Question 164
Let $ABC$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $BC$ such that $BE = EF = FC$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, so that $X$, $Y$, and $G$ are not collinear. If the line through $E$ parallel to $XG$ and the line through $F$ parallel to $YG$ intersect at $P \neq G$, prove that $GP$ passes through the midpoint of $XY$.