Question

If $z_1+z_2+z_3=0$ and $\left|z_1\right|=\left|z_2\right|=\left|z_2\right|=1$, then prove that the complex numbers $z_1,z_2,z_3$ are the vertices of an equilateral triangle inscribed in the unit circle.

Answer 1

$\begin{array}{c}{z}_1+z_2+z_3=0\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1\\ \frac{z_1+z_2+z_3}{3}=0\end{array}$

Centro id of triangle is $(0,0)$.

Also, cir cum center of triangle is $(0,0)$ as $z_i$ lies on circle $\left|z_1\right|=1$.

Circumcentre centro id of triangle coincide

Hence, triangle is equilateral.