On the Modified Fermat Problem

Abstract

$\newcommand{\p}[1]{#1'}$For a given positive real number $v$ smaller than $\sqrt{2}$, we consider the Fermat-like configuration consisting of a circle $k$ and a rectangle ${AB\p B\p A}$. A point ${P}$ is on ${k}$ if and only if the relation ${|AD|^2+|BC|^2=v^2\,|AB|^2}$ holds, where ${C}$ and ${D}$ are the intersections of the line ${AB}$ with the lines ${\p AP}$ and ${\p BP}$, respectively. There are four such rectangles with the side ${A\p A}$ parallel to any given line of symmetry of the circle. This property is shared by all ellipses. When ${v={\sqrt{2}}}$, analogous statements hold for parabolas. Finally, for ${v\gt{\sqrt{2}}}$, this is true for hyperbolas only for its line of symmetry containing the foci. We also show that many geometric properties of this configuration do not depend on a position of a point on the circle. The original Fermat problem corresponds to the case ${v=1}$.