Abstrakt
The locus of centers of circles tangent to two given circles in plane is known to be a pair of conic sections. The foci of these conic sections are the centers of the circles given.
As a generalization we get the Apollonius task with one missing element. In a spatial generalization of the problem mentioned we are to find a locus of centers of spheres tangent to three given elements (spheres, planes, or incident points).
This locus consists of the intersections of pairs of quadric surfaces of revolution, their foci being the centers of spheres given. These intersections are known to be composed of conics.
Some special configurations of elements given result in a task clear and easy even for high-school students. Sphere inversion helps to find the loci of points of tangency of the spheres of a parametric system to be found with elements given, whilst the locus of centers must be constructed using some other methods.