Miquel's theorem

If in the statement of Miquel's theorem the points A´, B´ and C´ form a triangle (that is, are not collinear) then the theorem was named the Pivot theorem in Forder (1960, p. 17).[4] (In the diagram these points are labeled P, Q and R.)

If A´, B´ and C´ are collinear then the Miquel point is on the circumcircle of ∆ABC and conversely, if the Miquel point is on this circumcircle, then A´, B´ and C´ are on a line.[5]

If the fractional distances of A´, B´ and C´ along sides BC (a), CA (b) and AB (c) are d_a, d_b and d_c, respectively, the Miquel point, in trilinear coordinates (x : y : z), is given by:

${\displaystyle x=a\left(-a^{2}d_{a}d_{a}^{\,'}+b^{2}d_{a}d_{b}+c^{2}d_{a}^{\,'}d_{c}^{\,'}\right)}$

${\displaystyle y=b\left(a^{2}d_{a}^{\,'}d_{b}^{\,'}-b^{2}d_{b}d_{b}^{\,'}+c^{2}d_{b}d_{c}\right)}$

${\displaystyle z=c\left(a^{2}d_{a}d_{c}+b^{2}d_{b}^{\,'}d_{c}^{\,'}-c^{2}d_{c}d_{c}^{\,'}\right),}$

where d'_a = 1 - d_a, etc.

In the case d_a = d_b = d_c = ½ the Miquel point is the circumcentre (cos α : cos β : cos γ).

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 C´ with another to give B (at the second intersection). B is then similarly connected, via intersection at A´ of the second and third circles, giving point C. Points C, A and the remaining point of intersection, B´, will then be collinear, and triangle ABC will always pass through the circle intersections A´, B´ and C´.

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

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.[7]

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).[9] 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.[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.[11] Wells refers to this as Miquel's theorem.[12]

The three-dimensional case: the four spheres intercepts other spheres on black circles.

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.[3]

• Clifford's circle theorems
• Bundle theorem
• Miquel configuration

• Coxeter, H.S.M.; Greitzer, S.L. (1967), Geometry Revisited, New Mathematical Library, 19, Washington, D.C.: Mathematical Association of America, ISBN 978-0-88385-619-2, Zbl 0166.16402
• Forder, H.G. (1960), Geometry, London: Hutchinson
• Ostermann, Alexander; Wanner, Gerhard (2012), Geometry by its History, Springer, ISBN 978-3-642-29162-3
• Pedoe, Dan (1988) [1970], Geometry / A Comprehensive Course, Dover, ISBN 0-486-65812-0
• Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, ISBN 0-534-35188-3
• Wells, David (1991), The Penguin Dictionary of Curious and Interesting Geometry, New York: Penguin Books, ISBN 0-14-011813-6, Zbl 0856.00005

Wikimedia Commons has media related to Miquel's theorem.

• Weisstein, Eric W. "Miquel's theorem". MathWorld.
• Weisstein, Eric W. "Miquel Five Circles Theorem". MathWorld.
• Weisstein, Eric W. "Miquel Pentagram Theorem". MathWorld.
• Weisstein, Eric W. "Pivot theorem". MathWorld.
• Miquels' Theorem as a special case of a generalization of Napoleon's Theorem at Dynamic Geometry Sketches