Modulus of Sum equals Modulus of Distance implies Quotient is Imaginary

Theorem
Let $z_1$ and $z_2$ be complex numbers such that:
 * $\cmod {z_1 + z_2} = \cmod {z_1 - z_2}$

Then $\dfrac {z_2} {z_1}$ is wholly imaginary.

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

Then:

Now we have:

But we have:
 * $x_1 x_2 + y_1 y_2 = 0$

Thus:
 * $\dfrac {z_1} {z_2} = \dfrac {i \paren {x_2 y_1 - x_1 y_2} } { {x_2}^2 + {y_2}^2}$

which is wholly imaginary.