Details

Let the point be the intersection of and the straight line through . Let be the tangent point of the sphere with . Since , lies on the circle with center and radius . The center of the sphere is the intersection of the line through that is orthogonal to and the plane orthogonal to through the midpoints of and .

So the locus of centers of spheres lies on a cylinder orthogonal to with radius and center .

Reference

[1] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.