Condition on Conjugate from Real Product of Complex Numbers

Theorem
Let $z_1, z_2 \in \C$ be complex numbers such that $z_1 z_2 \in \R_{\ne 0}$.

Then:
 * $\exists p \in \R: z_1 = p \overline {z_2}$

where $\overline {z_2}$ denotes the complex conjugate of $z_2$.

Proof
Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.

As $z_1 z_2$ is real:
 * $(1): \quad z_1 z_2 = x_1 x_2 - y_1 y_2$

and:
 * $(2): \quad x_1 y_2 + y_1 x_2 = 0$

So:

So $z_1 / \overline {z_2} = p$ where $p = \dfrac {x_1 x_2 - y_1 y_2}{x_2^2 + y_2^2}$, which is real.