Napoleon's Theorem in Spherical Geometry
Napoleon's theorem does not hold in spherical geometry, as can be seen be simple example.
For more details please consult the following applet:
Napoleons's Triangle (marked in red) is clearly not regular,
although there might be an upper bound for its edges's difference.
A very interesting approach to find a spherical counterpart of
Napoleon's Theorem is due to Mark Treuden:
Napoleon-Like Properties of Spherical Triangles
Use the left mouse button to rotate the sphere,
use the right mouse button to select on vertex
of the blue triangle and to drag it around on the sphere.
Lienhard Wimmer ( Impressum )