If z _ 1 , z _ 2 and z _ 3 represent the vertices of an equilateral...

Since, $\left|{z}_{1}\right| = \left|{z}_{2}\right| = \left|{z}_{3}\right| .</p><p>$⇒ \left|{z}_{1} - 0\right|$= \left|{z}_{2} - 0\right|$ = \left|{z}_{3} - 0\right|$.</p><p>Thus,$ the given points are equidistant from origin, and we know that the point which is equidistant from the vertices of a triangle is the circumcentre of the triangle.

∴ 0$is the circumcentre of an equilateral$∆ ABC$~and we also know that for an equilateral triangle the centroid and circumcentre are same.</p><p>$∴ \dfrac{{x}_{1} + {x}_{2} + {x}_{3}}{3} = 0 = \dfrac{{y}_{1} + {y}_{2} + {y}_{3}}{3}$</p><p>where,${z}_{1} = {x}_{1} + i{y}_{1}$.</p><p>$⇒ \dfrac{{x}_{1} + {x}_{2} + {x}_{3}}{3} + i\left(\dfrac{{y}_{1} + {y}_{2} + {y}_{3}}{3}\right) = 0$</p><p>$⇒ \left({x}_{1} + i{y}_{1}\right) + \left({x}_{2} + i{y}_{2}\right) + \left({x}_{3} + i{y}_{3}\right) = 0$</p><p>$⇒ {z}_{1} + {z}_{2} + {z}_{3} = 0 .