Dear Jean-Pierre,

> [ND]

> > > I tried to find for which points P the pedal

> triangle

> > > of P A'B'C' has vertices equidistant from A,B,C

> > > i.e. AA' = BB' = CC'.

> > > These points must be symmetric wrt H.

> > > I found that these points must be on these

> lines.

> > > From a sketch it seems to exist only two such

> points

> > > on one of these lines.

> > > Can we construct these points?

>

> [JPE]

> > Your points are the homothetic of the focii of the

> Steiner

> > circumellipse in (L,3/2) where L = de Longchamps

> point.

> > For both points, AA' = 3.T/2 where T = semimajor

> axis of the

> ellipse.

[JPE]

> A little explanation :

> it is well known that the focii of the Steiner

> circumellipse have

> cevian of equal lengths.

> As they lie on the Lucas cubic, their cevian

> triangles are the pedal

> triangles of the required points

Excellent!

Many thanks

Friendly

Nikos

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