Ex. 142. Angle $A$ in triangle $A B C$ is equal to $\alpha$. A circle passing through $A$ and $B$ and tangent to $B C$ intersects the median to side $B C$ (or its extension) at point $M$, different from $A$. Find $\angle B M C$.
Ex. 142. Answer: $180^{\circ}-\alpha$. Hint. Note that $\angle C B M=\angle B A D$, and $\angle B C M=\angle C A D$ (due to the similarity of triangles $C M D$ and $A D C$, where $D$ is the midpoint of $B C$).