Let $A B C$ be a triangle and $D, E, F$ be the midpoints of $\operatorname{arcs} B C, C A, A B$ on the circumcircle. Line $\ell_{a}$ passes through the feet of the perpendiculars from $A$ to $\overline{D B}$ and $\overline{D C}$. Line $m_{a}$ passes through the feet of the perpendiculars from $D$ to $\overline{A B}$ and $\overline{A C}$. Let $A_{1}$ denote the intersection of lines $\ell_{a}$ and $m_{a}$. Define points $B_{1}$ and $C_{1}$ similarly. Prove that triangles $D E F$ and $A_{1} B_{1} C_{1}$ are similar to each other.