Miquel Point


The Miquel point is the point of concurrence of the Miquel circles. It is therefore the radical center of these circles.

Let the points defining the Miquel circles be fractional distances k_a, k_b, and k_c along the sides BC, CA, and AB, respectively, and let k_i^'=1-k_i. Then the Miquel point has trilinear coordinates alpha:beta:gamma, where


In the special case k_a=k_b=k_c=1/2, the Miquel point becomes the circumcenter.

If DeltaA_1B_1C_1 and DeltaA_2B_2C_2 are inscribed in a reference triangle DeltaABC and also in the same circle, then their Miquel points M_1 and M_2 are isogonal conjugates. The angle that MA_1, MB_1 and MC_1 make to the respective sides of DeltaABC and the angle that MA_2, MB_2 and MC_2 make to these sides are supplementary. The pedal triangle is a special case.

See also

Miquel Circles, Miquel's Theorem, Miquel Triangle

Portions of this entry contributed by Floor van Lamoen

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Ayme, J.-L. "A Purely Synthetic Proof of the Droz-Farny Line Theorem." Forum Geom. 4, 219-224, 2004., J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 87-90, 1971.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 81, 1995.Miquel, A. "Mémoire de Géométrie." Journal de mathématiques pures et appliquées de Liouville 1, 485-487, 1838.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 151, 1991.

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Miquel Point

Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Miquel Point." From MathWorld--A Wolfram Web Resource.

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