CONO-SUR 2010 Problem 5

The incircle of triangle $ABC$ touches sides $BC$ , $AC$ , and $AB$ at $D, E$ , and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$ , and $DCE$ , respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$ , respectively. Let $r_A$ be the line $E_{a}F_a$ . Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$ , $r_B$ , and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$ .