OLYMPIC-REVENGE 2019 Problem 3

Let $\Gamma$ be a circle centered at $O$ with radius $R$ . Let $X$ and $Y$ be points on $\Gamma$ such that $XY<R$ . Let $I$ be a point such that $IX = IY$ and $XY = OI$ . Describe how to construct with ruler and compass a triangle which has circumcircle $\Gamma$ , incenter $I$ and Euler line $OX$ . Prove that this triangle is unique.