Dear Jean-Louis, dear Floor and dear friends

Very nice proof, Jean-Louis, written in your inimitable style!

As I cannot sleep, I ask myself if something can be generalized in this

Floor drawing.

So I start with our beloved triangle ABC and draw the cevian triangle A'B'C'

of some point P in the plane and look at the centers of the circles

circumscribed to PBA', PCA', PCB', PAB', PAC', PBC', maybe there was

something to look at and in fact Cabri tells me that these 6 points are on a

same conic <Gamma>.

Of course I was too tired to prove anything, so I let you folks to do the

job and please find for me:

1° the locus of the point P for which <Gamma> is a circle.

For the moment, I can only prove that the centroid G is on this locus,

that's better than nothing!

2° the locus of the point P for which <Gamma> is a rectangular hyperbola.

3° the locus of the point P for which <Gamma> degenerates in 2 lines.

Friendly

Francois

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