Brocard Line

A Line from any of the Vertices $A_i$ of a Triangle to the first $\Omega$ or second $\Omega'$ Brocard Point. Let the Angle at a Vertex $A_i$ also be denoted $A_i$, and denote the intersections of $A_1\Omega$ and $A_1\Omega'$ with $A_2A_3$ as $W_1$ and $W_2$. Then the Angles involving these points are

$$\angle A_1\Omega W_3=A_1$$ (1)

$$\angle W_3\Omega A_2=A_3$$ (2)

$$\angle A_2\Omega W_1=A_2.$$ (3)

Distances involving the points $W_i$ and $W_i'$ are given by

$$\overline{A_2\Omega}={a_3\over\sin A_2}\sin\omega$$ (4)

$${\overline{A_2\Omega}\over\overline{A_3\Omega}}={{a_3}^2\over a_1a_2}={\sin(A_3-\omega)\over\sin\omega}$$ (5)

$${\overline{W_3A_1}\over\overline{W_3A_2}}={a_2\sin\omega\over a_1\sin(A_3-\omega)} =\left({a_2\over a_3}\right)^2,$$ (6)

where $\omega$ is the Brocard Angle (Johnson 1929, pp. 267-268).

The Brocard line, Median $M$, and Lemoine Point $K$ are concurrent, with $A_1\Omega_1$, $A_2K$, and $A_3M$ meeting at a point $P$. Similarly, $A_1\Omega'$, $A_2M$, and $A_3K$ meet at a point which is the Isogonal Conjugate point of $P$ (Johnson 1929, pp. 268-269).

See also Brocard Axis, Brocard Diameter, Brocard Points, Isogonal Conjugate, Lemoine Point, Median (Triangle)

References

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.