If $\eta$ is a fifth root of unity and $i,j,k$ are integers, then
$$ |\eta^i+\eta^j+\eta^k|=|\eta^{i+j}+\eta^{i+k}+\eta^{j+k}| \tag{1} $$
I'm not satisfied with the proof I have so far, which is as follows : first, the identity is trivial if $\eta=1$, so we can assume that $\eta$ is a primitive fifth root of unity. Next, we may consider $i,j,k$ to be values modulo $5$. The identity at $(\eta,i,j,k)$ is equivalent to the identity at $(\eta,0,j-i,k-i)$, so we can assume $i=0$. Also, if $j\neq 0$, then the identity at $(\eta,0,j,k)$ is equivalent to the identity at $(\eta^j,0,1,\frac{k}{j})$, so we may assume $j\in \lbrace 0,1 \rbrace$.
We are now left with ten cases (since $i=0$, and there are two possible values for $j$, five possible values for $k$) which can be checked one by one.
But surely, a simple identity like (1) must have better and less painstaking proofs ? Perhaps it has illuminating generalizations also ?
One has $$|u+v+w|=|uv+uw+vw|$$ whenever $|u|=|v|=|w|=1$. This is because $$|uv+uw+vw|=|u||v||w||u^{-1}+v^{-1}+w^{-1}| =|\overline u+\overline v+\overline w|= |\overline{u+v+w}|=|u+v+w|.$$