KZ-MO 2021 Problem 4

Given acute triangle $ABC$ with circumcircle $\Gamma$ and altitudes $AD, BE, CF$ , line $AD$ cuts $\Gamma$ again at $P$ and $PF, PE$ meet $\Gamma$ again at $R, Q$ . Let $O_1, O_2$ be the circumcenters of $\triangle BFR$ and $\triangle CEQ$ respectively. Prove that $O_{1}O_{2}$ bisects $\overline{EF}$ .