D-Triangle

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 D-triangle lie on the respective Apollonius circles.

The following table gives the centers of the D-triangle in terms of the centers of the reference triangle that correspond to Kimberling centers .

	center of the D-triangle		center of reference triangle
	circumcenter		midpoint of and
	symmedian point		symmedian point
	first isodynamic point		first Fermat point
	second isodynamic point		second Fermat point
	far-out 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