Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. At point $A$ to $\omega_1$ and $\omega_2$ the tangents $\ell_1$ and $\ell_2$ are drawn respectively. The points $T_1$ and $T_2$ are chosen respectively on the circles $\omega_1$ and $\omega_2$ so that the angular measures of the arcs $T_1A$ and $AT_2$ are equal (the measure of the circular arc is calculated clockwise). The tangent $t_1$ at the point $ T_1$ to the circle $\omega_1$ intersects $\ell_2$ at the point $M_1$. Similarly, the tangent $t_2$ at the point $T_2$ to the circle $\omega_2$ intersects $\ell_1$ at point $M_2$. Prove that the midpoints of the segments $M_1M_2$ are on the same a straight line that does not depend on the position of points $T_1$, $T_2$.