First Brocard Triangle

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

(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).