For any two complex numbers z1z2 and any two real numbers a and b |az1−bz2|2+|bz1+az2|2 =
|az1−bz2|2+|bz1+az2|2
= (az1−bz2)(a¯¯¯¯¯z1−b¯¯¯¯¯z2)+(az1+bz2)(a¯¯¯¯¯z1+b¯¯¯¯¯z2)
= a2z1z2+b2z2¯¯¯¯¯z2−abz1¯¯¯¯¯z2−abz2¯¯¯¯¯z1+b2z1¯¯¯¯¯z1+a2z2¯¯¯¯¯z2+abz1¯¯¯¯¯z2+abz2¯¯¯¯¯z1
= (a2+b2)(|z1|2+|z2|2) ∴ z ¯¯¯z = |z|2