Editing Square (section)

===Inscribed and circumscribed circles===
[[File:Incircle and circumcircle of a square.png|thumb|The [[inscribed circle]] (purple) and [[circumscribed circle]] (red) of a square (black)]]
The [[inscribed circle]] of a square is the largest circle that can fit inside that square. Its center is the center point of the square, and its radius (the [[inradius]] of the square) is <math>r=\ell/2</math>. Because this circle touches all four sides of the square (at their midpoints), the square is a [[tangential quadrilateral]]. The [[circumscribed circle]] of a square passes through all four vertices, making the square a [[cyclic quadrilateral]]. Its radius, the [[circumradius]], is <math>R=\ell/\sqrt2</math>.{{sfnp|Rich|1963|p=[https://archive.org/details/in.ernet.dli.2015.131938/page/n149 133]}} If the inscribed circle of a square <math>ABCD</math> has tangency points <math>E</math> on <math>AB</math>, <math>F</math> on <math>BC</math>, <math>G</math> on <math>CD</math>, and <math>H</math> on <math>DA</math>, then for any point <math>P</math> on the inscribed circle,<ref>{{Cite web|url=http://gogeometry.com/problem/p331_square_inscribed_circle.htm|title=Problem 331. Discovering the Relationship between Distances from a Point on the Inscribed Circle to Tangency Point and Vertices in a Square|website=Go Geometry from the Land of the Incas|first=Antonio|last=Gutierrez|access-date=2025-02-05}}</ref><math display=block> 2(PH^2-PE^2) = PD^2-PB^2.</math> If <math>d_i</math> is the distance from an arbitrary point in the plane to the {{nowrap|<math>i</math>th}} vertex of a square and <math>R</math> is the [[circumradius]] of the square, then<ref>{{cite journal
 | last = Park | first = Poo-Sung
 | journal = [[Forum Geometricorum]]
 | mr = 3507218
 | pages = 227–232
 | title = Regular polytopic distances
 | url = http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
 | archive-url = https://web.archive.org/web/20161010184811/http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
 | archive-date = 2016-10-10
 | url-status = dead
 | volume = 16
 | year = 2016}}</ref><math display=block>\frac{d_1^4+d_2^4+d_3^4+d_4^4}{4} + 3R^4 = \left(\frac{d_1^2+d_2^2+d_3^2+d_4^2}{4} + R^2\right)^2.</math>
If <math>L</math>  and <math>d_i</math> are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices  respectively, then <math display=block>d_1^2 + d_3^2 = d_2^2 + d_4^2 = 2(R^2+L^2)</math> and <math display=block> d_1^2d_3^2 + d_2^2d_4^2 = 2(R^4+L^4), </math> where <math>R</math> is the circumradius of the square.<ref name=Mamuka >{{cite journal
 | last = Meskhishvili | first = Mamuka
 | issue = 1
 | journal = International Journal of Geometry
 | mr = 4193377
 | pages = 58–65
 | title = Cyclic averages of regular polygonal distances
 | url = https://ijgeometry.com/wp-content/uploads/2020/12/4.-58-65.pdf
 | volume = 10
 | year = 2021}}</ref>