Editing Circle (section)

==Locus of constant sum==
Consider a finite set of <math>n</math> 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.<ref>{{cite journal | last1 = Apostol | first1 = Tom | last2 = Mnatsakanian | first2 = Mamikon | date = 2003 | title = Sums of squares of distances in m-space | journal= American Mathematical Monthly | volume=110 | issue=6 | pages = 516–526 | doi = 10.1080/00029890.2003.11919989 | s2cid = 12641658}}</ref>
A generalisation for higher powers of distances is obtained if, instead of <math>n</math> points, the vertices of the regular polygon <math>P_n</math> are taken.<ref name="Mamuka">{{cite journal |last1=Meskhishvili |first1=Mamuka |date=2020 |title=Cyclic Averages of Regular Polygons and Platonic Solids |url=https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065 |journal=Communications in Mathematics and Applications |volume=11 |pages=335–355 |doi=10.26713/cma.v11i3.1420 |doi-broken-date=1 November 2024 |arxiv=2010.12340 |access-date=17 May 2021 |archive-date=22 April 2021 |archive-url=https://web.archive.org/web/20210422211229/https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065 |url-status=live }}</ref> The locus of points such that the sum of the <math>2m</math>-th power of distances <math>d_i</math> to the vertices of a given regular polygon with circumradius <math>R</math> is constant is a circle, if
<math display="block">\sum_{i=1}^n d_i^{2m} > nR^{2m} , \quad \text{  where } ~ m = 1, 2, \dots, n-1;</math>
whose centre is the centroid of the <math>P_n</math>.

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.