Napoleon Points

The inner Napoleon point $N$ is the Concurrence of lines drawn between Vertices of a given Triangle $\Delta ABC$ and the opposite Vertices of the corresponding inner Napoleon Triangle $\Delta N_{AB}N_{AC}N_{BC}$. The Triangle Center Function of the inner Napoleon point is

$$\alpha=\csc(A-{\textstyle{1\over 6}}\pi).$$

The outer Napoleon point $N'$ is the Concurrence of lines drawn between Vertices of a given Triangle $\Delta ABC$ and the opposite Vertices of the corresponding outer Napoleon Triangle $\Delta N_{AB}'N_{AC}'N_{BC}'$. The Triangle Center Function of the point is

$$\alpha=\csc(A+{\textstyle{1\over 6}}\pi).$$

See also Napoleon's Theorem, Napoleon Triangles

References

Casey, J. Analytic Geometry, 2nd ed. Dublin: Hodges, Figgis, & Co., pp. 442-444, 1893.

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