concyclicity

Concyclicity is a geometric relation in which a set of points lies on a single circle. In the Euclidean plane, a circle is the locus of points at a fixed distance (the radius) from a given point (the center). A finite set of points is concyclic if there exists a circle that passes through all of them. For three noncollinear points there is a unique circumcircle; for two points there are infinitely many circles through them; for four or more points concyclicity is a special condition to be verified.

A central criterion concerns quadrilaterals. A quadrilateral ABCD is concyclic if and only if the sum of opposite angles equals 180 degrees (angle ABC + angle ADC = 180°, equivalently angle BAD + angle BCD = 180°). Equivalently, angles subtended by the same chord are equal (angle ABC = angle ADC). Another way to test concyclicity is to ask whether all points lie at the same distance from some common center, i.e., there exists a point O such that OA = OB = OC = OD = radius. Algebraically, four points (x1, y1), (x2, y2), (x3, y3), (x4, y4) are concyclic if a certain determinant constructed from their coordinates vanishes; this provides a practical computational test.

Examples include the vertices of any regular polygon, which are all concyclic, and the vertices of a triangle together with its circumcenter. Concyclicity is a fundamental concept in circle geometry, with applications in angle chasing, chord properties, and the study of cyclic polygons. See also circumcircle, cyclic polygon, and power of a point.