\begin{definition}[Definition:Lemniscate of Bernoulli/Geometric Definition]
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:
:$P_1 M \times P_2 M = a^2$
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\end{definition}