The first Brocard triangle of a triangle 
has vertices 
, 
, and 
, where 
is the intersection of
and 
,
and 
are the Brocard points, and 
and 
are defined similarly. It is inversely
similar to 
(Honsberger 1995, p. 112) and 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 signed 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.
If 
is the unsigned area of the first Brocard triangle and 
is the area
of the pedal triangle of the symmedian
point 
,
then
 |
(6)
|
Morel (1883, §35, pp. 195-196) gives the individual areas as
 |  |  |
(7)
|
 |  |  |
(8)
|
where 
is the Brocard angle.
The triangle median from 
bisects the line
segment 
,
where 
is the symmedian point and 
is the corresponding vertex of
the first Brocard triangle (Morel 1883, §26, pp. 171-172).
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).
Among the 36 cataloged six-piece triangle dissections of a generic non-right triangle into similar triangles, nine have their interior junction at a vertex of the first Brocard triangle. They form three groups of three cyclically related layouts. Each group uses the same three straight-through cuts as the three layouts whose junction is the symmedian point. These cuts are parallel to the sides and pass through the symmedian point, so their six endpoints lie on the first Lemoine circle. In three of the nine first-Brocard layouts, all four boundary cut points lie on that circle (E. Pegg, Jr., pers. comm., Oct. 3, 2026).
Pegg, E. Jr. Mathematical Games. Episode 46: "Sporadic Similar
Triangles." Oct. 22, 2026.