In a triangle ABC with sides a, b, and c, where the vertices are labeled A, B and C in counterclockwise order, there is exactly one point P such that the line segments AP, BP, and CP form the same angle, ω, with the respective sides c, a, and b, namely that
Point P is called the first Brocard point of the triangle ABC, and the angle ω is called the Brocard angle of the triangle. The following applies to this angle:
There is also a second Brocard point, Q, in triangle ABC such that line segments AQ, BQ, and CQ form equal angles with sides b, c, and a respectively. In other words, the equations apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words, angle is the same as
The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in which the angles of triangle ABC are taken. So for example, the first Brocard point of triangle ABC is the same as the second Brocard point of triangle ACB.
The two Brocard points of a triangle ABC are isogonal conjugates of each other.
The most elegant construction of the Brocard points goes as follows. In the following example the first Brocard point is presented, but the construction for the second Brocard point is very similar.
Form a circle through points A and B, tangent to edge BC of the triangle (the center of this circle is at the point where the perpendicular bisector of AB meets the line through point B that is perpendicular to BC). Symmetrically, form a circle through points B and C, tangent to edge AC, and a circle through points A and C, tangent to edge AB. These three circles have a common point, the first Brocard point of triangle ABC. See also Tangent lines to circles.
The three circles just constructed are also designated as epicycles of triangle ABC. The second Brocard point is constructed in similar fashion.
Trilinears and the Brocard midpoint
Homogeneous trilinear coordinates for the first and second Brocard points are c/b : a/c : b/a, and b/c : c/a : a/b, respectively. The Brocard points are an example of a bicentric pair of points, but they are not triangle centers because neither Brocard point is invariant under similarity transformations: reflecting a scalene triangle, a special case of a similarity, turns one Brocard point into the other. However, the unordered pair formed by both points is invariant under similarities. The midpoint of the two Brocard points, called the Brocard midpoint, has trilinears
- sin(A + ω) : sin(B + ω) : sin(C + ω)
and is a triangle center. The third Brocard point, given in trilinear coordinates as a−3 : b−3 : c−3, or, equivalently, by
- csc(A − ω) : csc(B − ω) : csc(C − ω),
- Akopyan, A. V.; Zaslavsky, A. A. (2007), Geometry of Conics, Mathematical World 26, American Mathematical Society, pp. 48–52, ISBN 978-0-8218-4323-9.
- Honsberger, Ross (1995), "Chapter 10. The Brocard Points", Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Washington, D.C.: The Mathematical Association of America.
- Third Brocard Point at MathWorld
- Bicentric Pairs of Points and Related Triangle Centers
- Bicentric Pairs of Points
- Bicentric Points at MathWorld