Clifford's circle theorems
Encyclopedia
In geometry
, Clifford's theorems, named after the English geometer William Kingdon Clifford
, are a sequence of theorems relating to intersections of circle
s.
The first theorem considers any four circles passing through a common point M. Four new circles can be constructed to pass through their other intersection points taken in triplets, so that each intersection's triple touches three circles only. Then these four circles also pass through a single point P.
The second theorem considers five circles 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 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.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, Clifford's theorems, named after the English geometer William Kingdon Clifford
William Kingdon Clifford
William Kingdon Clifford FRS 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 honour, with interesting applications in contemporary mathematical physics...
, are a sequence of theorems relating to intersections of circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s.
The first theorem considers any four circles passing through a common point M. Four new circles can be constructed to pass through their other intersection points taken in triplets, so that each intersection's triple touches three circles only. Then these four circles also pass through a single point P.
The second theorem considers five circles 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 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.