Problems from Ukrainian Authors

$A$ is one of the points of intersection of circles ${\omega }_1$ and ${\omega }_2$, $T_1{T}_2$ is the external common tangent to these circles, which tangents ${\omega }_1$ at $T_1$ and tangents ${\omega }_2$ 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. Circumcircle of the $△A{B}_1{B}_2$ intersects lines $T_1{B}_1$, $T_2{B}_2$ at points $C_1,C_2$ respectively. Prove that $C_1{T}_2$ and $C_2{T}_1$ intersect on $\omega$.