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Re: [EMHL] Two Questions

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  • Bernard Gibert
    Dear Eisso, ... this is the Orion transform in Clark s ETC. See also the thread : Another stellar (or flowered) transformation, Hyacinthos #7999 & sq. The
    Message 1 of 4 , Mar 17, 2008
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      Dear Eisso,

      > [EJA] as I was playing with some Cevian triangles I noticed the
      > following:
      >
      > I) Let A'B'C' be the Cevian triangle of a point P w.r.t. a triangle
      > ABC. Now let P[A] be the reflection of P in B'C' and P[B] and P[C]
      > defined analogously. Then AP[A], BP[B], and CP[C] are concurrent in a
      > point P', thus defining a (birational) transformation P to P'.
      >
      > II) The pedal points of the altitudes of A'B'C' each lie on the
      > corresponding cevian XP[X], i.e. the pedal point of the altitude
      > from A'
      > on B'C' lies on AP[A] and so on.
      >
      > With regard to I), I would like to know if this transformation has
      > been
      > studied at all and perhaps even has a name. Regarding II), I am just
      > wondering if this property is known.


      this is the Orion transform in Clark's ETC.

      See also the thread :

      Another stellar (or flowered) transformation, Hyacinthos #7999 & sq.

      The fixed points of this transformation are the CPCC points in

      http://pagesperso-orange.fr/bernard.gibert/Tables/table11.html

      Best regards

      Bernard



      [Non-text portions of this message have been removed]
    • jpehrmfr
      Dear Eisso ... triangle ... in a ... from A ... been ... just ... Look at #8039; you ll see a proof of your results (your point P is the isogonal conjugate
      Message 2 of 4 , Mar 17, 2008
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        Dear Eisso
        > as I was playing with some Cevian triangles I noticed the following:
        >
        > I) Let A'B'C' be the Cevian triangle of a point P w.r.t. a
        triangle
        > ABC. Now let P[A] be the reflection of P in B'C' and P[B] and P[C]
        > defined analogously. Then AP[A], BP[B], and CP[C] are concurrent
        in a
        > point P', thus defining a (birational) transformation P to P'.
        >
        > II) The pedal points of the altitudes of A'B'C' each lie on the
        > corresponding cevian XP[X], i.e. the pedal point of the altitude
        from A'
        > on B'C' lies on AP[A] and so on.
        >
        > With regard to I), I would like to know if this transformation has
        been
        > studied at all and perhaps even has a name. Regarding II), I am
        just
        > wondering if this property is known.

        Look at #8039; you'll see a proof of your results (your point P' is
        the isogonal conjugate of P wrt the orthic triangle of A'B'C')
        Clark Kimberling named P->P' the Orion transformation and the first
        result above was named the Begonia theorem by Darij Grinberg who gave
        a proof available at
        http://de.geocities.com/darij_grinberg/Begonia.zip
        The transformation is rational but not birational (in fact there are
        7 points P -not necessarily real- with a given image P')
        Friendly. Jean-Pierre
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