structural:reformatting
Locus of constant sum

Consider a finite set of n {\displaystyle n} points in the plane. The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose centre is at the centroid of the given points. A generalization for higher powers of distances is obtained if under n {\displaystyle n} points the vertices of the regular polygon P n {\displaystyle P_{n}} are taken. The locus of points such that the sum of the 2 m {\displaystyle 2m} -th power of distances d i {\displaystyle d_{i}} to the vertices of a given regular polygon with circumradius R {\displaystyle R} is constant is a circle, if ∑ i = 1 n d i 2 m > n R 2 m, where m = 1, 2, …, n − 1; {\displaystyle \sum _{i=1}^{n}d_{i}^{2m}>nR^{2m},\quad {\text{ where }}~m=1,2,\dots,n-1;} whose centre is the centroid of the P n {\displaystyle P_{n}}.

In the case of the equilateral triangle, the loci of the constant sums of the second and fourth powers are circles, whereas for the square, the loci are circles for the constant sums of the second, fourth, and sixth powers. For the regular pentagon the constant sum of the eighth powers of the distances will be added and so forth.