6. (ROM) A plane $\epsilon$ is given and on one side of the plane three noncollinear points $A, B$, and $C$ such that the plane determined by them is not paralle] to $\epsilon$. Three arbitrary points $A^{\prime}, B^{\prime}$, and $C^{r}$ in $\epsilon$ are selected. Let $L, M$, and $N$ be the midpoints of $A A^{\prime}, B B^{\prime}$, and $C C^{\prime}$, and $G$ the centroid of $\triangle L M N$. Find the locus of all points obtained for $G$ as $A^{\prime}, B^{\prime}$, and $C^{\prime}$ are varied (independently of each other) across $\epsilon$.
6. Let $h(X)$ denote the distance of a point $X$ from $\epsilon, X$ restricted to being on the same side of $\epsilon$ as $A, B$, and $C$. Let $G_{1}$ be the (fixed) centroid of $\triangle A B C$ and $G_{1}^{\prime}$ the centroid of $\triangle A^{\prime} B^{\prime} C^{\prime}$. It is trivial to prove that $G$ is the midpoint of $G_{1} G_{1}^{\prime}$. Hence varying $G_{1}^{\prime}$ across $\epsilon$, we get that the locus of $G$ is the plane $\alpha$ parallel to $\epsilon$ such that
$$
X \in \alpha \Leftrightarrow h(X)=\frac{h\left(G_{1}\right)}{2}=\frac{h(A)+h(B)+h(C)}{6} .
$$