Abstract
The concepts of a Fermat base of a plane curve or a Fermat locus of a quadrangle come from an old geometric problem by Pierre de Fermat about a semicircle on a side of a rectangle with ratio of adjacent sides equal to \({\sqrt{2}}\) , which was resolved by synthetic methods first by Leonard Euler in 1750. An arbitrary quadrangle has a plane curve of order four as its Fermat locus. Conics are Fermat loci of trapeziums. Conversely, for every conic one can ask to find all of its Fermat bases. We give answers separately for parabolas, hyperbolas, ellipses and circles treating standard classes of special trapeziums: parallelograms, rhombi, rectangles and squares.
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