Thanks for this solution! Although the only thing I do not understand is how you go from |1-1/z|/|z-1| to |1/z||z-1|/|z-1|One way is by similar triangles.
We suppose that are distinct points (otherwise the question isn't very well defined).
In particular, . Then,
and we are done.
As a bonus, this doesn't require restrictions on and you also get equalities for the other two angles of the triangles.