1961 IMO 第 6 题

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 parallel to $\epsilon$. Three arbitrary points $A^{\prime}, B^{\prime}$, and $C^{\prime}$ 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 $\epsilon$.

给定一个平面$\epsilon$，在该平面的一侧有三个非共线点$A、B$和$C$，使得它们确定的平面不平行于$\epsilon$。选择$\epsilon$中的三个任意点$A^{\prime}、B^{\prime}$和$C^{\prime}$。设$L、M$和$N$为$A A^{\prime}、B B^{\prime}$和$C C^{\prime}$的中点，$G$为$\三角形L M N$的质心。求 $G$ 获得的所有点的轨迹，因为 $A^{\prime}、B^{\prime}$ 和 $C^{\prime}$ 在 $\epsilon$ 上变化(彼此独立)。