Problem

Source: RMO 2024 Q3

Tags: geometry, circumcircle, trigonometry



Let $ABC$ be an acute triangle with $AB = AC$. Let $D$ be the point on $BC$ such that $AD$ is perpendicular to $BC$. Let $O,H,G$ be the circumcenter, orthocenter and centroid of triangle $ABC$ respectively. Suppose that $2 \cdot OD = 23 \cdot HD$. Prove that $G$ lies on the incircle of triangle $ABC$.