Question

Let A and B be complex numbers such that $\frac{A}{B}+\frac{B}{A}=1$
Prove that the origin and the two points represented by A and B form vertices of an equilateral triangle.

Answer 1

$z_3$ is origin and $\Delta$ being equilateral we have angle at O = ${60}^{\circ }$

Hence rotation about O gives

OB = OA $e^{\pi i/3}$

or $z_2-0=(z_1-0)e^{\pi i/3}$ ...(1)

Rotation about A gives AO = AB$e^{\pi i/3}$

or $0-Z_1=(z_2-z_1)e^{\pi i/3}$ ....(2)

Dividing (1) by (2) , we get $\frac{z_2}{-z_1}=\frac{z_1}{z_2-z_1}$

or $z_2^2-z_1{z}_2=-z_1^2$ or $z_1^2+z_2^2=z_1{z}_2$

or $\frac{z_1}{z_2}+\frac{z_2}{z_1}=1$ or $\frac{A}{B}+\frac{B}{A}=1$