Crucial Point

The Homothetic Center of the Orthic Triangle and the triangular hull of the three Excircles. It has Triangle Center Function

$$\alpha=\tan A=\sin(2B)+\sin(2C)-\sin(2A).$$

References

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

Lyness, R. and Veldkamp, G. R. Problem 682 and Solution. Crux Math. 9, 23-24, 1983.