AMO-Bench 2025 Problem 37

In $\triangle ABC$ with $AB < AC < BC$, let $O$ and $I$ be its circumcenter and incenter, respectively. Let $M_b$ and $M_c$ be the midpoints of the arc $AC$ (not containing $B$) and the arc $AB$ (not containing $C$), respectively, on the circumcircle $\odot O$. Let $S_b$ be a point on $\odot O$ such that $IS_b \perp BS_b$, and let $S_c$ be a point on $\odot O$ such that $IS_c \perp CS_c$. Let $H_a$ be the intersection of lines $M_bS_c$ and $M_cS_b$. Points $H_b$ and $H_c$ are defined analogously.It is given that the lines $AH_a, BH_b, CH_c$ are pairwise non-parallel. Let $G$ be the centroid of the triangle formed by these lines (if the three lines are concurrent, $G$ is defined as their point of concurrency). Let the side lengths of $\triangle ABC$ be $BC = a$, $CA = b$, and $AB = c$. Find the product of the distances from point $G$ to the lines $IH_a, IH_b, IH_c$. (Expressed in terms of $a, b, c$ and simplified completely.)