On the Inscribed Sphere and Concurrent Lines through the Centers of Apollonius Spheres in $\mathbb{R}^n$

Abstract

The Apollonius problem asks for a sphere tangent to $n+1$ given spheres or hyperplanes in $\mathbb{R}^n$. This problem has been widely studied for an isolated configuration of $n+1$ spheres. In this paper, we study relations among the solutions of the Apollonius problem arising from a common family of spheres within the framework of Lie sphere geometry. More precisely, we consider a configuration of $n+2$ spheres in $\mathbb{R}^n$ and the solutions of the Apollonius problem corresponding to all its subsets of size $n+1$. The first main result concerns lines passing through the centers of pairs of solutions to the Apollonius problem. We prove that all these lines intersect at a single point $P_X$. We then introduce a two--step construction of further Apollonius spheres and show that the lines determined by their centers also pass through $P_X$. This yields numerous applications in two and three dimensions and, at the same time, automatically extends them to $\mathbb{R}^n$. The second main result is an $n$--dimensional generalization of K. Morita's three-dimensional theorem on the inscribed sphere in a configuration of mutually tangent spheres. We show that Morita's theorem is a special case of our result for an arbitrary configuration of $n+2$ spheres in $\mathbb{R}^n$, not necessarily mutually tangent. Moreover, we connect this result with the preceding ones by proving that the center of the corresponding inscribed sphere is again the point $P_X$.