For n+2 spheres in R^n the centers of Apollonius solutions lie on lines concurrent at a single point P_X that is also the center of the generalized inscribed sphere.
Kimberling, Encyclopedia of Triangle Centers University of Evansville (2023)
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On the Inscribed Sphere and Concurrent Lines through the Centers of Apollonius Spheres in $\mathbb{R}^n$
For n+2 spheres in R^n the centers of Apollonius solutions lie on lines concurrent at a single point P_X that is also the center of the generalized inscribed sphere.