The second Brocard triangle of a reference triangle 
is the triangle 
obtained by the following
construction. Let 
be the circle tangent at 
to 
and passing through 
, and let 
be the circle tangent at 
to 
and passing through 
. Their second intersection
is 
.
Define 
and 
similarly by cyclically permuting the indices.
The second Brocard triangle is also the triangle obtained as the intersections of the lines 
,
,
and 
with the Brocard circle, where 
is the symmedian point.
Let 
,
,
and 
be the intersections of the lines 
,
,
and 
with the circumcircle of 
. Then 
, 
, and 
are the midpoints of 
, 
, and 
, respectively (Lachlan 1893).
The second Brocard triangle has trilinear vertex matrix
![[2bccosA ab ac; ab 2accosB bc; ac bc 2abcosC].](/images/equations/SecondBrocardTriangle/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 focus 
of the Kiepert parabola
of the second Brocard triangle is 
of the reference
triangle, the midpoint of its Kiepert
parabola focus 
and Parry point 
(O. Walsh, pers. comm., Nov. 16, 2025; Kimberling).
The following table gives the centers of the second Brocard triangle in terms of the centers of the
reference triangle that correspond to Kimberling
centers 
.
 | center of second Brocard triangle |  | center of reference triangle |
 | circumcenter |  | midpoint of Brocard diameter |
 | symmedian point |  | harmonic of  |
 | first isodynamic point |  | first isodynamic point |
 | second isodynamic point |  | second isodynamic point |
 | focus of Kiepert parabola |  | midpoint of  and  |
 | Schoute center |  | Schoute center |
 | isogonal
conjugate of  |  | isogonal
conjugate of  |
 | isogonal
conjugate of  |  | isogonal
conjugate of  |
 | first Brocard-axis intercept of circumcircle |  | symmedian point |
 | second Brocard-axis intercept of circumcircle |  | circumcenter |
 | second Brocard-axis-Moses-circle intersection |  | Brocard midpoint |
 | inverse-in-circumcircle of  |  | inverse-in-circumcircle of  |
 | inverse-in-circumcircle of  |  | inverse-in-circumcircle of  |
The first Brocard triangle and second Brocard triangle are in perspective with perspector
at the triangle centroid 
of 
.
