  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.
