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.

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." Mathesis6, Suppl. 3, 1-22, 1886.