In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, Clifford's theorems, named after the English geometer
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
, are a sequence of theorems relating to intersections of
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s.
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 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 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
*
Cox's chain
*
Five circles theorem
In geometry, the five circles theorem states that, given five circles centered on a common sixth circle and intersecting each other chainwise on the same circle, the lines joining their second intersection points forms a pentagram whose points li ...
*
Miquel's six circles theorem
Miquel's theorem is a result in geometry, named after Auguste Miquel, concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circles ...
References
* W. K. Clifford (1882)
Mathematical Papers pages 51,2 via
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
*
H. S. M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
(1965). ''Introduction to Geometry'', page 262,
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in p ...
*
Further reading
* H. Martini & M. Spirova (2008) "Clifford’s chain of theorems in strictly convex Minkowski planes",
Publicationes Mathematicae Debrecen
''Publicationes Mathematicae Debrecen'' is a Hungarian mathematical journal, edited and published in Debrecen, at the Mathematical Institute of the University of Debrecen. It was founded by Alfréd Rényi, Tibor Szele
Tibor Szele (Debrecen, 21 J ...
72: 371–83
External links
* {{MathWorld, title=Clifford's Circle Theorem, urlname=CliffordsCircleTheorem
Theorems about circles