US-TST 2007 Problem 5

Triangle $ABC$ is inscribed in circle $\omega$ . The tangent lines to $\omega$ at $B$ and $C$ meet at $T$ . Point $S$ lies on ray $BC$ such that $AS \perp AT$ . Points $B_1$ and $C_1$ lie on ray $ST$ (with $C_1$ in between $B_1$ and $S$ ) such that $B_1T = BT = C_1T$ . Prove that triangles $ABC$ and $AB_1C_1$ are similar to each other.