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 .

The trilinear vertex matrix is

(1)

It has area

(2)

where
is the area of the reference triangle , and
side lengths

where ,
,
and
are the side lengths of the reference triangle .

The following table gives the centers of the first Brocard triangle in terms of the centers of the reference triangle for Kimberling
centers
with .

The triangles ,
,
and
are isosceles triangles with base angles , where is the Brocard angle .
The sum of the areas of the isosceles triangles
is ,
the area of triangle .

The first Brocard triangle is in perspective with with perspector
at the third Brocard point of .

Let perpendiculars be drawn from the midpoints , , and of each side of the first Brocard triangle to the opposite
sides of the triangle . Then the extensions of these lines concur
in the nine-point center of (Honsberger 1995, pp. 116-118).

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

The triangle centroid of the first Brocard triangle
is also the triangle centroid of the original triangle (Honsberger 1995, pp. 112-116).

See also D-Triangle ,

Brocard Triangles ,

Second Brocard Triangle ,

Third Brocard Triangle
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References Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 100, 1913. 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. Referenced on Wolfram|Alpha First Brocard Triangle
Cite this as:
Weisstein, Eric W. "First Brocard Triangle."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/FirstBrocardTriangle.html

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