Dear Wilson

Sorry to have misunderstood you and your notations.

I like the way you look at orthology but I fear you can't generalize it to

other centers than the centroid and the orthocenter.

Of course I knew that the centroid was the only affine center but I wanted

to emphasize this G-logic property was just a special case of a more general

affine property.

Forget the euclidian structure for a while and look at 2 non homothetic

triangles ABC and A'B'C' in the affine plane and call f the affine map

sending ABC to A'B'C'.

Then the locus of the point M such that the lines through A', B', C'

respectively parallel to the lines AM, BM, CM are on a point M' is a conic

Gamma through A, B, C, the locus of M' beeing the conic Gamma' = f(Gamma)

through A', B', C'. Moreover Gamma and Gamma' have the same points at

infinity, to say the fixed points at infinity of the extended projective map

of f.

Now back to the euclidian structure, if ABC and A'B'C' were X(n)-logic, that

is equivalent to say that X(n)(ABC) is on Gamma and X(n)(A'B'C') is on

Gamma' .

This is the case for G-logic and H-logic triangles for different reasons:

1° In case of the centroid, if G is on Gamma, then G' = f(G) is on Gamma' =

f(Gamma).

2° In case of the orthocenter, if H is on Gamma, then Gamma is a rectangular

hyperbola and Gamma' is also a rectangular hyperbola and H' is on Gamma'.

but I find it hard to swallow for other centers with no special geometrical

properties as affine property for f in case of G or symmetric property of

the associated linear map of f in case of H.

Of course, this is not a proof and I would be happy if there is such a

proof!

Friendly

François

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