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November 2013 On the Modified Fermat Problem
Zvonko Cerin
Missouri J. Math. Sci. 25(2): 153-166 (November 2013). DOI: 10.35834/mjms/1384266200


$\newcommand{\p}[1]{#1'}$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 ${v=1}$.


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Zvonko Cerin. "On the Modified Fermat Problem." Missouri J. Math. Sci. 25 (2) 153 - 166, November 2013.


Published: November 2013
First available in Project Euclid: 12 November 2013

zbMATH: 1286.51008
MathSciNet: MR3161631
Digital Object Identifier: 10.35834/mjms/1384266200

Primary: 54H01

Keywords: distance , point , rectangle , semicircle , square

Rights: Copyright © 2013 Central Missouri State University, Department of Mathematics and Computer Science


Vol.25 • No. 2 • November 2013
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