IIT-JEE 2001 ScreeningQuestion 1
The complex numbers $z_1$, $z_2$ and $z_3$ satisfying $\dfrac{z_1-z_3}{z_2-z_3}=\dfrac{1-i\sqrt3}{2}$ are the vertices of a triangle which is
Complex Numbers
Solution
$\frac{z_1-z_3}{z_2-z_3}=e^{-i\pi/3}$ has modulus $1$ and argument $-60^\circ$: $|z_1-z_3|=|z_2-z_3|$ with the angle at $z_3$ equal to $60^\circ$. So the triangle is equilateral.