Concyclic points are points lying on a common circle. 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.
Guy (1994, p. 241; 2004, p. 368) considers the largest subset of the 
lattice
points 
,
, with no four concyclic
points. A single row provides 
such points, since a circle
meets a line in at most two points.
For four distinct points on a noncircular ellipse, a Gröbner basis calculation can eliminate
the center and radius of a circle
to give a condition for concyclicity (Trott 2006). Let 
with 
, and use distinct real parameters 
in
 |
where 
, 2, 3, 4. These points
are concyclic iff
 |
Substituting the parameterization into a circle equation gives a quartic equation whose cubic and linear coefficients are equal. The displayed condition follows from Vieta's formulas.
Equivalently, writing 
,
the four distinct points are concyclic iff 
,
where the 
are eccentric anomalies. This angular form also
includes the point 
excluded by the rational parameterization.
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.
