CR-MO 2010 Problem 5

Let $C_1$ be a circle with center $O$ and let $B$ and $C$ be points in $C_1$ such that $BOC$ is an equilateral triangle. Let $D$ be the midpoint of the minor arc $BC$ of $C_1$ . Let $C_2$ be the circle with center $C$ that passes through $B$ and $O$ . Let $E$ be the second intersection of $C_1$ and $C_2$ . The parallel to $DE$ through $B$ intersects $C_1$ for second time in $A$ . Let $C_3$ be the circumcircle of triangle $AOC$ . The second intersection of $C_2$ and $C_3$ is $F$ . Show that $BE$ and $BF$ trisect the angle $\angle ABC$ .