Complex Numbers: Parallel/Collinearity Condition

How do we know if two segments $AB,CD$ are parallel?

Let $a=A, b=B, c=C, d=D$ be complex numbers.

Translate $AB, CD$ so a, c are at the origin.

Then $b'=b-a, d'=d-c$.

Now, if AB, CD are truly parallel, then $AB'CD'$ is all one line with A,C both at the origin. This means that $d'$ is a multiple of $b'$ or that

$\frac{d'}{b'}=\frac{d-c}{b-a}$ is a real number.

This can be rewritten as $\frac{d-c}{b-a}=\overline{\frac{d-c}{b-a}}$, where the bar covers the whole fraction, not just the numerator.