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Re: Two Questions

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  • 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 1 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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