The complex numbers $z_1,z_2$  and $z_3$  satisfying $\frac{z_1-z_3}{z_2-z_3}=\frac{1-i\sqrt{3}}{2}$  are the vertices of a triangle which is:

$\left|\frac{z_1-z_3}{z_2-z_3}\right|=\left|\frac{1}{2}-i\frac{\sqrt{3}}{2}\right|=\sqrt{\frac{1}{4}+\frac{3}{4}}=1$ ,

So,       $\left|z_1-z_3\right|=\left|z_2-z_3\right|$

Also, $amp\left(\frac{z_1-z_3}{z_2-z_3}\right)={\tan }^{-1}\left(\frac{-\sqrt{3}/2}{1/2}\right)$

$={\tan }^{-1}\left(-\sqrt{3}\right)=-\frac{\pi }{3}$

or     $amp\left(\frac{z_2-z_3}{z_1-z_3}\right)=\frac{\pi }{3}$

or       $\angle z_2{z}_3{z}_1={60}^0$

$\therefore$      The triangle has two sides equal and the angle between the equal sides ${60}^0$ . So, it is equilateral.