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 \(\displaystyle p\) and \(\displaystyle q\)). Without loss of generality, let the radius of \(\displaystyle p\) be smaller than the radius of \(\displaystyle q\). Choose a point \(\displaystyle C\) on the constructed axis so that \(\displaystyle p\) is between point \(\displaystyle C\) and \(\displaystyle q\) and that \(\displaystyle CD \cdot CD' = CE \cdot CE'\), where \(\displaystyle DE\) is the diameter of \(\displaystyle p\) and \(\displaystyle E'D'\) is the diameter of \(\displaystyle q\), both of which are the diameters we chose to lie of the constructed axis (where \(\displaystyle E\) lies between \(\displaystyle D\) and \(\displaystyle E'\) and \(\displaystyle E'\) lies between \(\displaystyle E\) and \(\displaystyle D'\)). Then we can verify that \(\displaystyle I_{C, \sqrt{CD\cdot CD'}}\) maps \(\displaystyle p\) onto \(\displaystyle q\). 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.