In space, two perpendicular intersecting lines are given. Find the set of midpoints of all segments of a given length, the ends of which lie on these lines.

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Let $a$ be the distance between the given lines, $d$ be the length of the considered segments. Choose a coordinate system so that the points of the first line have coordinates $(x, 0,0)$, and the points of the second line have coordinates $(0, y, a)$. We are interested in pairs of points for which $x^{2}+a^{2}+y^{2}=d^{2}$, i.e., $x^{2}+y^{2}=d^{2}-a^{2}$. The midpoint of the segment with endpoints at points with coordinates $(x, 0,0)$ and $(0, y, a)$ has coordinates $(x / 2, y / 2, a / 2)$, so the desired set is a circle of radius $\sqrt{d^{2}-a^{2}} / 2$ centered at $(0,0, a / 2)$, located in the plane $z=a / 2$.