KMO 2022 Problem 5

For a scalene triangle $ABC$ with an incenter $I$ , let its incircle meets the sides $BC, CA, AB$ at $D, E, F$ , respectively. Denote by $P$ the intersection of the lines $AI$ and $DF$ , and $Q$ the intersection of the lines $BI$ and $EF$ . Prove that $\overline{PQ}=\overline{CD}$ .