Question 166
$A$ is one of the points of intersection of circles $\omega_1$ and $\omega_2$, $T_1T_2$ is the external common tangent to these circles, which touches $\omega_1$ at $T_1$ and touches $\omega_2$ at $T_2$. $B_1$ is any point on the circle $\omega_1$, $B_2$ is any point on the circle $\omega_2$, such that $A$, $B_1$ and $B_2$ are not collinear. The circumcircle $\omega$ of $\triangle AB_1B_2$ intersects lines $T_1B_1$, $T_2B_2$ at points $C_1, C_2$ respectively. Prove that $C_1T_2$ and $C_2T_1$ intersect on $\omega$.