Problem 1493

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 $T_1T_2T_3$.
Proposed by: Bálint Bíró, Eger
(5 pont)