Editing Concyclic points (section)

== Triangles ==
{{main|Circumcircle}}

The vertices of every [[triangle]] fall on a circle called the [[circumcircle]]. (Because of this, some authors define "concyclic" only in the context of four or more points on a circle.)<ref>{{citation
 | last = Elliott | first = John
 | page = 126
 | publisher = Swan Sonnenschein & co.
 | title = Elementary Geometry
 | url = https://books.google.com/books?id=9psBAAAAYAAJ&pg=PA126
 | year = 1902}}.</ref> Several other sets of points defined from a triangle are also concyclic, with different circles; see [[Nine-point circle]]<ref>{{citation
 | last = Isaacs | first = I. Martin
 | author-link=Martin Isaacs
 | isbn = 9780821847947
 | page = 63
 | publisher = American Mathematical Society
 | series = Pure and Applied Undergraduate Texts
 | title = Geometry for College Students
 | url = https://books.google.com/books?id=0ahK8UneO3kC&pg=PA63
 | volume = 8
 | year = 2009}}.</ref> and [[Lester's theorem]].<ref>{{citation
 | last = Yiu | first = Paul
 | journal = Forum Geometricorum
 | mr = 2868943
 | pages = 175–209
 | title = The circles of Lester, Evans, Parry, and their generalizations
 | url = http://forumgeom.fau.edu/FG2010volume10/FG201020.pdf
 | volume = 10
 | year = 2010}}.</ref>

The [[radius]] of the circle on which lie a set of points is, by definition, the radius of the circumcircle of any triangle with vertices at any three of those points. If the pairwise distances among three of the points are ''a'', ''b'', and ''c'', then the circle's radius is

:<math>R = \sqrt{\frac{a^2b^2c^2}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}.</math>

The equation of the circumcircle of a triangle, and expressions for the radius and the coordinates of the circle's center, in terms of the Cartesian coordinates of the vertices are given [[Circumcircle#Circumcircle equations|here]].

=== Other concyclic points ===

In any triangle all of the following nine points are concyclic on what is called the [[nine-point circle]]: the midpoints of the three edges, the feet of the three [[altitude (geometry)|altitudes]], and the points halfway between the [[orthocenter]] and each of the three vertices.

[[Lester's theorem]] states that in any [[scalene triangle]], the two [[Fermat point]]s, the [[nine-point center]], and the [[circumcenter]] are concyclic.

If [[Line (mathematics)|lines]] are drawn through the [[Lemoine point]] [[parallel (geometry)|parallel]] to the sides of a triangle, then the six points of intersection of the lines and the sides of the triangle are concyclic, in what is called the [[Lemoine circle]].

The [[van Lamoen circle]] associated with any given triangle <math>T</math> contains the [[circumcenter]]s of the six triangles that are defined inside <math>T</math> by its three [[median (geometry)|median]]s.

A triangle's [[circumcenter]], its [[Lemoine point]], and its first two [[Brocard points]] are concyclic, with the segment from the circumcenter to the Lemoine point being a [[diameter]].<ref>Scott, J. A. "Some examples of the use of areal coordinates in triangle geometry", ''[[Mathematical Gazette]]'' 83, November 1999, 472–477.</ref>