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Under what circumstances does $|z_1 + z_2| = |z_1| + |z_2|$?

My attempt:

Let $z_1, z_2\in \mathbb{C}$ then

$|z_1+z_2|=|z_1|+|z_2|\iff |z_1+z_2|^2=(|z_1|+|z_2|)^2\iff (z_1+z_2)\overline{(z_1+z_2)}=|z_1|^2+2|z_1||z_2|+|z_2|^2\iff (z_1+z_2)(\overline {z_1}+\overline{z_2})=z_1\overline{z_1}+2|z_1z_2|+z_2\overline{z_2}\iff z_1\overline z_2+ z_2\overline{z_1}=2|z_1z_2|$

Here i'm stuck. Can someone help me?

rcoder
  • 4,545
  • Hint: $\overline{a} + a = 2\Re{a}$. (That's the real part of $a$.) Apply this to $z_1\overline{z_2}$. That won't finish the problem for you but it might get you unstuck. – JonathanZ Oct 12 '18 at 02:28
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    Also, it might help to think about $z_1$ and $z_2$ as the sides of a triangle to get a geometric intuition as to what the answer is, even though you should prove it using the properties of complex numbers. – JonathanZ Oct 12 '18 at 02:32
  • Spoiler:

    ! This happens if and only if $z_1$ and $z_2$ have the same polar angle.

     – AlkaKadri Oct 12 '18 at 02:39

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