Given triangle Brocard points ,
and similarly define first Brocard triangle , and
is inversely similar to Brocard circle .
Let circles through the vertices intersect
in the first Brocard point Brocard
point second
Brocard triangle .
The first and second Brocard triangles are in perspective
with perspector at the triangle
centroid
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. 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. Johnson, R. A. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Lachlan, R. An
Elementary Treatise on Modern Pure Geometry. Lalesco, T. La
géometrie du triangle. Referenced on
Wolfram|Alpha Brocard Triangles
Cite this as:
Weisstein, Eric W. "Brocard Triangles."
From MathWorld https://mathworld.wolfram.com/BrocardTriangles.html
Subject classifications
, 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.
,
,
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.
of 
.
