View source for Clifford's circle theorems

{{short description|A sequence of theorems relating to sets of circles intersecting at a common point}}
[[Image:Clifford circle theorems.svg|thumb|right|400px]]

In [[Geometry|geometry]], '''Clifford's theorems''', named after the English geometer [[Biography:William Kingdon Clifford|William Kingdon Clifford]], are a sequence of theorems relating to intersections of [[Circle|circle]]s.

== Statement ==

The first theorem considers any four circles passing through a common point ''M'' and otherwise in [[General position|general position]], meaning that there are six additional points where exactly two of the circles cross and that no three of these crossing points are collinear.  Every set of three of these four circles has among them three crossing points, and (by the assumption of non-collinearity) there exists a circle passing through these three crossing points. The conclusion is that, like the first set of four circles, the second set of four circles defined in this way all pass through a single point&nbsp;''P'' (in general not the same point as ''M'').

The second theorem considers five circles in general position passing through a single point ''M''.  Each subset of four circles defines a new point ''P'' according to the first theorem.  Then these five points all lie on a single circle&nbsp;''C''.

The third theorem considers six circles in general position that pass through a single point ''M''.  Each subset of five circles defines a new circle by the second theorem.  Then these six new circles ''C'' all pass through a single point.

The sequence of theorems can be continued indefinitely.

==See also==
* [[Biography:Homersham Cox (mathematician)#Cox's chain|Cox's chain]]
* [[Five circles theorem]]
* Miquel's six circles theorem

==References==
* W. K. Clifford (1882). [https://archive.org/details/mathematicalpap00smitgoog/page/n133 Mathematical Papers], pages 51,2 via [[Internet Archive]]
* H. S. M. Coxeter (1965). ''Introduction to Geometry'', page 262, {{wipe|John Wiley & Sons}}
* {{cite book | author = Wells, D. | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | url = https://archive.org/details/penguindictionar0000well | url-access = registration | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = [https://archive.org/details/penguindictionar0000well/page/32 32], 33}}

==Further reading==
* H. Martini & M. Spirova (2008) "Clifford’s chain of theorems in strictly convex Minkowski planes", [[Publicationes Mathematicae Debrecen]] 72: 371–83 {{mr|id=2406927}}

==External links==
* {{MathWorld|title=Clifford's Circle Theorem|urlname=CliffordsCircleTheorem}}

[[Category:Circles]]
[[Category:Theorems in plane geometry]]

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