If  $z_1$ and $z_2$ are two non zero complex numbers such that ${|z_1+z_2|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$ then 

${|z_1+z_2|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$

$(z_1+z_2)(\overline{z_1}+\overline{z_2})={\left|z_1\right|}^2+{\left|z_2\right|}^2$

$z_1\overline{z_1}+\overline{z_1}{z}_2+z_1\overline{z_2}+z_2\overline{z_2}={\left|z_1\right|}^2+{\left|z_2\right|}^2$

${\left|z_1\right|}^2+\overline{z_1}{z}_2+z_1\overline{z_2}+{\left|z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$

$\overline{z_1}{z}_2+z_1\overline{z_2}=0$

$z_1\overline{z_2}$ is purely imaginary

$\frac{z_1}{z_2}{\left|z_2\right|}^2$ is purely imaginary

$\frac{z_1}{z_2}$ is purely imaginary.

$arg\left(\frac{z_1}{z_2}\right)=\frac{\pi }{2}$

$0,z_1,z_2$ form a right angled triangle.