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 $l$, 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 $T_1 T_2 T_3$.
I found that $ABC \sim T_1 T_2 T_3$ , and now we are learning the Feuerbach's circle.