This is from a collection book of problems on complex variables (Volkovyskii, Lunts, Aramanovich).
I don't know how to tackle it without involving heavy unpromising calculations:
Prove that both values of $\sqrt{z^2-1}$ lie on the straight line passing through the coordinate origin and parellel to the bisector of the internal angle of the triangle with vertices at points $-1$ and $1$ and $z$, which passes through the vertex z.
What would be an elegant solution to this problem? Or a promising way to approach it?
Thank you.