Miquel's theorem

In a complete quadrilateral, the circumcircles of its four constituent triangles pass through one common point, called the Miquel point.

Complete quadrilateral: A configuration of four lines with no three concurrent, together with their six pairwise intersection points. Its four triples of lines define the triangles whose circumcircles concur.
Euclidean geometry: The geometry of points, lines, distances, and angles in a plane or higher-dimensional space. Miquel's theorem is a classical incidence result in the Euclidean plane.
Auguste Miquel: A French mathematician associated with the theorem on circles of a complete quadrilateral. His name is attached to the common point and the theorem describing its existence.
Miquel configuration: A geometric arrangement in which circles associated with several triangles meet at a common point. This broader configuration language applies the theorem's concurrence pattern in constructed diagrams.
Miquel's theorem for complete quadrilaterals: The classical circle-concurrency statement for the four triangles determined by four lines in general position. This alternate formulation names the precise configuration underlying the theorem's standard statement.
Circumcircle: The unique circle passing through the three vertices of a nondegenerate triangle. Each constituent triangle contributes one circumcircle to the concurrency.
Triangle: A polygon with three sides and three vertices. The four circles arise as circumcircles of triangles formed by triples of quadrilateral lines.
Complete quadrilateral theorem: A classical projective-geometric result about the three pairs of opposite sides of a complete quadrilateral. It is a separate named theorem about the same underlying quadrilateral configuration.
Inversion in a circle: A transformation sending each point to a point on the same ray whose distances from a fixed center have constant product. Inversion can turn intersecting circles through a Miquel point into lines or simpler circle arrangements.
Pappus's theorem: A projective theorem stating that intersections formed from two triples of collinear points yield three collinear points. Both results reveal unexpected incidence consequences from a small arrangement of lines.

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