International Mathematical Olympiad 1961 Problem 6

Consider a plane $\varepsilon$ and three non-collinear points $A, B, C$ on the same side of $\varepsilon$; suppose the plane determined by these three points is not parallel to $\varepsilon$. In plane $\varepsilon$ take three arbitrary points $A', B', C'$. Let $L, M, N$ be the midpoints of segments $AA', BB', CC'$; let $G$ be the centroid of triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L, M, N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\varepsilon$?