Missouri Journal of Mathematical Sciences

On the Modified Fermat Problem

Zvonko Cerin

Full-text: Open access

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 ${v=1}$.

Article information

Source
Missouri J. Math. Sci., Volume 25, Issue 2 (2013), 153-166.

Dates
First available in Project Euclid: 12 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1384266200

Digital Object Identifier
doi:10.35834/mjms/1384266200

Mathematical Reviews number (MathSciNet)
MR3161631

Zentralblatt MATH identifier
1286.51008

Subjects
Primary: 54H01

Keywords
point semicircle rectangle square distance

Citation

Cerin, Zvonko. On the Modified Fermat Problem. Missouri J. Math. Sci. 25 (2013), no. 2, 153--166. doi:10.35834/mjms/1384266200. https://projecteuclid.org/euclid.mjms/1384266200


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References

  • E. C. Catalan, Théorèmes et problèmes de Géométrie 6e ed., Paris, 1879.
  • Z. Čerin, On the Fermat geometric problem, Poučak, 49 (2012), 1–10 (in Croatian).
  • L. Euler, Various Geometric Demonstrations, New Commentaries of the Petropolitan Academy of Sciences I, (1747/48), 1750, 49–66. (English translation prepared in 2005 by Adam Glover.)
  • F. G. -M., Exercices de Géométrie (6e ed.), Éditions Jacques Gabay, Paris 1991. (Reprint of the 6th edition published by Mame and De Gigord, Paris, 1920.)
  • M. L. Hacken, Sur un théorème de Fermat, Mathesis, 27 (1907), 181, 264.
  • E. Lionnet, Solutions des questions proposées, Nouvelles Annales de Mathématiques, Series 2, Vol 9 (1870), 189–191, \ttwww.numdam.org.
  • E. Sandifer, A forgotten Fermat problem, How Euler Did It, MAA Online, Washington, DC, Dec. 2008, \ttwww.maa.org/news/howeulerdidit.html.