i can do first part of this question. i need help with the last part

find roots of $\displaystyle z^3 - 1 = 0 $

show if one complex root is $\displaystyle w $ show the other is $\displaystyle w^2$

show $\displaystyle Re(\frac{1}{1+w^r}) = 1/2 $ for r=1,2,3

i need help for next part

if $\displaystyle z_1^2 +z_2^2+z_3^2- z_1 z_2-z_2 z_3-z_3 z_1 = 0 $

show

$\displaystyle z_1 = -wz_2 -w^2 z_3 $ or $\displaystyle z_1 = -wz_3 -w^2 z_2 $

deduce three complex numbers are in a equaliteral triangle