In triangle $ABC$, $M$ is the intersection point of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.
Problem
Source: 2004 Oral Moscow Geometry Olympiad grade 10 p4
Tags: geometry, Centroid, Circumcenter, equal segments