In triangle $ABC$ the circumcircle has radius $R$ and center $O$ and the incircle has radius $r$ and center $I\ne O$ . Let $G$ denote the centroid of triangle $ABC$. Prove that $IG \perp BC$ if and only if $AB = AC$ or $AB + AC = 3BC$.
Source: 2010 Saudi Arabia IMO TST V p1
Tags: perpendicular, Centroid, geometry
In triangle $ABC$ the circumcircle has radius $R$ and center $O$ and the incircle has radius $r$ and center $I\ne O$ . Let $G$ denote the centroid of triangle $ABC$. Prove that $IG \perp BC$ if and only if $AB = AC$ or $AB + AC = 3BC$.