Cocircularity

Cocircularity, also called concyclicity, is a geometric property of a set of points in the plane: the points are cocircular if there exists a single circle that contains all of them. The term is commonly used for finite sets, especially four or more points.

In the plane, three non-collinear points determine a unique circle. If a fourth point lies on that same circle, the four points are cocircular. More generally, a set of points is cocircular if one circle passes through every point in the set.

Algebraically, cocircularity can be tested in several equivalent ways. For four points A(x1, y1), B(x2, y2), C(x3, y3), D(x4, y4), they are cocircular if the determinant of the 4x4 matrix formed by rows [xi, yi, xi^2+yi^2, 1] is zero. Another formulation is that there exist constants D, E, F such that the equation x^2 + y^2 + Dx + Ey + F = 0 is satisfied by all four points, indicating a common circle. A geometric construction checks whether the center, found as the intersection of the perpendicular bisectors of two chords (for example AB and AC), is equidistant from all four points; if so, the points share a circle.

In higher dimensions, cocircularity remains a planar concept: the points must lie in the same plane and on a common circle. If points lie in 3D but not in a single plane, they cannot be cocircular. Applications include solving geometry problems, circle fitting in data analysis, and various computational geometry algorithms.