I have the following exercise: Given two points $P,Q$ inside the unit disk $D:=\{z\in \mathbb{C}:|z|< 1\}$ whose boundary is the unit circle $S^1$. Prove that there exists an inversion with respect to a circle perpendicular to $S^1$ that sends $P$ to $Q$. Note that here we can consider the reflection through a line as a special inversion.
My trying until now: Let $I$ and $R$ be the center and the radius of the circle we want to find. By assumption, we have $IP\cdot IQ=R^2=OI^2-1$ where $O$ is the center of $S^1$. Then since $P,Q$ are given, this equality gives an equation for the coordinates of $I$ (and this equation probably has a lot of solutions).
I am wandering if there is a geometric way to construct such a circle $(I,R)$. Can someone help me? Thanks a lot!
Think I cracked it will work out details later
Just construct the circle (hyperbolic line) through P and Q (see wikipedia poincare disk model)
The points where the line PQ intersects with the line through both ideal points of the hyperbolic line is the center of the circle you are looking for
Then with the geometric mean construction you can find the radius
GOOD Question and a remarkable easy answer (after long puzzling)