Napoleon's original theorem refers to arbitrary triangles in the
Euclidean plane. If equilateral triangles are externally erected on the sides
of a given triangle, then their three corresponding circumcenters form an
equilateral triangle. We present some analogous theorems and related statements
for the isotropic (Galilean) plane.
Some theorems from inversive and Euclidean circle geometry are extended to all affine Cayley-Klein planes. In particular,
we obtain an analogue to the first step of Clifford’s chain of theorems, a statement related to Napoleon’s theorem, extensions
of Wood’s theorem on similar-perspective triangles and of the known fact that the three radical axes of three given circles
are parallel or have a point in common. For proving these statements, we use generalized complex numbers.