Second Brocard Triangle

Let , , and be the circles through the vertices and , and , and and , respectively, which intersect in the first Brocard point . Similarly, define , , and with respect to the second Brocard point . Let the two circles and tangent at to and , and passing respectively through and , meet again at , and similarly for and . Then the triangle is called the second Brocard triangle.

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

(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 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
	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 and second Brocard triangles are in perspective with perspector at the triangle centroid of .