ELMO SL 2017 G2

Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$ . Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$ , respectively, and let $I$ be the incenter of $\triangle ABC$ . Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$ , show that $\overline{OI}\perp \overline{BC}$ .

[i]Proposed by Vincent Huang