If a point is marked on each side of a Triangle , then the three Miquel Circles (each through a Vertex and the two marked points on the adjacent sides) are Concurrent at a point called the Miquel Point. This result is a slight generalization of the so-called Pivot Theorem.
If lies in the interior of the triangle, then it satisfies
Given four lines , ..., each intersecting the other three, the four Miquel Circles passing through each subset of three intersection points of the lines meet in a point known as the 4-Miquel point . Furthermore, the centers of these four Miquel Circles lie on a Circle (Johnson 1929, p. 139). The lines from to given points on the sides make equal Angles with respect to the sides.
Similarly, given lines taken by s yield Miquel Circles like passing through a point , and their centers lie on a Circle .
See also Miquel Circles, Miquel Equation, Miquel Triangle, Nine-Point Circle, Pedal Circle, Pivot Theorem
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.
Boston, MA: Houghton Mifflin, pp. 131-144, 1929.