Lemniscate of Bernoulli as Locus in Complex Plane

Theorem
The locus of $z$ on the complex plane such that:
 * $\cmod {z - a} \cmod {z - a} = a^2$

is a lemniscate of Bernoulli.

Proof
which is the equation defining a lemniscate of Bernoulli with major semiaxis $a \sqrt 2$.