  Planar central configurations can be seen as critical points of the reduced
potential or solutions of a system of equations. By the homogeneity and
invariance of the potential with respect to SO(2), it is possible to see that
the SO(2)-orbits of central configurations are fixed points of a suitable map
f. The purpose of the paper is to define this map and to derive some properties
using topological fixed point theory. The generalized Moulton-Smale theorem for
collinear configurations is proved, together with some estimates on the number
of central configurations in the case of 3 bodies, using fixed point indexes.
Well-known results such as the compactness of the set of central configuration
can also be proved in an easy way in this topological framework. At the end of
the paper some tables of (numerical) planar central configurations of n equal
masses with Newtonian potential are given, for n=3,..., 10. They have been
computed as the fixed points of a suitable self-map of R^{2(n-2)}.
