2009 IMO Shortlist G6

UKR Let the sides $A D$ and $B C$ of the quadrilateral $A B C D$ (such that $A B$ is not parallel to $C D$ ) intersect at point $P$. Points $O_{1}$ and $O_{2}$ are the circumcenters and points $H_{1}$ and $H_{2}$ are the orthocenters of triangles $A B P$ and $D C P$, respectively. Denote the midpoints of segments $O_{1} H_{1}$ and $O_{2} H_{2}$ by $E_{1}$ and $E_{2}$, respectively. Prove that the perpendicular from $E_{1}$ on $C D$, the perpendicular from $E_{2}$ on $A B$ and the line $H_{1} H_{2}$ are concurrent.

UKR 让四边形 $A B C D$ 的边 $A D$ 和 $B C$ (使得 $A B$ 不与 $C D$ 平行)相交于点 $P$。点$O_{1}$和$O_{2}$是三角形$A B P$和$D C P$的外心，点$H_{1}$和$H_{2}$分别是三角形$A B P$和$D C P$的垂心。分别用 $E_{1}$ 和 $E_{2}$ 表示线段 $O_{1} H_{1}$ 和 $O_{2} H_{2}$ 的中点。证明 $E_{1}$ 在 $C D$ 上的垂线、$E_{2}$ 在 $A B$ 上的垂线和线 $H_{1} H_{2}$ 是并发的。