Question

If $z_1,z_2,z_3,z_4$ be the vertices of rhombus in argand palne and $\angle CBA=\pi /3$ , then prove that
$2z_2=z_1(1+i\sqrt{3})+z_3(1-i\sqrt{3})$
and $2z_4=z_1(1-i\sqrt{3})+z_3(1+i\sqrt{3})$

Answer 1

Rotation about B in anti-clockwise sense
$z_1-z_2=(z_3-z_2)e^{i\pi /3}$
or $z_1-z_2=(z_3-z_2)\frac{1+i\sqrt{3}}{2}$
$z_1-z_3.$ $\frac{1+i\sqrt{3}}{2}=z_2(1-\frac{1+i\sqrt{3}}{2})$
or $z_1-\frac{z_3}{2}(1+i\sqrt{3})=\frac{z_2}{2}(1-i\sqrt{3})$

Multiplying both sides by $1+i\sqrt{3}$, we get

$\therefore$ $z_1(1+i\sqrt{3})-\frac{z_3}{2}(-2+2i\sqrt{3})=2z_2$
or $z_1(1+i\sqrt{3})+z_3(1-i\sqrt{3})=2z_2$
Similarly we get $2z_4$ by rotation about D.