Question 36
Let $ABCD$ be a convex quadrilateral with $AD$ and $BC$ not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at two points $E,F$ inside $ABCD$. Let $\Gamma_E$ be the circle through the perpendicular projections of $E$ onto $AB,BC,CD$, and $\Gamma_F$ the circle through the perpendicular projections of $F$ onto $CD,DA,AB$. Prove that the midpoint of $EF$ lies on the line through the two intersection points of $\Gamma_E$ and $\Gamma_F$.