Miquel point

A point where the circumcircles of the four triangles formed by four lines in general position concur. The concurrence is an instance of Miquel's theorem.

Miquel's theorem: A theorem stating that the four circumcircles associated with four lines in general position meet at one point. It gives the concurrence that defines the Miquel point.
Triangle: A polygon with three sides and three vertices. Each defining circle is the circumcircle of a triangle.
Miquel's five-circle theorem: A configuration theorem in which five circles associated with a complete quadrilateral have a further concurrence relation. It extends the four-circle concurrence to a richer cyclic configuration.
Augustus De Morgan: An English mathematician and logician who worked on algebra, logic, and geometry. De Morgan published an early account of the concurrence now associated with Miquel.
Cyclic quadrilateral: A quadrilateral whose four vertices lie on one circle. Angle relations in cyclic quadrilaterals establish the circles' shared point.
Incidence geometry: The study of which points and lines lie on or pass through one another. The configuration begins with intersections among four lines.
Miquel configuration: A geometric arrangement of circles and lines whose incidences satisfy Miquel-type concurrence. The point is the central concurrence in this family of configurations.
Michel Chasles: A French mathematician known for foundational work in projective geometry. Chasles helped establish the geometric tradition in which Miquel's result was developed.
Inscribed angle theorem: A theorem stating that an inscribed angle measures half the central angle subtending the same arc. It converts equal angles at line intersections into cyclicity.
Concyclic points: Points that all lie on a single circle. The theorem's conclusion is that each circle contains the same point.

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