2D. Explanation of Invariance of the sum of squares of distances by using the Steiner theorem.
n/2 points of A1,...,An may be moving in a circle of radius r. Each point has antipodal point on the circle. Thus, the center of gravity Cm is the centre of the circle.
It is shown, that the sum of squares of distances from any point D of a circle to → {Ai } , divided by radius squared, is always equal of twice the number of points.
https://en.wikipedia.org/wiki/Parallel_axis_theorem
https://en.wikipedia.org/wiki/Antipodal_point