Equivalence of Definitions of Lemniscate of Bernoulli

Geometric Definition implies Cartesian Definition
Let $M$ be a lemniscate of Bernoulli by the geometric definition.

Then by definition:

Let $P_1 = \tuple {a, 0}$ and $P_2 = \tuple {-a, 0}$.

Let $p = \tuple {x, y}$ be an arbitrary point of $M$.

We have:

Thus $M$ is a lemniscate of Bernoulli by the Cartesian definition.

Cartesian Definition implies Geometric Definition
Let $M$ be a lemniscate of Bernoulli by the Cartesian definition.

Then by definition:



Thus $M$ is a lemniscate of Bernoulli by the geometric definition.