Brocard points

Construction

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.

As in the diagram above, 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 △ABC. See also Tangent lines to circles.

The three circles just constructed are also designated as epicycles of △ABC. The second Brocard point is constructed in similar fashion.

Trilinears and barycentrics of the first two Brocard points

Homogeneous trilinear coordinates for the first and second Brocard points are: $${\displaystyle \begin{array}{cccccc}P=& \frac{c}{b}& :& \frac{a}{c}& :& \frac{b}{a}\\ Q=& \frac{b}{c}& :& \frac{c}{a}& :& \frac{a}{b}\end{array}}$$ Thus their barycentric coordinates are: $${\displaystyle \begin{array}{cccccc}P=& c^2{a}^2& :& a^2{b}^2& :& b^2{c}^2\\ Q=& a^2{b}^2& :& b^2{c}^2& :& c^2{a}^2\end{array}}$$

The segment between the first two Brocard points

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 trilinear coordinates

$${\displaystyle \sin (A+\omega ):\sin (B+\omega ):\sin (C+\omega )=a(b^2+c^2):b(c^2+a^2):c(a^2+b^2),}$$ and is a triangle center; it is center X(39) in the Encyclopedia of Triangle Centers. The third Brocard point, given in trilinear coordinates as

$${\displaystyle \csc (A-\omega ):\csc (B-\omega ):\csc (C-\omega )=a^{-3}:b^{-3}:c^{-3},}$$

is the Brocard midpoint of the anticomplementary triangle and is also the isotomic conjugate of the symmedian point. It is center X(76) in the Encyclopedia of Triangle Centers.

The distance between the first two Brocard points P and Q is always less than or equal to half the radius R of the triangle's circumcircle:

$${\displaystyle \overline{PQ}=2R\sin \omega \sqrt{1-4{\sin }^2\omega }\le \frac{R}{2}.}$$

The segment between the first two Brocard points is perpendicularly bisected at the Brocard midpoint by the line connecting the triangle's circumcenter and its Lemoine point. Moreover, the circumcenter, the Lemoine point, and the first two Brocard points are concyclic—they all fall on the same circle, of which the segment connecting the circumcenter and the Lemoine point is a diameter.

Distance from circumcenter

The Brocard points P and Q are equidistant from the triangle's circumcenter O:

$${\displaystyle \overline{PO}=\overline{QO}=R\sqrt{\frac{a^4+b^4+c^4}{a^2{b}^2+b^2{c}^2+c^2{a}^2}-1}=R\sqrt{1-4{\sin }^2\omega }.}$$

Similarities and congruences

The pedal triangles of the first and second Brocard points are congruent to each other and similar to the original triangle.

If the lines AP, BP, CP, each through one of a triangle's vertices and its first Brocard point, intersect the triangle's circumcircle at points L, M, N, then the triangle △LMN is congruent with the original triangle △ABC. The same is true if the first Brocard point P is replaced by the second Brocard point Q.