CYCLIC AVERAGES OF REGULAR POLYGONAL DISTANCES.

We consider a regular plane polygon with n vertices and an arbitrary point in the plane. Let R be the circumscribed radius of the polygon and L a distance from the point to the centroid of the polygon. Then the averages of the (2m)-th powers of distances from the point to the polygon vertices satisfy the relations Sn(2) = R² + L²; Sn(2m) = (R²> + L²)m + bm 2 ∑ k=1 (m 2k) (2k k) (R² + L²)m-2k (RL)2k, where m = 2,..., n - 1. [ABSTRACT FROM AUTHOR]