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 .

See also 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

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