In an acute triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
the points
![D,E](/media/m/1/b/3/1b378dbc2087b834ee8ae2b49675e20e.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
are the feet of the altitudes through
![A,B](/media/m/7/1/7/7174f8a9f33236ee137c01b144237389.png)
and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
respectively. The incenters of the triangles
![AEF](/media/m/6/e/8/6e87ef6678dbdc845347a6471c34f83a.png)
and
![BDF](/media/m/f/5/0/f5003fffaa52534b09542e985cfb81b0.png)
are
![I_1](/media/m/4/1/6/416fe38d243ecd6d66747e4d88b6518d.png)
and
![I_2](/media/m/c/b/0/cb0739259063636b82f7b6b3e9dd3de0.png)
respectively; the circumcenters of the triangles
![ACI_1](/media/m/5/d/9/5d9aed9c84860a3e71bfbdb8a6637b5d.png)
and
![BCI_2](/media/m/a/7/e/a7e1f34cb7e5a19af35bd780a898577e.png)
are
![O_1](/media/m/7/2/b/72b270d556043f6f393afbf50620eb57.png)
and
![O_2](/media/m/f/2/d/f2de7ab4fb5625160a4d2f2ac2dd707d.png)
respectively. Prove that
![I_1I_2](/media/m/b/0/e/b0efa9a1b8480f63a30ccb52811bfd25.png)
and
![O_1O_2](/media/m/d/f/d/dfdefa801f9dce3b3b6138913c5ea15c.png)
are parallel.
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In an acute triangle $ABC$ the points $D,E$ and $F$ are the feet of the altitudes through $A,B$ and $C$ respectively. The incenters of the triangles $AEF$ and $BDF$ are $I_1$ and $I_2$ respectively; the circumcenters of the triangles $ACI_1$ and $BCI_2$ are $O_1$ and $O_2$ respectively. Prove that $I_1I_2$ and $O_1O_2$ are parallel.