Circle geometry in affine Cayley-Klein planes

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. Supported by a grant D01-761/24.10.06 from the Ministry of Education and Sciences, and by a grant 108/2007 from Sofia University.