2012 IMO Shortlist G7

Let $A B C D$ be a convex quadrilateral with non-parallel sides $B C$ and $A D$. Assume that there is a point $E$ on the side $B C$ such that the quadrilaterals $A B E D$ and $A E C D$ are circumscribed. Prove that there is a point $F$ on the side $A D$ such that the quadrilaterals $A B C F$ and $B C D F$ are circumscribed if and only if $A B$ is parallel to $C D$.

令 $A B C D$ 为具有非平行边 $B C$ 和 $A D$ 的凸四边形。假设边 $B C$ 上有一点 $E$，使得四边形 $A B E D$ 和 $A E C D$ 外接。证明$A D$ 边上有一点$F$，使得四边形$A B C F$ 和$B C D F$ 外接当且仅当$A B$ 与$C D$ 平行。