Lemniscate of Bernoulli is Special Case of Ovals of Cassini/Geometric Proof

Theorem

The lemniscate of Bernoulli is a special case of the ovals of Cassini.

Proof

The ovals of Cassini are defined as follows:

Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$.

The ovals of Cassini are the loci of points $M$ in the plane such that:

$P_1 M \times P_2 M = b^2$

for a real constant $b$.

The lemniscate of Bernoulli is defined geometrically as:

Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$.

The lemniscate of Bernoulli is the locus of points $M$ in the plane such that:

$P_1 M \times P_2 M = a^2$

It follows that the lemniscate of Bernoulli is an oval of Cassini where $b = a$.

$\blacksquare$