Miquel's theorem

The set of Miquel, named after Auguste Miquel, makes a statement about intersections of three circles through one corner of a triangle in the real plane (see figure):

• It is a triangle ABC with vertices A, B, C, the sides BC, AC, AB and three points A ' BC, B' and C to AC ' in AB. Then: The 3 circles by AB'C ', A'BC ', and A'B'C intersect at a point M.

The proof follows by 3- fold application of the theorem on a circle square: four points lie on a circle only if opposite angles in a square add to 180 degrees. Let M be the intersection of the two circles A'BC ' and A'B'C and are the angles of a triangle A, B, C. Then, the angle at M in the circle square A'MBC ' and the angle at M in the circle square A'MB'C Is. So the angle in the quadrilateral AB'MC ' in M is the same, ie the four points A, B ', M, C' lie on a circle.

Describing the real plane in the usual manner with the complex numbers (see Gauss number plane ) and complements the complex numbers to the icon that should be on all lines, we obtain a model of classical geometry of the circuits that are Möbius is called level. The fractional linear transformations, the Möbius transformations form circles and straight lines from completed towards precisely the same. If one forms the above Miquel character with a suitable Möbius transformation down so that the sides of the triangle are transferred to correct circles, one obtains the set of Miquel in general form:

• Can you assign 8 points so the corners of a cube that each side surface points assigned are 5 times on a circle, so this is also the case for the 6th square (see image ).

Meaning of the sentence by Miquel:

Note: Using a stereographic projection is satisfied that the classical Möbius plane to the geometry of the circles on the unit sphere is isomorphic. Here there are only circles ( not straight ) and the general form of the theorem of Miquel is a statement of 6 circles in the.