Brocard Points

The first Brocard point is the interior point $\Omega$ (or $\tau_1$ or $Z_1$) of a Triangle for which the Angles $\angle\Omega AB$, $\angle\Omega BC$, and $\angle\Omega CA$ are equal. The second Brocard point is the interior point $\Omega'$ (or $\tau_2$ or $Z_2$) for which the Angles $\angle\Omega'AC$, $\angle\Omega'CB$, and $\angle\Omega'BA$ are equal. The Angles in both cases are equal to the Brocard Angle $\omega$,

$$\omega = \angle\Omega AB=\angle\Omega BC=\angle\Omega CA$$

$$= \angle\Omega'AC=\angle\Omega'CB=\angle\Omega'BA.$$

The first two Brocard points are Isogonal Conjugates (Johnson 1929, p. 266).

Let $C_{BC}$ be the Circle which passes through the vertices $B$ and $C$ and is Tangent to the line $AC$ at $C$, and similarly for $C_{AB}$ and $C_{BC}$. Then the Circles $C_{AB}$, $C_{BC}$, and $C_{AC}$ intersect in the first Brocard point $\Omega$. Similarly, let $C_{BC}'$ be the Circle which passes through the vertices $B$ and $C$ and is Tangent to the line $AB$ at $B$, and similarly for $C_{AB}'$ and $C_{AC}'$. Then the Circles $C_{AB}'$, $C_{BC}'$, and $C_{AC}'$ intersect in the second Brocard points $\Omega'$ (Johnson 1929, pp. 264-265).

The Pedal Triangles of $\Omega$ and $\Omega'$ are congruent, and Similar to the Triangle $\Delta ABC$ (Johnson 1929, p. 269). Lengths involving the Brocard points include

$$\overline{O\Omega}=\overline{O\Omega'}=R\sqrt{1-4\sin^2\omega}$$ (1)

$$\overline{\Omega\Omega'}=2R\sin\omega\sqrt{1-4\sin^2\omega}.$$ (2)

Brocard's third point is related to a given Triangle by the Triangle Center Function

$$\alpha=a^{-3}$$ (3)

(Casey 1893, Kimberling 1994). The third Brocard point $\Omega''$ (or $\tau_3$ or $Z_3$) is Collinear with the Spieker Center and the Isotomic Conjugate Point of its Triangle's Incenter.

See also Brocard Angle, Brocard Midpoint, Equi-Brocard Center, Yff Points

References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 66, 1893.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 263-286, 1929.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.

Stroeker, R. J. ``Brocard Points, Circulant Matrices, and Descartes' Folium.'' Math. Mag. 61, 172-187, 1988.