TITLE: How to Construct orthogonal circles?
QUESTION [7 upvotes]: Let $C_{1}$ be a circle of unit radius. Let A and B be two points inside $C_{1}$. Now I want to construct another circle $C_{2}$ such that A and B lie on $C_{2}$ and $C_{2}$ is orthogonal to $C_{1}$ at their point of intersection(I want $C_{2}$ in such a way that it intersects $C_{1}$). I tried and failed to find a way to construct such a $C_{2}$.
Any help is appreciated.

REPLY [5 votes]: Take the inverse of $A$ with respect to the circle $C_1$ (see here) to obtain the third point $C$. Now construct the circle containing $A,B,C$.