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# Miquel Triangle

Given a point and a triangle , the Miquel triangle is the triangle connecting the side points , , and of with respect to which is the Miquel point.

Let the points defining the Miquel circles be fractional distances , , and along the sides , , and , respectively, and let and . The Miquel triangle has side lengths

 (1) (2) (3)

and area

 (4)

where is the area of the reference triangle.

In the special case , the Miquel triangle becomes the medial triangle.

All Miquel triangles of a given point are directly similar, and is the similitude center in every case.

Miquel Circles, Miquel Point, Miquel's Theorem

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## References

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.

Miquel Triangle

## Cite this as:

Weisstein, Eric W. "Miquel Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MiquelTriangle.html