z and w are two complex numbers prove the relationship

Question

If $z$ and $w$ are complex numbers such that $|z+w| = |z-w|$

Prove that $\arg z - \arg w = \pm \ \pi/2$

Can someone please help me?

Answer 1

$|z+w|^2 = |z|^2+2 \operatorname{re} \overline{z} w + |w|^2$.

$|z-w|^2 = |z|^2-2 \operatorname{re} \overline{z} w + |w|^2$.

If the two are equal, we must have $\operatorname{re} \overline{z} w = 0$.

What does that say about $\operatorname{arg} \overline{z} w$?