US-TST 2004 Problem 4

Let $ABC$ be a triangle. Choose a point $D$ in its interior. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ be a circle passing through $C$ and $D$ so that the other point of intersection of the two circles lies on $AD$ . Let $\omega_1$ and $\omega_2$ intersect side $BC$ at $E$ and $F$ , respectively. Denote by $X$ the intersection of $DF$ , $AB$ and $Y$ the intersection of $DE, AC$ . Show that $XY \parallel BC$ .