Question

Show that the points in the Argand diagram represented by the complex numbers $2+2i,-2-2i,-2\sqrt{3}+2\sqrt{3}i$ are the vertices of an equilateral triangle.

Answer 1

Let points be ,

$z_1=2+2i$ , $z_2=-2-2i$ , $z_3=-2\sqrt{3}+2\sqrt{3}i$

For equilateral triangle,

By rotation concept,

$\frac{z_1-z_2}{z_1-z_3}=e^{\frac{i\pi }{3}}$ ........... ....... ........ (1)

So,

$⟹\frac{2+2i+2+2i}{2+2i+2\sqrt{3}-2\sqrt{3}i}$

$⟹\frac{2+2i}{(1+\sqrt{3})+(1-\sqrt{3})i}$

$⟹\frac{1+i}{(1/2+\sqrt{3}/2)-(-1/2+\sqrt{3}/2)i}$

$⟹\frac{e^{\frac{i\pi }{4}}}{e^{\frac{-i\pi }{12}}}$

$⟹e^{\frac{\pi }{3}}$

So, eq. (1) , satisfies.

Hence, given points are the vertices of an equilateral triangle.