Napoleon Triangles

The inner Napoleon triangle is the Triangle $\Delta N_{AB}N_{AC}N_{BC}$ formed by the centers of internally erected Equilateral Triangles $\Delta ABE_{AB}$, $\Delta ACE_{AC}$, and $\Delta BCE_{BC}$ on the sides of a given Triangle $\Delta ABC$. It is an Equilateral Triangle.

The outer Napoleon triangle is the Triangle $\Delta N_{AB}'N_{AC}'N_{BC}'$ formed by the centers of externally erected Equilateral Triangles $\Delta ABE_{AB}'$, $\Delta ACE_{AC}'$, and $\Delta BCE_{BC}'$ on the sides of a given Triangle $\Delta ABC$. It is also an Equilateral Triangle.

See also Equilateral Triangle, Napoleon Points, Napoleon's Theorem

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 60-65, 1967.