Just to elaborate on nadeem's post:

If z₁+z₂+z₃ = 0 then we can find complex numbers a, b and c such that z₁=a-b; z₂ = b-c and z₃ = c-a. On the Argand Plane, let these be represented by the points A, B and C. Now, it is easy to see that the angle between any two z_i is the exterior angle at the vertices A, B and C.

 

Let the interior angles be ₁, ₂ and ₃. Now let's assume that none of the external angles are greater than 2/3

Then ₁+₂ < 2/3

       ₂+₃ < 2/3

       ₁+₃< 2/3

 

Summing up, ₁+₂+₃ <  which contradicts ₁+₂+₃ = .

Hence atleast one exterior angle is greater than 2/3.