## Gergonne problem locus?

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• 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.
• 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
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