Multiply everything by some unimodular complex number (i.e. rotate everything) so that the line becomes the real axis. The result should be easy to show in that setting--then divide by the same number to get the result you want.
I need help with this problem:
Let Zk (k = 1, ... , n) be complex numbers lying on the same side of a straight line passing through the origin. Show that
z1 + z2 + ... + zn ≠ 0 and
1/z1 + 1/z2 + ... + 1/zn ≠ 0