Question 22
Let $M$ be the midpoint of side $AB$ in triangle $ABC$. $B_1$ is a point on segment $AC$ such that $CB = CB_1$. The circumcircles of triangles $ABC$ and $BMB_1$, $\omega$ and $\omega_1$, intersect for the second time at point $K$. Let $Q$ be the midpoint of arc $ACB$ of $\omega$. Lines $B_1Q$ and $BC$ intersect at point $E$. Prove that line $KC$ passes through the midpoint of segment $B_1E$.