Prove that $z_1^2 + z_2^2 + z_3^2 = 0$

Question

Let $z_1,\,z_2$ and $z_3$ with the following properties: $z_1+z_2+z_3=0$ and $|z_1|=|z_2|=|z_3|=1$.

Prove that $z_1^2 + z_2^2 + z_3^2 = 0$.

Answer 1

HINT:

Another way: $$0=(z_1+z_2+z_3)^2=z_1^2+z_2^2+z_3^2+2z_1z_2z_3\left(\dfrac1{z_1}+\dfrac1{z_2}+\dfrac1{z_3}\right)$$

Now $\dfrac1{z_1}=\dfrac{\bar{z_1}}{|z_1|^2}=\bar{z_1}$

Also use the fact that if $z=0$ , then $\ \bar{z}=0$.

Answer 2

Hint:

$$ \require{cancel} \begin{align} z_1^2 + z_2^2 + z_3^2 & = \bcancel{(z_1 + z_2 + z_3)^2} - 2(z_1z_2+z_2z_3+z_3z_1) \\[3px] & = -2z_1z_2z_3\left(\frac{1}{z_1}+\frac{1}{z_3}+\frac{1}{z_3}\right) \\[3px] & = -2z_1z_2z_3(\bar z_1+\bar z_2 + \bar z_3) \\[3px] & = -2z_1z_2z_3(\overline{z_1+z_2+z_3}) \end{align} $$