Let $T \in SL(2,\mathbb{C})$ be a normalised Möbius transformation. Then, $$|T(z) - T(w)| =|z-w||T'(z)^{\frac{1}{2}}||T'(w)^{\frac{1}{2}}|$$.
The above is an exercise from Outer Circles by Marden (ex. 1.1).
I was wondering if there is any geometrical significance to the question?
