Clifford's circle theorems

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

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.
Bibliography

Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 32–33. ISBN 0-14-011813-6.

External links

Weisstein, Eric W., "Clifford's Circle Theorem" from MathWorld.

Mathematics Encyclopedia