Question

prove that for any distinct, nonzero complex numbers u and v, the points 0,u,v, and u+v, are the vertices of a parrallelogram

Answer 1

write u = (a,b) and v = (c,d). Then the side 0 to u is a vector
u - 0 = <a,b>
and the opposite side is the vector from v to u+v which is
(u+v) - v = <a,b>

Hence since the two sides have exactly the same vector direction, they must be parallel.

You can now make the same argument for the other two sides: 0 to v, and u to u+v

Answer 2

I should also add that each member of a pair of sides are of equal length