UA-UMO 2021 Problem 104

Let $O, I, H$ be the circumcenter, the incenter, and the orthocenter of $\triangle ABC$ . The lines $AI$ and $AH$ intersect the circumcircle of $\triangle ABC$ for the second time at $D$ and $E$ , respectively. Prove that if $OI \parallel BC$ , then the circumcenter of $\triangle OIH$ lies on $DE$ .

(Fedir Yudin)