Let the circles and used in the construction of the Brocard points which are tangent to at and , respectively, meet again at . The points then define the triangle, also known as the fourth Brocard triangle (Gibert).
It has trilinear vertex matrix
(1)

The vertices of the triangle are the isogonal conjugates of the second Brocard triangle, and is inversely similar to the medial triangle (Johnson 1929, p. 285). In addition, the vertices lie on the respective medians of the reference triangle. The circumcircle of the triangle is the orthocentroidal circle, which has diameter , where is the triangle centroid and is the orthocenter.
The vertices satisfy
(2)
 
(3)
 
(4)

(correcting Johnson 1929, p. 285).
The vertices of the Dtriangle lie on the respective Apollonius circles.
The following table gives the centers of the Dtriangle in terms of the centers of the reference triangle that correspond to Kimberling centers .
center of the Dtriangle  center of reference triangle  
circumcenter  midpoint of and  
symmedian point  symmedian point  
first isodynamic point  first Fermat point  
second isodynamic point  second Fermat point  
farout point  Parry point  
Parry point  triangle centroid  
Schoute center  center of Kiepert hyperbola  
isogonal conjugate of  direction of vector , where  
isogonal conjugate of  crossdifference of line and  
Collings transform of  orthocenter  
radical center of (circumcircle, Brocard circle, Parry circle)  tripolar triangle centroid of  
isogonal conjugate of  point Biham  
isogonal conjugate of  isogonal conjugate of 