Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that $$a_1^3+a_2^3+a_3^3+a_4^3=0$$ $$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$ $$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^2+2|a_2|^4)a_2+(1+2|a_3|^2+2|a_3|^4)a_3+(1+2|a_4|^2+2|a_4|^4)a_4=0.$$
Could someone help me to prove that $|a_1|=|a_2|=|a_3|=|a_4|=R$ and $\{a_1,a_2,a_3,a_4\}$ forms a regular polygon on $|z|=R?$
I believe it is true. However, could you give me an example if it is not true? Thank you so much,
Best Regards, Mash