The relation between symmetries and coincidence and collinearity of polygon centers and centers of multisets of points in the plane

TL;DR

This paper explores how symmetries of polygons and point multisets in the plane determine the coincidence and collinearity of their centers, establishing a formal link between symmetry groups and centers.

Contribution

It provides a formal framework connecting symmetries of polygons and multisets to the properties of their centers, proving that centers are exactly the fixed points of symmetry groups.

Findings

Centers coincide with fixed points of symmetry groups.

Symmetry properties determine collinearity of centers.

Formal characterization of centers via symmetry groups.

Abstract

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.