2012 IMO Shortlist G3

In an acute triangle $A B C$ the points $D, E$ and $F$ are the feet of the altitudes through $A$, $B$ and $C$ respectively. The incenters of the triangles $A E F$ and $B D F$ are $I_{1}$ and $I_{2}$ respectively; the circumcenters of the triangles $A C I_{1}$ and $B C I_{2}$ are $O_{1}$ and $O_{2}$ respectively. Prove that $I_{1} I_{2}$ and $O_{1} O_{2}$ are parallel.

在锐角三角形 $A B C$ 中，点 $D、E$ 和 $F$ 分别是经过 $A$、$B$ 和 $C$ 的高度的脚。三角形$A E F$和$B D F$的内心分别为$I_{1}$和$I_{2}$；三角形$A C I_{1}$ 和$B C I_{2}$ 的外心分别为$O_{1}$ 和$O_{2}$。证明 $I_{1} I_{2}$ 和 $O_{1} O_{2}$ 平行。