TUYMAADA 2011 S2

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$ , and $M$ is the midpoint of $AB$ . Points $S_1$ and $S_2$ lie on the line $AB$ (but not between $A$ and $B$ ). The tangents drawn from $S_1$ to $\omega_1$ touch it at $X_1$ and $Y_1$ , and the tangents drawn from $S_2$ to $\omega_2$ touch it at $X_2$ and $Y_2$ . Prove that if the line $X_1X_2$ passes through $M$ , then line $Y_1Y_2$ also passes through $M$ .