Miquel's theorem is a result in geometry, named after Auguste Miquel, 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 Aô, Bô and Cô on sides BC, AC, and AB respectively (or their extensions). Draw three circumcircles (Miquel's circles) to triangles ABôCô, AôBCô, and AôBôC. Miquel's theorem states that these circles intersect in a single point M, called the Miquel point. In addition, the three angles MAôB, MBôC and MCôA (green in the diagram) are all equal, as are the three supplementary angles MAôC, MBôA and MCôB.
The theorem (and its corollary) follow from the properties of cyclic quadrilaterals. Let the circumcircles of A'B'C and AB'C' meet at Then hence BA'MC' is cyclic as desired.
If in the statement of Miquel's theorem the points Aô, Bô and Cô form a triangle (that is, are not collinear) then the theorem was named the Pivot theorem in . (In the diagram these points are labeled P, Q and R.)
If Aô, Bô and Cô are collinear then the Miquel point is on the circumcircle of âÂÂABC and conversely, if the Miquel point is on this circumcircle, then Aô, Bô and Cô are on a line.
If the fractional distances of Aô, Bô and Cô along sides BC (a), CA (b) and AB (c) are d<sub>a</sub>, d<sub>b</sub> and d<sub>c</sub>, respectively, the Miquel point, in trilinear coordinates (x : y : z), is given by:
where d<sub>a</sub> = 1 - d<sub>a</sub>, etc.
In the case d<sub>a</sub> = d<sub>b</sub> = d<sub>c</sub> = ý the Miquel point is the circumcenter .
The theorem can be reversed to say: for three circles intersecting at M, a line can be drawn from any point A on one circle, through its intersection Cô with another to give B (at the second intersection). B is then similarly connected, via intersection at Aô of the second and third circles, giving point C. Points C, A and the remaining point of intersection, Bô, will then be collinear, and triangle ABC will always pass through the circle intersections Aô, Bô and Cô.
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.
The circumcircles of all four triangles of a complete quadrilateral meet at a point M. In the diagram above these are âÂÂABF, âÂÂCDF, âÂÂADE and âÂÂBCE.
This result was announced, in two lines, by Jakob Steiner in the 1827/1828 issue of Gergonne's Annales de Mathématiques, but a detailed proof was given by Miquel.
Let ABCDE be a convex pentagon. 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. Then the second intersection points (other than A,B,C,D,E), namely the new points M,N,P,R and Q are concyclic (lie on a circle). See diagram.
The converse result is known as the Five circles theorem.
Given points, A, B, C, and D on a circle, and circles passing through each adjacent pair of points, the alternate intersections of these four circles at W, X, Y and Z then lie on a common circle. This is known as the six circles theorem. 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. David G. Wells refers to this as Miquel's theorem.
There is also a three-dimensional analog, in which the four spheres passing through a point of a tetrahedron and points on the edges of the tetrahedron intersect in a common point.