Problem

Source: 2022 Nigerian Senior MO Round 2/Problem 2

Tags: geometry, Concyclic, cyclic quadrilateral



Let $G$ be the centroid of $\triangle ABC $ and let $D, E $ and $F$ be the midpoints of the line segments $BC, CA $ and $AB$ respectively. Suppose the circumcircle of $\triangle ABC $ meets $AD $ again at $X$, the circumcircle of $\triangle DEF $ meets $BE$ again at $Y$ and the circumcircle of $\triangle DEF $ meets $CF$ again at $Z$. Show that $G, X, Y $ and $Z$ are concyclic.