Question 118
Let $ABC$ be a triangle and $D$ a point inside it. Let $E$ be a point different from $D$ on the line $AD$. Let $\omega_{1}$ and $\omega_{2}$ be the circumferences of triangles $BDE$ and $CDE$, respectively. $\omega_{1}$ and $\omega_{2}$ intersect the side $BC$ at the interior points $F$ and $G$, respectively. Let the intersection of $DG$ and $AB$ be $X$ and the intersection of $DF$ and $AC$ be $Y$. Show that $XY$ is parallel to $BC$.