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Miquel Point

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MiquelPoint

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

alpha=a(-a^2k_ak_a^'+b^2k_ak_b+c^2k_a^'k_c^')
(1)
beta=b(a^2k_a^'k_b^'-b^2k_bk_b^'+c^2k_bk_c)
(2)
gamma=c(a^2k_ak_c+b^2k_b^'k_c^'-c^2k_ck_c^').
(3)
MiquelPointCircumcircle

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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References

Ayme, J.-L. "A Purely Synthetic Proof of the Droz-Farny Line Theorem." Forum Geom. 4 , 219-224, 2004. http://forumgeom.fau.edu/FG2004volume4/FG200426index.html . Coolidge, 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. https://mathworld.wolfram.com/MiquelPoint.html

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