  Central configurations have been of great interest over many years, with the
earliest examples due to Euler and Lagrange. There are numerous results in the
literature demonstrating the existence of central configurations with specific
symmetry properties, using slightly different techniques in each. The aim here
is to describe a uniform approach by adapting to the symmetric case the
well-known variational argument showing the existence of central
configurations. The principal conclusion is that there is a central
configuration for every possible symmetry type, and for any symmetric choice of
masses. Finally the same argument is applied to the class of balanced
configurations introduced by Albouy and Chenciner.
