Consider a continous map from $S^2$ to $C$.

Is it true that there exists 3 points equially spaced on a big circle, $x_1,x_2,x_3$, such that if $w$ is the third root of unity, $f(x_1)+wf(x_2)+w^2f(x_3)=0$?

More generally I'm asking this if we take nth unity roots.