A circle $\omega_1$ with centre $O_1$ passes through the centre $O_2$ of a second circle $\omega_2$. The tangent lines to $\omega_2$ from a point $C$ on $\omega_1$ intersect $\omega_1$ again at points $A$ and $B$ respectively. Prove that $AB$ is perpendicular to $O_1O_2$. (5 points)