# Generalization of the Apollonius theorem for simplices and related problems

arXiv: https://arxiv.org/abs/2401.03232 · HTML: https://mlsanity.com/paper/2401.03232/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jan 2024.

## Built on by (most cited first, 2 of 2)

- [Towards the mathematical foundation of the minimum enclosing ball and related problems](https://mlsanity.com/paper/2402.06629/) (2024)
- [Towards the methodology for solving the minimum enclosing ball and related problems](https://mlsanity.com/paper/2410.11063/) (2024)

## Abstract (arXiv)

The Apollonius theorem gives the length of a median of a triangle in terms of the lengths of its sides. The straightforward generalization of this theorem obtained for m-simplices in the n-dimensional Euclidean space for n greater than or equal to m is given. Based on this, generalizations of properties related to the medians of a triangle are presented. In addition, applications of the generalized Apollonius' theorem and the related to the medians results, are given for obtaining: (a) the minimal spherical surface that encloses a given simplex or a given bounded set, (b) the thickness of a simplex that it provides a measure for the quality or how well shaped a simplex is, and (c) the convergence and error estimates of the root-finding bisection method applied on simplices.