Miquel's theorem

Miquel's theorem is a result in geometry, named after Auguste Miquel,[1] concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides.

It is one of several results concerning circles in Euclidean geometry due to Miquel, whose work was published in Liouville's newly founded journal Journal de mathématiques pures et appliquées.

Formally, let ABC be a triangle, with arbitrary points A´, B´ and C´ on sides BC, AC, and AB respectively (or their extensions).

[2][3] The theorem (and its corollary) follow from the properties of cyclic quadrilaterals.

[5] If the fractional distances of A´, B´ and C´ along sides BC (a), CA (b) and AB (c) are da, db and dc, respectively, the Miquel point, in trilinear coordinates (x : y : z), is given by: where d'a = 1 - da, etc.

If the inscribed triangle XYZ is similar to the reference triangle ABC, then the point M of concurrence of the three circles is fixed for all such XYZ.[6]: p.

257 The circumcircles of all four triangles of a complete quadrilateral meet at a point M.[7] In the diagram above these are ∆ABF, ∆CDF, ∆ADE and ∆BCE.

This result was announced, in two lines, by Jakob Steiner in the 1827/1828 issue of Gergonne's Annales de Mathématiques,[8] but a detailed proof was given by Miquel.

Extend all sides until they meet in five points F,G,H,I,K and draw the circumcircles of the five triangles CFD, DGE, EHA, AIB and BKC.

[10] It is also known as the four circles theorem and while generally attributed to Jakob Steiner the only known published proof was given by Miquel.

A diagram showing circles passing through the vertices of a triangle ABC and points A´ , B´ and C´ on the adjacent sides of the triangle intersecting at a common point, M .
The Pivot Theorem for various triangles
Miquel and Steiner's Quadrilateral Theorem
Miquel's Pentagon Theorem
Miquel's six circles theorem states that if five circles share four triple-points of intersection then the remaining four points of intersection lie on a sixth circle
The three-dimensional case: the four spheres intercepts other spheres on black circles.