21. A segment of constant length "slides" with its ends along two mutually perpendicular intersecting lines. What line does the midpoint of the segment describe?
21. Let us draw two parallel planes $\alpha$ and $\beta$ through two intersecting lines $a$ and $b$. The plane $\gamma$, parallel to $\alpha$ and $\beta$ and passing between them at equal distances from them, bisects any segment passing between them (see problem $12, \S 4$). Let $A$ and $B$ be the endpoints of a given segment, with $A \in a, B \in b$. The projections $a^{\prime}$ and $b^{\prime}$ of the lines $a$ and $b$ onto the plane $\gamma$ are mutually perpendicular, since $a^{\prime} \| a$ and $b^{\prime} \| b$. If $A^{\prime}$ and $B^{\prime}$ are the projections of $A$ and $B$ onto $\gamma$, then the midpoint $M$ of the segment $A B$ is also the midpoint of the segment $A^{\prime} B^{\prime}$. The length $\left|A^{\prime} B^{\prime}\right|$ is constant, since $\left|A^{\prime} B^{\prime}\right|^{2} = |A B|^{2} - p^{2}$, where $p$ is the distance between the planes $\alpha$ and $\beta$. Thus, we have arrived at a planimetric problem: a segment $A^{\prime} B^{\prime}$ of constant length slides with its ends along two mutually perpendicular lines. What line does the midpoint of the segment describe? Let $O = a^{\prime} \cap b^{\prime}$. The length $|O M|$ is constant, since $[O M]$ is the median of the right triangle $A^{\prime} B^{\prime} O$, the length of which is half the length of the hypotenuse $\left[A^{\prime} B^{\prime}\right]$. Therefore, the point $M$ describes a circle with center $O$ and radius $\left.|O M| = \frac{1}{2} \left|A^{\prime} B^{\prime}\right| \right\,$.