Clifford's circle theorems - Revision history

Qwerfjkl: Removed 'a(n)' from the beginning of the short description per WP:SDFORMAT, from a request at Wikipedia:Reward board#Remove "A" as the first word from short descriptions. (via WP:JWB)

2021-07-11T19:30:04Z

Removed 'a(n)' from the beginning of the short description per WP:SDFORMAT, from a request at Wikipedia:Reward board#Remove "A" as the first word from short descriptions. (via WP:JWB)

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

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External links: cat.

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

	[[Category:~~Circles~~]]		[[Category:Theorems about circles]]
	[[Category:Theorems in plane geometry]]

1234qwer1234qwer4: context is clear, grammar, design

2020-06-09T16:04:45Z

context is clear, grammar, design

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

	== Statement ~~of the theorem~~ ==		== Statement ==

	The first theorem considers any four circles passing through a common point ''M'' and otherwise in [[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 ''P'' (in general not the same point as ''M'').		The first theorem considers any four circles passing through a common point ''M'' and otherwise in [[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 ''P'' (in general not the same point as ''M'').
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	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 ''C''.		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 ''C''.

	The third theorem ~~consider~~ 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 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.		The sequence of theorems can be continued indefinitely.
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	==References==		==References==
	* W. K. Clifford (1882) [https://archive.org/details/mathematicalpap00smitgoog/page/n133 Mathematical Papers], pages 51,2 via [[Internet Archive]]		* 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, [[John Wiley & Sons]]		* [[H. S. M. Coxeter]] (1965). ''Introduction to Geometry'', page 262, [[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}}		* {{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}}

ArnoldReinhold: Adding short description: "A sequence of theorems relating to sets of circles intersecting at a common point" (Shortdesc helper)

2020-05-15T20:44:02Z

Adding short description: "A sequence of theorems relating to sets of circles intersecting at a common point" (Shortdesc helper)

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

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2020-02-07T12:33:22Z

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	* W. K. Clifford (1882) [https://archive.org/details/mathematicalpap00smitgoog/page/n133 Mathematical Papers], pages 51,2 via [[Internet Archive]]		* 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, [[John Wiley & Sons]]		* [[H. S. M. Coxeter]] (1965) ''Introduction to Geometry'', page 262, [[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}}		* {{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==		==Further reading==

InternetArchiveBot: Bluelink 1 book for verifiability.) #IABot (v2.0) (GreenC bot

2019-11-17T13:12:15Z

Bluelink 1 book for verifiability.) #IABot (v2.0) (GreenC bot

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

David Eppstein: Category:Theorems in plane geometry

2019-11-16T08:17:48Z

Category:Theorems in plane geometry

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

Tomo: /* Further reading */

2019-05-01T12:15:43Z

Rgdboer: /* See also */ lk Cox's chain

2018-11-26T23:47:47Z

Michael Hardy at 19:02, 25 November 2018

2018-11-25T19:02:44Z

← Previous revision		Revision as of 19:02, 25 November 2018
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	In [[geometry]], '''Clifford's theorems''', named after the English geometer [[William Kingdon Clifford]], are a sequence of theorems relating to intersections of [[circle]]s.		In [[geometry]], '''Clifford's theorems''', named after the English geometer [[William Kingdon Clifford]], are a sequence of theorems relating to intersections of [[circle]]s.

			== Statement of the theorem ==
⚫	The first theorem considers any four circles passing through a common point ''M'' and otherwise in [[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. ~~Like~~ the first set of four circles, the second set of four circles defined in this way all pass through a single point ''P''.

		⚫	The first theorem considers any four circles passing through a common point ''M'' and otherwise in [[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 ''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 ''C''.		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 ''C''.

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	==Further reading==		==Further reading==
	* H. Martini & M. Spirova (~~20008~~) "Clifford’s chain of theorems in strictly convex Minkowski planes", [[Publicationes Mathematicae Debrecen]] 72: 371–83 {{mr\|id=2406927}}		* 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==		==External links==

← Previous revision		Revision as of 23:47, 26 November 2018
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	==See also==		==See also==
			* [[Homersham Cox (mathematician)#Cox's chain\|Cox's chain]]
	* [[Five circles theorem]]		* [[Five circles theorem]]
	* [[Miquel's six circles theorem]]		* [[Miquel's six circles theorem]]