If the complex numbers z1,z2 and z3 represent the vertices of an equilateral triangle such that z1=z2=z3, then

Let z1=z2=z3=k (says),⇒z1,z2,z3 lie on a circle with centre at the origin and radius k. As z1,z2,z3 are vertices of an equilateral triangle, the circumcentre and the centroid of the triangle coincide. Therefore 13z1+z2+z3=0⇒z1+z2+z3=0