Rotation in complex plane

If, for complex z and w, $\displaystyle |z+w|=|z-w|$, show that $\displaystyle \arg z$ and $\displaystyle \arg w$ differ by $\displaystyle \pi /2$.

I have an idea how to solve this. Drive out $\displaystyle a/b=-y/x$ from the initial equation (where $\displaystyle z=a+bi,w=x+yi$), then do something with rotating tangents around 90 degrees blahblah. The problem is I don't really know how to put this down on paper as I get kinda lost when it comes to these funky substitutions into arctan functions. Or maybe there is another way to do this?

(We didn't do vectors yet, so I should perhaps do without them, but any hint you can drop on dot and cross products and whatnot will be appreciated.) Thanks!