TSTST 2012 Problem 4

In scalene triangle $A B C$, let the feet of the perpendiculars from $A$ to $\overline{B C}, B$ to $\overline{C A}$, $C$ to $\overline{A B}$ be $A_{1}, B_{1}, C_{1}$, respectively. Denote by $A_{2}$ the intersection of lines $B C$ and $B_{1} C_{1}$. Define $B_{2}$ and $C_{2}$ analogously. Let $D, E, F$ be the respective midpoints of sides $\overline{B C}, \overline{C A}, \overline{A B}$. Show that the perpendiculars from $D$ to $\overline{A A_{2}}, E$ to $\overline{B B_{2}}$ and $F$ to $\overline{C C_{2}}$ are concurrent.