First Brocard Point


The first Brocard point Omega is the interior point Omega (also denoted tau_1 or Z_1) of a triangle DeltaABC with points labeled in counterclockwise order for which the angles ∠OmegaAB, ∠OmegaBC, and ∠OmegaCA are equal, with the unique such angle denoted omega and called the Brocard angle. The first Brocard point Omega fails to be a triangle center because it is bicentric with the second Brocard point Omega^', but it has trilinear coordinates


(Kimberling 1998, p. 47).

Note that extreme care is needed when consulting the literature, since reversing the order in which the points of the triangle are labeled results in exchanging the Brocard points.

Distances involving the second Brocard point include


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

See also

Brocard Angle, Brocard Midpoint, Brocard Points, Second Brocard Point, Third Brocard Point

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Honsberger, R. "The Brocard Points." Ch. 10 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 98-124, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 19-21, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lemoine, É. "Propriétés relatives a deux points Omega, Omega^' du plan d'un triangle ABC qui se déduisent d'un point K quelconque di plan comme les points de Brocard de déduisent du point de Lemoine." Mathesis 6, Suppl. 3, 1-22, 1886.

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First Brocard Point

Cite this as:

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

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