Concentric circles in inversive-geometry

Question

Is it possible to obtain two concentric generalised circles(parallel straight lines) through inversion of two circles w.r.t a third circle?

Answer 1

Some parallel line segments are shown reflected about a unit circle centered at origin as mirror ( feature available in Geogebra). When parallel lines go to $ \infty,$ then their images inside the inversion circle tend to go to the origin, being tangent only at an infinite distance from the farthest points .. a feature of hyperbolic geometry. The images can never be concentric just as in both the Poincare's models ( semi-infinite plane or disk.)

EDIT 1:

Inversions of a set of parallel lines bear same label in the figure above.

A set of parallel lines have co-tangential circular inversions.

These inversions have either to be co-tangential or con-centric but not both at the same time !!

Answer 2

can the two straight lines in the figure be concentric circles in any case ? Or if we have two parallel lines is it possible that they are concentric generalised circles?