Let $ABC$ be a triangle. Circle $\Gamma$ passes through point $A$, meets segments $AB$ and $AC$ again at $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $(BDF)$ at $F$ and the tangent to circle $(CEG)$ at $G$ meet at $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.