I am trying to prove that

$\displaystyle |z_1+z_2+...+z_n| = |z_1| + |z_2| + ... |z_n|$

iff $\displaystyle z_i/z_j$ is a positive real number $\displaystyle \forall$ integers i and j, s.t. $\displaystyle i,j \in $ $\displaystyle \left\{ 1,...,n \right\}$

I really don't see how these two ideas imply each other. After looking at this for several hours the only thing I managed to come up with (which I'm sure could be extended to n variables) is

$\displaystyle |z_1|^2 + m|z_2|^2 = |z_1+z_2|^2$ where $\displaystyle m \in \mathbb{R}$

however, that real number m is not always $\displaystyle z_i/z_j$

Can anyone offer me advice please?