  There are several remarkable points, defined for polygons and multisets of
points in the plane, called centers (such as the centroid). To make possible
their study, there exists a formal definition for the concept of center in both
cases. In this paper, the relation between symmetries of polygons and multisets
of points in the plane and the coincidence and collinearity of their centers is
studied. First, a precise statement for the problem is given. Then, it is
proved that, given a polygon or a multiset of points in the plane, a given
point in the plane is a center for this object if and only if it belongs to the
set of points fixed by its group of symmetries.
