Generalized Apollonius Circles As Equioptic Curves Of Circles In Constants Curvature Geometries

We extend the old definition of the Apollonius circle in such a way that it results in the same curve in Euclidean geometry but will be more convenient in hyperbolic and spherical geometries. We show that there exists an Apollonius circle of the centers of two circles that coincides with their equioptic curves, as in Euclidean geometry.