2011 IMO Shortlist G3

Let $A B C D$ be a convex quadrilateral whose sides $A D$ and $B C$ are not parallel. Suppose that the circles with diameters $A B$ and $C D$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_{E}$ be the circle through the feet of the perpendiculars from $E$ to the lines $A B, B C$, and $C D$. Let $\omega_{F}$ be the circle through the feet of the perpendiculars from $F$ to the lines $C D, D A$, and $A B$. Prove that the midpoint of the segment $E F$ lies on the line through the two intersection points of $\omega_{E}$ and $\omega_{F}$.

设 $A B C D$ 为凸四边形，其边 $A D$ 和 $B C$ 不平行。假设直径为 $A B$ 和 $C D$ 的圆在四边形内部交于点 $E$ 和 $F$。设 $\omega_{E}$ 为通过从 $E$ 到线 $A B、B C$ 和 $C D$ 的垂线脚的圆。设 $\omega_{F}$ 为通过从 $F$ 到线 $C D、D A$ 和 $A B$ 的垂线脚的圆。证明线段$E F$的中点位于$\omega_{E}$和$\omega_{F}$两个交点的直线上。