Conditions on centroids implying Euclideanity

We look at the conditions on a center of mass operator, acting on finite sets of mass points (P, m) (with P being the point and m the mass concentrated at P), that entail that the metric of the plane is Euclidean, in the sense that there exists a rectangle, or that the sum of the angles of any triangle is \(180^{\circ }\) ${180}^{\circ }$ . It turns out that the independence of the mass concentrated in the center of gravity of a system of two mass points (A, m) and (B, m), of the distance between A and B, even if applied to systems of at most three mass points, implies the Euclideanity of the metric.