MathNet 2018-part-06 — 181 problems (EN–VI) Question 37

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