Question 110
Let $ABC$ be a non-isosceles triangle with $M$ as the midpoint of $BC$. Take a point $D \neq M$ on the segment $BC$ and the circumcircles of triangles $ABD$, $ACD$ intersect $AC$, $AB$ again at $E$, $F$ respectively. Line $BE$, $CF$ meet at $K$ and $AK$, $EF$ meet at $I$. Denote $O$ as the circumcenter of triangle $AEF$ and suppose that $(\omega)$ is some circle passing through $B$, $C$ and external tangent to $(O)$. Prove that the common tangent of $(\omega)$ and $(O)$ passes through the orthocenter of triangle $IMO$.