ELMO SL 2013 G7

Let $ABC$ be a triangle inscribed in circle $\omega$ , and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$ , and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$ . Prove that $O_1$ , $O_2$ , and the nine-point center of $ABC$ are collinear.

Proposed by Michael Kural