FKMO 2015 Problem 2

In a triangle $\triangle ABC$ with incenter $I$ , the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively. Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively. Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$ . Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$ .