Here is the equation to prove:

$\displaystyle |a-b|^2=|a|^2-2 re \hat{a} b+|b|^2$

I tried with something like this:

$\displaystyle |z|^2=z \hat{z} $

$\displaystyle z+\hat{z}=2rez $

but i still can not get the right result.

a,b are different complex numbers.