How can $|z_1-z_2|^2 = |z_1^2+z_2^2-2z_1z_2| $?

Question

My book states that

$$|z_1-z_2|^2 = |z_1^2+z_2^2-2z_1z_2|$$

How is this true? Isn't the property $$ |z_1-z_2|^2 = |z_1|^2+|z_2|^2-2\Re(z_1z_2)$$

Answer 1

Note that for a complex number, $|z|^2=|z^2|$.

Here $z=z_1-z_2$.

More generally, $|z_1z_2|=|z_1||z_2|$. A quick proof can be seen using polar form.

Answer 2

HINT: $|z^2|=|z|^2$. In your case $z=z_1-z_2$.

Answer 3

$$|z_1-z_2|^2 = |(z_1-z_1)^2|=|z_1^2+z_2^2-2z_1z_2|$$

Note that both identities hold, while they look different they are the same expression.