perpendicular feet from $E$ to the lines $AB,BC,CD$, and let $\omega_E$ be their circumcircle. Similarly, let $\omega_F$ be the circle through the perpendicular feet from $F$ to $CD,DA,AB$. Prove that the midpoint of $EF$ lies on the line joining the two intersection points of $\omega_E$ and $\omega_F$.