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Pivot theorem
In geometry, the pivot theorem states that, given any three points P, Q, and R on each respective side of a triangle ABC, the three circles through the points AQR, BPR and CPQ share a common point M. Conversely, this is equivalent to a porism: given any three circles sharing a common point M, there are an infinite number of triangles such that one point lies on each circle and the sides of the triangle pass through the intersection points of the 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.
See also
Miquel's theorem
Bibliography
Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. pp. 227–228. ISBN 0-14-011813-6.
External links
Weisstein, Eric W., "Pivot theorem" from MathWorld.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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