Brocard Triangles


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.


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

Gibert defines the third Brocard triangle as the isogonal conjugate of the first Brocard triangle.

Gibert also defines the fourth Brocard triangle, more commonly known as the D-triangle, as the isogonal conjugate of the second Brocard triangle.

See also

Brocard Circle, Circle-Circle Intersection, D-Triangle, First Brocard Triangle, McCay Circles, Nine-Point Center, Second Brocard Triangle, Steiner Points, Tarry Point, Third Brocard Triangle

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Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 75, 1971.Emmerich, A. Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks. Berlin: Reimer, 1891.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.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 277-281, 1929.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 78-81, 1893.Lalesco, T. La géometrie du triangle. Paris: Gabay, 1987.

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Brocard Triangles

Cite this as:

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

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