Seven circles theorem

A Euclidean geometry theorem stating that, in a particular configuration of seven circles, three associated points lie on one straight line.

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Miquel's theorem: A theorem in plane geometry linking the intersections of circles through the vertices of a complete quadrilateral. Its circle-intersection principle underlies the configuration’s chain of related points.
Euclidean geometry: The geometry of points, lines, distances, and angles in flat space. The theorem assumes the ordinary plane and its Euclidean circle properties.
Clifford's chain theorem: A theorem describing incidence patterns generated by iterated circle inversions and intersections. It belongs to the broader family of circle-incidence results related to seven-circle configurations.
Coordinate geometry: The use of numerical coordinates and algebraic equations to solve geometric problems. Coordinates provide an alternative to a purely synthetic proof of the collinearity.
Radical axis: The line containing points with equal powers with respect to two circles. Radical axes turn pairs of circles in the configuration into lines with shared intersection points.
Collinearity: The property of three or more points lying on a single straight line. Collinearity is the theorem’s stated conclusion.
Miquel: Auguste Miquel was a nineteenth-century French mathematician associated with classical circle and incidence theorems. Miquel’s work established a key circle-intersection result used in this theorem’s geometric lineage.
Complex plane geometry: The study of plane geometry using complex numbers to represent points and transformations. Complex coordinates can encode circle equations and test the resulting collinearity.
Power of a point: A signed quantity measuring a point’s relation to a circle, equal to the product of secant segment lengths. Equal powers provide an algebraic route from the circle incidences to collinearity.
Circle-circle intersection: The set of points shared by two circles, containing at most two points unless the circles coincide. The configuration is built by repeatedly taking intersections of circles.