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**Pinkk** Let $\displaystyle p$ and $\displaystyle q$ be circles with unequal radii. Prove there exists an inversion that maps $\displaystyle p$ onto $\displaystyle q$.

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 $\displaystyle \color{red}E$ lies between $\displaystyle \color{red}D$ and $\displaystyle \color{red}E'$ and $\displaystyle \color{red}E'$ lies between $\displaystyle \color{red}E$ and $\displaystyle \color{red}D'$). 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.