Question

True or false : If complex numbers $Z_1,Z_2\text{and}{Z}_3$ represent the vertices of an equilateral triangle such that $|Z_1|=|Z_2|=|Z_3|$, then $Z_1+Z_2+Z_3=0$.

Answer 1

$Z_1,Z_2\text{and}{Z}_3$ represent the vertices of an equilateral triangle such that $|Z_1|=|Z_2|=|Z_3|$$\therefore Z_1,Z_2\text{and}{Z}_3$ are eqidistant from origin.

So, O is circumcentre of $\Delta ABC$. As $\Delta ABC$ is an equilateral triangle ,so circumcentre and centroid coincide.

Centroid of $\Delta ABC=0$
$\Rightarrow \frac{Z_1+Z_2+Z_3}{3}=0$
$\therefore Z_1+Z_2+Z_3=0$

Hence, the given statement is true .