The lemniscate of Bernoulli, without formulas (Q2341292)

``A polynomial lemniscate with foci \(F_1,F_2,\dots,F_n\) is a locus of points \(X\) such that the product of distances from \(X\) to the foci is constant (\(\prod_{i=1,\dots,n}|F_iX|=\mathrm{const}\)). The \(n\)-th root of this value is called the \textit{radius} of the lemniscate. It is clear that a lemniscate is an algebraic curve of degree (at most) \(2n\)'' (from the text). Using purely synthetic arguments, the author presents three constructions of the Bernoulli lemniscate (\(n=2\), \(\mathrm{const}=(1/4)|F_1F_2|^2\)), one is based on a three-bar linkage invented by James Watt. In the same way it is proved that the Bernoulli lemniscate is an inversion image of an equilateral hyperbola. Finally, a very simple construction of the normal of the Bernoulli lemniscate is described.