Let $\displaystyle z_1, z_2 $ be nonzero complex numbers such that

$\displaystyle z_1^2 + z_2^2 = \sqrt{2}z_1 z_2 $.

Show that $\displaystyle | z_1 | = | z_2 |$ and find $\displaystyle Arg{(\frac{z_1}{z_2})}$.

I've tried substituting $\displaystyle z = x+yi$ and $\displaystyle z = re^{i\theta}$ to prove this but it didnt work out. I assume there is a geometric or inequality approach to this question?