Let $C_1, C_2$ be two concentric circles in the extended complex plane. Is it true that if another "circle" $C$ is orthogonal to both $C_1$ and $C_2$, then $C$ must be a line?
I think that this should be true, because at the point of intersection between $C_1$ and $C$, the tangent of $C$ must intersect the center of $C_1$. Similarly, this must also hold for $C_2$. But since the centers of concentric circles are the same, I think that this might constrain $C$ to taking the form of a line. I'm not sure if this is actually true though, or how one would show the statement.