TOPICS

# 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 , , and along the sides , , and , respectively, and let . Then the Miquel point has trilinear coordinates , where

 (1) (2) (3)

In the special case , the Miquel point becomes the circumcenter.

If and are inscribed in a reference triangle and also in the same circle, then their Miquel points and are isogonal conjugates. The angle that , and make to the respective sides of and the angle that , and make to these sides are supplementary. The pedal triangle is a special case.

Miquel Circles, Miquel's Theorem, Miquel Triangle

Portions of this entry contributed by Floor van Lamoen

## 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.

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