Question 5
Acute triangle $ABC$ with $AB \lt AC$ has circumcircle $\omega$. Let $D$, $E$, $F$ be the midpoints of sides $BC$, $CA$, $AB$ correspondingly. Ray $AD$ intersects $\omega$ at point $K$ ($K \neq A$) and the circumcircle of triangle $AEF$ at point $M$ ($M \neq A$). On circle $\omega$ a point $T$ such that $AT \parallel BC$ is chosen. The circumcircles of triangles $KDT$ and $AEF$ intersect at point $L$ on the shorter arc $EF$. Prove that circle $\omega$ and the circumcircle of triangle $KLM$ are tangent to each other.