Cocircular

Cocircular, also spelled concyclic, is an adjective used in geometry to describe a set of points that lie on a common circle. A circle can pass through all the points, and the points are said to be cocircular or concyclic.

In the plane, any two distinct points are cocircular, since infinitely many circles pass through them. Three non-collinear points determine a unique circle; a fourth or more point is cocircular with them only if it lies on that same circle.

Determining cocircularity can be done algebraically by fitting a circle to the points. A circle has equation x^2+y^2+ax+by+c=0; solving for a, b, c that satisfy all point coordinates confirms cocircularity when a solution exists. Geometrically, points are cocircular if their circumcenter is equidistant from all the points.

Examples include the vertices of a square or any regular polygon, which all lie on a circumscribed circle. In problems about cyclic quadrilaterals, the term cocircular is used to indicate that all four vertices lie on a circle.

Cocircular is a common synonym of concyclic, and the concept is used in Euclidean geometry, computer-aided design, and graphics to check or enforce circularity of a set of points.