Editing Concyclic points (section)

==Variations==

In contexts where lines are taken to be a type of [[generalised circle]] with infinite radius, [[collinear points]] (points along a single line) are considered to be concyclic. This point of view is helpful, for instance, when studying [[circle inversion|inversion through a circle]] or more generally [[Möbius transformation]]s (geometric transformations generated by reflections and circle inversions), as these transformations preserve the concyclicity of points only in this extended sense.<ref>{{citation
 | last = Zwikker | first = C.
 | authorlink = Cornelis Zwikker
 | isbn = 9780486442761
 | page = 24
 | publisher = Courier Dover Publications
 | title = The Advanced Geometry of Plane Curves and Their Applications
 | url = https://books.google.com/books?id=25tMEYTik-AC&pg=PA24
 | year = 2005}}.</ref>

In the [[complex plane]] (formed by viewing the [[real and imaginary parts]] of a [[complex number]] as the ''x'' and ''y'' [[Cartesian coordinates]] of the plane), concyclicity has a particularly simple formulation: four points in the complex plane are either concyclic or collinear if and only if their [[cross-ratio]] is a [[real number]].<ref>{{citation
 | last = Hahn | first = Liang-shin
 | edition = 2nd
 | isbn = 9780883855102
 | page = 65
 | publisher = Cambridge University Press
 | series = MAA Spectrum
 | title = Complex Numbers and Geometry
 | url = https://books.google.com/books?id=s3nMMkPEvqoC&pg=PA65
 | year = 1996}}.</ref>