1965 IMO 第 3 题

A tetrahedron $A B C D$ is given. The lengths of the edges $A B$ and $C D$ are $a$ and $b$, respectively, the distance between the lines $A B$ and $C D$ is $d$, and the angle between them is equal to $\omega$. The tetrahedron is divided into two parts by the plane $\pi$ parallel to the lines $A B$ and $C D$. Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane $\pi$ from $A B$ and $C D$ is equal to $k$.

给出一个四面体$A B C D$。边$A B$ 和$C D$ 的长度分别为$a$ 和$b$，线$A B$ 和$C D$ 之间的距离为$d$，它们之间的角度等于$\omega$。四面体被平行于线 $A B$ 和 $C D$ 的平面 $\pi$ 分为两部分。如果平面 $\pi$ 距 $A B$ 和 $C D$ 的距离之比等于 $k$，则计算零件的体积比。