Isosceles triangle with complex number

Question

We have to Prove that the complex numbers z1 and z2 and the origin form an isosceles triangle with vertical angle 2$\pi$/3 if $z_{1}^{2} + z_{2}^{2}+z_{1}z_{2}=0$

I tried as

$|z_1|=|z_2|$

And

arg(z1)-arg(z2)=2$\pi$/3

Answer 1

Hint: other than the trivial solution $z_1=z_2=0$ we can assume that $z_{1,2} \ne 0$. Multiplying by $z_1-z_2$ gives $z_1^3 - z_2^3 = 0\,$ $\iff$ $\left(\cfrac{z_1}{z_2}\right)^3=1$. It follows that $\left|\cfrac{z_1}{z_2}\right|=1$ and $\arg z_1-\arg z_2=\cdots$