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Concyclic

Concyclic

Four or more points P_1, P_2, P_3, P_4, ... which lie on a circle C are said to be concyclic. Three points are trivially concyclic since three noncollinear points determine a circle (i.e., every triangle has a circumcircle). Ptolemy's theorem can be used to determine if four points are concyclic.

The number of the n^2 lattice points x,y in [1,n] which can be picked with no four concyclic is o(n^(2/3)-epsilon) (Guy 1994).

A theorem states that if any four consecutive points of a polygon are not concyclic, then its area can be increased by making them concyclic. This fact arises in some proofs that the solution to the isoperimetric problem is the circle.

See also

Antiparallel, Circle, Circumcircle, Collinear, Concentric, Cyclic Hexagon, Cyclic Pentagon, Cyclic Quadrilateral, Eccentric, N-Cluster, Ptolemy's Theorem

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References

Coolidge, J. L. "Concurrent Circles and Concyclic Points." §1.6 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 85-95, 1971.Guy, R. K. "Lattice Points, No Four on a Circle." §F3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 241, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

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Concyclic

Cite this as:

Weisstein, Eric W. "Concyclic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Concyclic.html

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