- 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,

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