KöMaL 2015 A648

In the acute angled triangle $ABC$ , the midpoints of the sides $BC$ , $CA$ and $AB$ are $D$ , $E$ and $F$ , respectively. The foot of the altitude of the triangle starting from $C$ is $T_1$. On some line, passing through point $C$ but not containing $T_1$, the feet of the perpendiculars starting from $A$ and $B$ are $T_2$ and $T_3$, respectively. Prove that the circle $DEF$ passes through the center of the circle $T_1T_2T_3$.