Neuberg Circles

The Locus of the Vertex $A_1$ of a Triangle on a given base $A_2A_3$ and with a given Brocard Angle $\omega$ is a Circle on either side of $A_2A_3$. From the center $N_1$, the base $A_2A_3$ subtends the Angle $2\omega$. The Radius of the Circle is

$$r={\textstyle{1\over 2}}a_1\sqrt{\cot^2\omega-3}.$$

See also Brocard Angle

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 287-290, 1929.