IMO 2019 Problem 6

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$ . The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$ , and $AB$ at $D, E,$ and $F$ , respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$ . Line $AR$ meets $\omega$ again at $P$ . The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$ .

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$ .

Proposed by Anant Mudgal, India