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# First Brocard Point

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

 (1)

(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

 (2) (3) (4) (5)

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

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

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## References

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 , du plan d'un triangle qui se déduisent d'un point quelconque di plan comme les points de Brocard de déduisent du point de Lemoine." Mathesis 6, Suppl. 3, 1-22, 1886.

## Referenced on Wolfram|Alpha

First Brocard Point

## Cite this as:

Weisstein, Eric W. "First Brocard Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstBrocardPoint.html