Miquel's theorem

Miquel's theorem: triangle and 3 circles

The set of Miquel , named after Auguste Miquel , makes a statement about points of intersection of three circles by a respective corner of a triangle in the real plane (s image.):

• It should be a triangle with vertices , sides and three points on on and on . Then the following applies: The 3 circles go through and intersect at one point . ${\displaystyle ABC}$ ${\displaystyle A,B,C}$ ${\displaystyle BC,AC,AB}$ ${\displaystyle A^{\prime }}$ ${\displaystyle BC,B^{\prime }}$ ${\displaystyle AC}$ ${\displaystyle C^{\prime }}$ ${\displaystyle AB}$ ${\displaystyle AB^{\prime }C^{\prime },A^{\prime }BC^{\prime }}$ ${\displaystyle A^{\prime }B^{\prime }C}$ ${\displaystyle M}$

The proof is obtained by applying the theorem about a square circle three times : Four points are only located on a circle if opposing angles in the square add up to 180 degrees. It is the intersection of two circles and and are the angles of a triangle . Then, the angle at is in the circle square and the angle at the circular square is . So the angle in the square is equal to , i.e. H. the four points lie on a circle. ${\displaystyle M}$ ${\displaystyle A^{\prime }BC^{\prime }}$ ${\displaystyle A^{\prime }B^{\prime }C}$ ${\displaystyle \alpha ,\beta ,\gamma }$ ${\displaystyle A,B,C}$ ${\displaystyle M}$ ${\displaystyle A^{\prime }MBC^{\prime }}$ ${\displaystyle 180^{\circ }-\beta }$ ${\displaystyle M}$ ${\displaystyle A^{\prime }MB^{\prime }C}$ ${\displaystyle 180^{\circ }-\gamma }$ ${\displaystyle AB^{\prime }MC^{\prime }}$ ${\displaystyle M}$ ${\displaystyle 360^{\circ }-(180^{\circ }-\beta +180^{\circ }-\gamma )=\beta +\gamma =180^{\circ }-\alpha }$ ${\displaystyle A,B^{\prime },M,C^{\prime }}$

Miquel's set: 6-circle shape

If you describe the real plane in the usual way with the complex numbers (see Gaussian plane of numbers ) and add the symbol to the complex numbers that should lie on all straight lines, you get a model of the classic geometry of the circles , which is also Möbius Level is called. The broken linear images , the Möbius transformations , map circles and completed straight lines onto them. If you depict the Miquel figure above with a suitable Möbius transformation so that the sides of the triangle merge into correct circles, you get Miquel's theorem in a general form : ${\displaystyle \mathbb {C} }$ ${\displaystyle \infty }$ ${\displaystyle PGL(2,\mathbb {C} )}$

• If you can assign 8 points to the corners of a cube in such a way that the points assigned to each side are located 5 times on a circle, this is also the case for the 6th square (see picture). ${\displaystyle P_{1},...,P_{8}}$

Meaning of Miquel's sentence:

1. Miquel's theorem plays an important role in the classification of axiomatic Möbius planes .
2. Miquel's theorem is also available for parabolas and hyperbolas and plays an important role in the classification of the Laguerre and Minkowski planes .

classic Möbius level: 2D / 3D model

Comment: With the help of a stereographic projection one can convince oneself that the classical Möbius plane is isomorphic to the geometry of the circles on the unit sphere. There are only circles (no straight lines) and the general form of Miquel's theorem is a statement about 6 circles in the . ${\displaystyle \mathbb {R} ^{3}}$

literature

• W. Benz: Lectures on the geometry of algebras . Springer, 1973
• F. Buekenhout (ed.): Handbook of Incidence Geometry . Elsevier, 1995, ISBN 0-444-88355-X
• P. Dembowski: Finite Geometries . Springer-Verlag, 1968, ISBN 3-540-61786-8
• M. Koecher, A. Krieg: Level geometry . 3. Edition. Springer-Verlag, 2007, ISBN 978-3-540-49327-3

Web links

• Eric W. Weisstein : Miquel's theorem . In: MathWorld (English).
• Miquels' Theorem as a special case of a generalization of Napoleon's Theorem at Dynamic Geometry Sketches

Individual evidence

1. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 47.
2. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), pp. 70 and 93.