Brocard Line


A Brocard line is a line from any of the vertices A_i of a triangle DeltaA_1A_2A_3 to the first Omega or second Omega^' Brocard point. Let the angle at a vertex A_i also be denoted A_i, and denote the intersections of A_1Omega and A_1Omega^' with A_2A_3 as W_1 and W_2. Then the angles involving these points are


Distances involving the point W_i satisfy


where omega is the Brocard angle (Johnson 1929, pp. 267-268).

If G is the triangle centroid and K is the symmedian point of a triangle DeltaA_1A_2A_3, then the lines A_1Omega, A_2K, and A_3G meet at a point P. Similarly, A_1Omega^', A_2G, and A_3K meet at a point P^' which is the isogonal conjugate point of P (Johnson 1929, pp. 268-269).

See also

Brocard Axis, Brocard Circle, Brocard Diameter, Brocard Points, Brocard Triangles, Isogonal Conjugate, Symmedian Point, Triangle Median

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Emmerich, A. Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks. Berlin: Reimer, 1891.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263-286, 1929.

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Brocard Line

Cite this as:

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

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