SHARYGIN 2013 Problem 21

Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$ . The line through $D$ parallel to $BC$ meets $\omega$ again at $F$ , and $FA$ meets $\omega$ again at $T$ . Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$ . Show that $(DEN)$ passes through the midpoint of $BC$ .