First Brocard Triangle


Given triangle DeltaA_1A_2A_3, let the point of intersection of A_2Omega and A_3Omega^' be B_1, where Omega and Omega^' are the Brocard points, and similarly define B_2 and B_3. Then DeltaB_1B_2B_3 is called the first Brocard triangle, and is inversely similar to DeltaA_1A_2A_3 (Honsberger 1995, p. 112). It is inscribed in the Brocard circle.

The trilinear vertex matrix is

 [abc c^3 b^3; c^3 abc a^3; b^3 a^3 abc].

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 first Brocard triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=100.

The triangles DeltaB_1A_2A_3, DeltaB_2A_3A_1, and DeltaB_3A_1A_2 are isosceles triangles with base angles omega, where omega is the Brocard angle. The sum of the areas of the isosceles triangles is Delta, the area of triangle DeltaA_1A_2A_3.


The first Brocard triangle is in perspective with DeltaA_1A_2A_3 with perspector at the third Brocard point Omega^('') of DeltaA_1A_2A_3.


Let perpendiculars be drawn from the midpoints M_1, M_2, and M_3 of each side of the first Brocard triangle to the opposite sides of the triangle DeltaA_1A_2A_3. Then the extensions of these lines concur in the nine-point center N of DeltaA_1A_2A_3 (Honsberger 1995, pp. 116-118).


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


The triangle centroid of the first Brocard triangle DeltaB_1B_2B_3 is also the triangle centroid G of the original triangle DeltaA_1A_2A_3 (Honsberger 1995, pp. 112-116).

See also

D-Triangle, Brocard Triangles, Second Brocard Triangle, Third Brocard Triangle

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Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 100, 1913.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.

Referenced on Wolfram|Alpha

First Brocard Triangle

Cite this as:

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

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