Problem

Source: 2018 Taiwan TST Round 3

Tags: geometry, incenter, circumcircle, Taiwan



Let $I,G,O$ be the incenter, centroid and the circumcenter of triangle $ABC$, respectively. Let $X,Y,Z$ be on the rays $BC, CA, AB$ respectively so that $BX=CY=AZ$. Let $F$ be the centroid of $XYZ$. Show that $FG$ is perpendicular to $IO$.