1962 IMO 第 3 题

A cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is given. The point $X$ is moving at a constant speed along the square $A B C D$ in the direction from $A$ to $B$. The point $Y$ is moving with the same constant speed along the square $B C C^{\prime} B^{\prime}$ in the direction from $B^{\prime}$ to $C^{\prime}$. Initially, $X$ and $Y$ start out from $A$ and $B^{\prime}$ respectively. Find the locus of all the midpoints of $X Y$.

给出一个立方体 $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$。点$X$沿着正方形$A B C D$以恒定速度从$A$到$B$的方向移动。点$Y$沿着正方形$B C C^{\prime} B^{\prime}$以相同的恒定速度从$B^{\prime}$向$C^{\prime}$移动。最初，$X$和$Y$分别从$A$和$B^{\prime}$开始。求$X Y$所有中点的轨迹。