KöMaL 2011 A540

Let $A_1 A_2 A_3$ be a non-equilateral triangle, let point $M$ be its orthocenter, let $F$ be its Feuerbach point, and let $k$ be the circumcircle of the triangle. For $i$=1,2,3 denote by $k_i$ the circle that is internally tangent to $k$ and tangent to the sides $A_i A_{i+1}$ and $A_i A_{i+2}$. (The indices are considered modulo 3, i.e. $A_4$=$A_1$ and $A_5$=$A_2$.) Let $T_i$ be the point of tangency between $k$ and $k_i$. Prove that the lines $A_1 T_1$, $A_2 T_2$, $A_3 T_3$ and $MF$ are concurrent.