Isogonic Centers

The first isogonic center $F_1$ of a Triangle is the Fermat Point. The second isogonic center $F_2$ is constructed analogously with the first isogonic center except that for $F_2$, the Equilateral Triangles are constructed on the same side of the opposite Vertex. The second isogonic center has Triangle Center Function

$$\alpha=\csc(A-{\textstyle{1\over 3}}\pi).$$

Its Antipedal Triangle is Equilateral and has Area

$$2\Delta=[-1+\cot\omega\cot({\textstyle{1\over 3}}\pi)],$$

where $\omega$ is the Brocard Angle.

The first and second isogonic centers are Isogonal Conjugates of the Isodynamic Points.

See also Brocard Angle, Equilateral Triangle, Fermat Point, Isodynamic Points, Isogonal Conjugate

References

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 107, 1913.

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