I'm doing self study with the book "Continuous Symmetry," and I had one question about proving the incidence axioms for the Poincare disk.
I'm to prove, using analytic geometry, that the circle $C$ meets the unit circle orthogonally iff $$ C := \{(x,y) : (x - a)^2 + (y - b)^2 = a^2 + b^2 - 1\} $$
I was able to prove the necessary portion, yet the authors say I need to use some possibly messy algebra to get the orthogonality from $$ r^2 = a^2 + b^2 -1. $$ I see I could use the converse of the pythagorean theorem to get the right angles, but I feel like I'm missing an easier solution that the authors intended.
Any hints would be appreciated.
I don't know where the "possibly messy algebra" comes in. This seems pretty straightforward:
$$\overline{OT}\perp\overline{PT} \quad\iff\quad r^2 + 1^2 = a^2 + b^2$$