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14373Re: [EMHL] Orthologic Triangles

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  • Francois Rideau
    Oct 29, 2006
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      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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