Lemniscate of Bernoulli as Locus in Complex Plane

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

is a lemniscate of Bernoulli.

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