Three unit complex numbers with zero sum

1. Let $u=\dfrac{z+w}{1+zw}$. Using $\bar z=\frac1z$ and $\bar w=\frac1w$:
$$\bar u=\frac{\frac1z+\frac1w}{1+\frac1{zw}}=\frac{z+w}{1+zw}=u.$$
So $u=\bar u$, i.e. $u$ is real.

2. From $z_1+z_2=-z_3$ we get $|z_1+z_2|^2=1$. Expanding,
$$|z_1+z_2|^2=2+2\,\mathrm{Re}(z_1\bar z_2)=1,$$
so $\mathrm{Re}(z_1\bar z_2)=-\frac12$. Then
$$|z_1-z_2|^2=2-2\,\mathrm{Re}(z_1\bar z_2)=3.$$
By symmetry, $|z_2-z_3|^2=|z_3-z_1|^2=3$. The three sides all have length $\sqrt3$: equilateral.

Lesson. For points on the unit circle use $\bar z=1/z$: it turns geometric conditions into algebraic identities.