Five circles theorem

The five circles theorem states that, in a suitable configuration of five circles, certain intersection points formed by their pairs lie on a single circle.

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Circle: A plane curve whose points are all the same distance from a fixed center. The five objects in the configuration are circles.
Radical center: The common intersection point of the three radical axes of three circles. It captures a compatibility condition among triples of circles in the configuration.
Miquel's five-circle theorem: A circle-geometry result concerning concyclicity in a configuration of five circles. This named formulation is a prominent version associated with the five circles theorem.
Euclidean geometry: The geometry of points, lines, distances, and angles in a flat plane or space. The theorem is a classical incidence result within plane Euclidean geometry.
Concyclic points: Points that lie on one circle. The theorem’s conclusion is a concyclicity relation among selected intersection points.
Miquel point: A point associated with a cyclic incidence configuration, where several circles pass through the same point. Miquel’s theorem is a standard engine for proving concyclicity in circle configurations.
Miquel's theorem for a complete quadrilateral: The result that the four circumcircles of a complete quadrilateral meet at a common Miquel point. It supplies a basic local configuration used in many five-circle arguments.
Cyclic quadrilateral: A quadrilateral whose four vertices lie on one circle. Cyclic quadrilaterals frequently appear as intermediate figures in the theorem’s proofs.
Intersection of circles: The points shared by two circles; distinct circles meet in at most two points. The configuration’s candidate points arise where pairs of its circles meet.
Miquel's theorem: A theorem stating that, under specified cyclic incidences, four circles have a common point. Its repeated-circle concurrency principle can yield the five-circle concyclicity relation.