ELMO SL 2019 G3

Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$ . The incircle touches sides $BC,CA,$ and $AB$ at $D,E,$ and $F$ respectively, and $A'$ is the reflection of $A$ over $O$ . The circumcircles of $ABC$ and $A'EF$ meet at $G$ , and the circumcircles of $AMG$ and $A'EF$ meet at a point $H\neq G$ , where $M$ is the midpoint of $EF$ . Prove that if $GH$ and $EF$ meet at $T$ , then $DT\perp EF$ .

Proposed by Ankit Bisain