I have been stuck on this problem for hours.
I have an ideal hyperbolic triangle with vertices at $A=(a,0), B=(b,0), C=(c,0)$ where $0<a,b,c$.
The question asks to prove that there exists a circle $q$ orthogonal to the x-axis so that $I_q(A)=(0,0)$ and $I_q(C) = \infty$.
Here is what I can deduce from this. The center of $q$ must be $C$. I don't know what I can do with the fact that $I_q(A)=(0,0)$ because $A$ here is arbitrary.