Loading ...
Sorry, an error occurred while loading the content.
 

Gergonne problem locus?

Expand Messages
  • Floor van Lamoen
    Dear all, In his paper on the Gergonne problem in FG Nikolaos Dergiades showed how to find a point O in a given plane pi such that AO+BO+CO was minimal. Now
    Message 1 of 2 , Jul 4, 2001
      Dear all,

      In his paper on the Gergonne problem in FG Nikolaos Dergiades showed how
      to find a point O in a given plane \pi such that AO+BO+CO was minimal.
      Now if we take for \pi different planes parallel to the plane of ABC,
      and we let O' be the orthogonal projection of O to ABC, can anything be
      said about the locus of O', except of course that it passes through the
      Fermat point?

      Kind regards,
      Sincerely,
      Floor.
    • jean-pierre.ehrmann@wanadoo.fr
      Dear Floor and other Hyacinthists, ... showed how ... minimal. ... ABC, ... anything be ... the ... O must be barycentric (1/OA, 1/OB, 1/OC); As OA^2 - O A^2
      Message 2 of 2 , Jul 4, 2001
        Dear Floor and other Hyacinthists,
        Floor wrote :

        > In his paper on the Gergonne problem in FG Nikolaos Dergiades
        showed how
        > to find a point O in a given plane \pi such that AO+BO+CO was
        minimal.
        > Now if we take for \pi different planes parallel to the plane of
        ABC,
        > and we let O' be the orthogonal projection of O to ABC, can
        anything be
        > said about the locus of O', except of course that it passes through
        the
        > Fermat point?
        >

        O' must be barycentric (1/OA, 1/OB, 1/OC);
        As OA^2 - O'A^2 = OB^2 - O'B^2 = OC^2 - O'C^2,
        a little elimination leads to the quintic
        a^2.y.z.(y-z).(y.z + x^2) + circular = 0
        Friendly. Jean-Pierre
      Your message has been successfully submitted and would be delivered to recipients shortly.