Brocard Axis

The Line $KO$ passing through the Lemoine Point $K$ and Circumcenter $O$ of a Triangle. The distance $\overline{OK}$ is called the Brocard Diameter. The Brocard axis is Perpendicular to the Lemoine Axis and is the Isogonal Conjugate of Kiepert's Hyperbola. It has equations

$$\sin(B-C)\alpha+\sin(C-A)\beta+\sin(A-B)\gamma=0$$

$$bc(b^2-c^2)\alpha+ca(c^2-a^2)\beta+ab(a^2-b^2)\gamma=0.$$

The Lemoine Point, Circumcenter, Isodynamic Points, and Brocard Midpoint all lie along the Brocard axis. Note that the Brocard axis is not equivalent to the Brocard Line.

See also Brocard Circle, Brocard Diameter, Brocard Line