Problem

Source: 2023 Turkey TST D2 P5

Tags: geometry, circumcircle, incenter



Let $ABC$ be a scalene triangle with circumcentre $O$, incentre $I$ and orthocentre $H$. Let the second intersection point of circle which passes through $O$ and tangent to $IH$ at point $I$, and the circle which passes through $H$ and tangent to $IO$ at point $I$ be $M$. Prove that $M$ lies on circumcircle of $ABC$.