Brocard points

The Brocard point of a triangle, constructed at the intersection point of three circles

In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician.

Definition

In a triangle Template:Math with sides Template:Mvar, where the vertices are labeled Template:Mvar in counterclockwise order, there is exactly one point Template:Mvar such that the line segments Template:Mvar form the same angle, Template:Mvar, with the respective sides Template:Mvar, namely that

$$\mathrm{\angle }PAB=\mathrm{\angle }PBC=\mathrm{\angle }PCA=\omega .$$

Point Template:Mvar is called the first Brocard point of the triangle Template:Math, and the angle Template:Mvar is called the Brocard angle of the triangle. This angle has the property that

$$\cot \omega =\cot \left(\mathrm{\angle }CAB\right)+\cot \left(\mathrm{\angle }ABC\right)+\cot \left(\mathrm{\angle }BCA\right).$$

There is also a second Brocard point, Template:Mvar, in triangle Template:Math such that line segments Template:Mvar form equal angles with sides Template:Mvar respectively. In other words, the equations $$\mathrm{\angle }QCB=\mathrm{\angle }QBA=\mathrm{\angle }QAC$$ apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words, angle $$\mathrm{\angle }PBC=\mathrm{\angle }PCA=\mathrm{\angle }PAB$$ is the same as $$\mathrm{\angle }QCB=\mathrm{\angle }QBA=\mathrm{\angle }QAC.$$

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 Template:Math are taken. So for example, the first Brocard point of Template:Math is the same as the second Brocard point of Template:Math.

The two Brocard points of a triangle Template:Math are isogonal conjugates of each other.

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 Template:Mvar and Template:Mvar, tangent to edge Template:Mvar of the triangle (the center of this circle is at the point where the perpendicular bisector of Template:Mvar meets the line through point Template:Mvar that is perpendicular to Template:Mvar). Symmetrically, form a circle through points Template:Mvar and Template:Mvar, tangent to edge Template:Mvar, and a circle through points Template:Mvar and Template:Mvar, tangent to edge Template:Mvar. These three circles have a common point, the first Brocard point of Template:Math. See also Tangent lines to circles.

The three circles just constructed are also designated as epicycles of Template:Math. 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: $$\begin{array}{cccccc}P=& \frac{c}{b}& :& \frac{a}{c}& :& \frac{b}{a}\\ Q=& \frac{b}{c}& :& \frac{c}{a}& :& \frac{a}{b}\end{array}$$ Thus their barycentric coordinates are: $$\begin{array}{cccccc}P=& c^2{a}^2& :& a^2{b}^2& :& b^2{c}^2\\ Q=& a^2{b}^2& :& b^2{c}^2& :& c^2{a}^2\end{array}$$

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

$$\sin (A+\omega ):\sin (B+\omega ):\sin (C+\omega )=a(b^2+c^2):b(c^2+a^2):c(a^2+b^2),$$ 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

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

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 Template:Mvar and Template:Mvar is always less than or equal to half the radius Template:Mvar of the triangle's circumcircle:

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

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 Template:Mvar and Template:Mvar are equidistant from the triangle's circumcenter Template:Mvar:

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

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 Template:Mvar, each through one of a triangle's vertices and its first Brocard point, intersect the triangle's circumcircle at points Template:Mvar, then the triangle Template:Math is congruent with the original triangle Template:Math. The same is true if the first Brocard point Template:Mvar is replaced by the second Brocard point Template:Mvar.

References

• Template:Citation.
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External links

• Third Brocard Point at MathWorld
• Bicentric Pairs of Points and Related Triangle Centers
• Bicentric Pairs of Points
• Bicentric Points at MathWorld