Let

and

be circles with unequal radii. Prove there exists an inversion that maps

onto

.

So my idea was this:

Proof:

Choose an axis that passes through both centers of the circles (so that the axis contains the diameters of

and

). Without loss of generality, let the radius of

be smaller than the radius of

. Choose a point

on the constructed axis so that

is between point

and

and that

, where

is the diameter of

and

is the diameter of

, both of which are the diameters we chose to lie of the constructed axis (

where lies between and and lies between and ). Then we can verify that

maps

onto

. Q.E.D.

This seems to make sense to me, but I want to know if this is a viable proof or if I have left some aspects or cases out. If the circles share a common center then the proof for that case is easy so I omitted it. Thanks.