Ukrainian Mathematical Olympiad

Let a point $A$ lay outside a given circle $\omega$. Through $A$ two lines are drawn, the first intersect $\omega$ at $B$ and $C$, the second, at $D$ and $E$ ($D$ is between $A$ and $E$). The line going through $D$ and parallel to $BC$ intersects $\omega$ at $F\ne D$, and the line $AF$ intersects $\omega$ at $T\ne F$. Let $BC$ and $ET$ intersect at $M$, $N$ be symmetric to $A$ with respect to $M$, and $K$ be the midpoint of $BC$. Prove that the points $D$, $E$, $K$, $N$ are cyclic.