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 element [1, n] which can be picked with no four concyclic is o(n^(2/3) - ϵ). 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.

antiparallel | circle | circumcircle | collinear | concentric | cyclic hexagon | cyclic pentagon | cyclic quadrilateral | eccentric | N-cluster | Ptolemy's theorem