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
![[abc c^3 b^3; c^3 abc a^3; b^3 a^3 abc].](/images/equations/FirstBrocardTriangle/NumberedEquation1.svg) |
(1)
|
It has area
 |
(2)
|
where 
is the area of the reference triangle, and
side lengths
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
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 
.
 | center of first Brocard triangle |  | center of reference triangle |
 | triangle centroid |  | triangle centroid |
 | circumcenter |  | midpoint of Brocard diameter |
 | orthocenter |  | reflection of  in  |
 | Exeter point |  | inverse of  in the Brocard circle |
 | far-out point |  | focus of Kiepert parabola |
 | Euler infinity point |  | direction of vector  , where  |
 | symmedian point of the anticomplementary triangle |  | external similitude center
of Moses circle and ( ,  ) |
 | Tarry point |  | circumcenter |
 | Steiner
point  |  | symmedian point |
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).
