## 14373Re: [EMHL] Orthologic Triangles

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• Oct 29, 2006
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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