Second Brocard Triangle


Let c_1, c_2, and c_3 be the circles through the vertices A_2 and A_3, A_1 and A_3, and A_1 and A_2, respectively, which intersect in the first Brocard point Omega. Similarly, define c_1^', c_2^', and c_3^' with respect to the second Brocard point Omega^'. Let the two circles c_1 and c_1^' tangent at A_1 to A_1A_2 and A_1A_3, and passing respectively through A_3 and A_2, meet again at C_1, and similarly for C_2 and C_3. Then the triangle DeltaC_1C_2C_3 is called the second Brocard triangle.


The second Brocard triangle is also the triangle obtained as the intersections of the lines A_1K, A_2K, and A_3K with the Brocard circle, where K is the symmedian point. Let P_1, P_2, and P_3 be the intersections of the lines A_1K, A_2K, and A_3K with the circumcircle of DeltaA_1A_2A_3. Then C_1, C_2, and C_3 are the midpoints of A_1P_1, A_2P_2, and A_3P_3, respectively (Lachlan 1893).

The second Brocard triangle has trilinear vertex matrix

 [2bccosA ab ac; ab 2accosC bc; ac bc 2abcosC].

It has area


where Delta is the area of the reference triangle, and side lengths


where a^', b^', and c^' 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 X_n.

X_ncenter of second Brocard triangleX_ncenter of reference triangle
X_3circumcenterX_(182)midpoint of Brocard diameter
X_6symmedian pointX_(574)harmonic of X_(187)
X_(15)first isodynamic pointX_(15)first isodynamic point
X_(16)second isodynamic pointX_(16)second isodynamic point
X_(187)Schoute centerX_(187)Schoute center
X_(511)isogonal conjugate of X_(98)X_(511)isogonal conjugate of X_(98)
X_(512)isogonal conjugate of X_(99)X_(512)isogonal conjugate of X_(99)
X_(1379)first Brocard-axis intercept of circumcircleX_6symmedian point
X_(1380)second Brocard-axis intercept of circumcircleX_3circumcenter
X_(2029)second Brocard-axis-Moses-circle intersectionX_(39)Brocard midpoint
X_(2459)inverse-in-circumcircle of X_(371)X_(2459)inverse-in-circumcircle of X_(371)
X_(2460)inverse-in-circumcircle of X_(372)X_(2460)inverse-in-circumcircle of X_(372)

The first and second Brocard triangles are in perspective with perspector at the triangle centroid G of DeltaA_1A_2A_3.

See also

Brocard Triangles, D-Triangle, First Brocard Triangle, Third Brocard Triangle

Explore with Wolfram|Alpha


Gibert, B. "Brocard Triangles.", R. "The Brocard Triangles." §10.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 110-118, 1995.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 78-81, 1893.

Referenced on Wolfram|Alpha

Second Brocard Triangle

Cite this as:

Weisstein, Eric W. "Second Brocard Triangle." From MathWorld--A Wolfram Web Resource.

Subject classifications