Does $|z_1+z_2|=|z_1|+|z_2|\implies z_1=kz_2$? Let $z_1,z_2\in \mathbb C$. I was wondering if 

$$|z_1+z_2|=|z_1|+|z_2|\implies z_1=k z_2\ \ or\ \ z_2=0,\quad k\in\mathbb R^+.$$

Is it true? 
I am having problems trying to show it. I'm sure it's easy to show using brute force, but if it's true, is there an elegant way to show it?  

Squaring both sides, we see this is equivalent to:
$$(z_1+z_2)(\overline z_1+\overline z_2)=z_1\overline z_1 + 2|z_1||z_2| + z_2\overline z_2$$
which is equivalent to:
$$\mathrm{Re}(z_1\overline z_2)=\frac{1}{2}\left(z_1\overline z_2+z_2\overline z_1\right)=|z_1 \overline z_2|$$
So $z_1\overline z_2=\alpha$ must be  non-negative real. Then you get:
$$z_1|z_2|^2 = \alpha z_2$$
So either $z_2=0$ or $z_1=\frac{\alpha}{|z_2|^2}z_2$.
