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# Brocard Triangles

Given triangle , let the point of intersection of and be , where and are the Brocard points, and similarly define and . Then is called the first Brocard triangle, and is inversely similar to (Honsberger 1995, p. 112). It is inscribed in the Brocard circle.

Let , , and be the circles through the vertices and , and , and and , respectively, which intersect in the first Brocard point . Similarly, define , , and with respect to the second Brocard point . Let the two circles and tangent at to and , and passing respectively through and , meet again at , and similarly for and . Then the triangle is called the second Brocard triangle.

The first and second Brocard triangles are in perspective with perspector at the triangle centroid of .

Gibert defines the third Brocard triangle as the isogonal conjugate of the first Brocard triangle.

Gibert also defines the fourth Brocard triangle, more commonly known as the D-triangle, as the isogonal conjugate of the second Brocard triangle.

Brocard Circle, Circle-Circle Intersection, D-Triangle, First Brocard Triangle, McCay Circles, Nine-Point Center, Second Brocard Triangle, Steiner Points, Tarry Point, Third Brocard Triangle

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## References

Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 75, 1971.Emmerich, A. Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks. Berlin: Reimer, 1891.Gibert, B. "Brocard Triangles." http://perso.wanadoo.fr/bernard.gibert/gloss/brocardtriangles.html.Honsberger, R. "The Brocard Triangles." §10.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 110-118, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 277-281, 1929.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 78-81, 1893.Lalesco, T. La géometrie du triangle. Paris: Gabay, 1987.

## Referenced on Wolfram|Alpha

Brocard Triangles

## Cite this as:

Weisstein, Eric W. "Brocard Triangles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrocardTriangles.html