Apollonius problem in terms of oriented circles

Abstract: The solution of Apollonius' problem on constructing a circle (line), tangent to three given circles (lines), is presented in terms of oriented circles and inversive invariants. Tangency is understood as the coincidence of tangent vectors at the common point, in contrast to counter-tangency. The problem has 0, 1 or 2 solutions. By reversing each of the given circles one by one, we obtain the remaining solutions of the classical non-oriented problem.