If some complex numbers lie on the one side of a line which goes through 0, then the sum of them are not 0 Let me describe the problem more specifically.
Suppose that $z_{1},z_{2},\cdots,z_{n}$ are all complex number and they all lie on one side of a straight line passing through $0$. Then $z_{1}+z_{2}+\cdots+z_{n}\neq0$.
I am asked to prove it analytically, even thought it seems quite clear geometrically. So I am thinking that I could use some inequality to prove, but don't have any idea yet..
Thank you very much for any hints or explanations!

Well, you have to give a characterization of those numbers. Since the lines passes through $0$, let's suppose it forms with the positive $x$ semiaxis and angle $\alpha$, what can you say of the $\arg(z_i)$ for all the points on one side and what for those on the other side?
