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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.

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up vote 5 down vote accepted

This follows directly from the arguments of the numbers involved:


This is the angle at which the bisector at $z$ between the lines from $z$ to $1$ and $-1$ intersects the real axis, since these lines intersect the real axis at angles $\arg(z-1)$ and $\arg(z+1)$, respectively.

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