Problem

Source: FKMO 2021 Problem 1

Tags: FKMO, geometry, trivial



An acute triangle $\triangle ABC$ and its incenter $I$, circumcenter $O$ is given. The line that is perpendicular to $AI$ and passes $I$ intersects with $AB$, $AC$ in $D$,$E$. The line that is parallel to $BI$ and passes $D$ and the line that is parallel to $CI$ and passes $E$ intersects in $F$. Denote the circumcircle of $DEF$ as $\omega$, and its center as $K$. $\omega$ and $FI$ intersect in $P$($\neq F$). Prove that $O,K,P$ is collinear.