2012 IMO Shortlist G2

Let $A B C D$ be a cyclic quadrilateral whose diagonals $A C$ and $B D$ meet at $E$. The extensions of the sides $A D$ and $B C$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $E C G D$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $A D$. Prove that $D, H, F, G$ are concyclic.

令 $A B C D$ 为循环四边形，其对角线 $A C$ 和 $B D$ 交于 $E$。 $A$ 和 $B$ 之外的边 $A D$ 和 $B C$ 的延伸在 $F$ 处相交。令$G$为使得$E C G D$为平行四边形的点，并令$H$为$E$在$A D$中的反射下的图像。证明$D、H、F、G$是同循环的。