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This is Exercise EP $8$ from Fernandez and Bernardes's book Introdução às Funções de uma Variável Complexa (in Portuguese).

If $z,w\in \mathbb{C}^{\times}$, then $|z+w|=|z|+|w|$ iff $w=tz$ for some $t\gt 0.$

I supposed that $|z+w|=|z|+|w|$ and by squaring it I got $\operatorname{Re}(z\bar{w})=|z||w|$ and I am stuck. I would appreciate a hint in this question.

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up vote 2 down vote accepted

The condition $w=tz$ can be restated as a claim about the arguments of $z$ and $w$.

Take $\mathrm{Re}(z\bar{w})=|z| |w|$ and divide by the RHS, then write $z/|z|=e^{i\theta}$ and $w/|w|=e^{i\phi}$.

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