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The lemniscate of Bernoulli, without formulas. (English)
Math. Intell. 36, No. 4, 47-50 (2014).
“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 {\it 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.
Reviewer: Rolf Riesinger (Wien)
Classification: G70