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[CEMI] Re: [EMHL] [CEMI] projective theorem

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  • Alexey.A.Zaslavsky
    Dear Francois! You are right, I proved the theorem by this way. Sincerely Alexey Dear Alexey That reminds me something
    Message 1 of 3 , Nov 1, 2008
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      Dear Francois!
      You are right, I proved the theorem by this way.

      Sincerely Alexey



      Dear Alexey
      That reminds me something already met here but I don't remember who and
      when!
      Maybe Jean-Pierre ?
      If we choose a projective frame ABCP in which P is the unit point that is to
      say an affine chart of the projective plane for which P is center of gravity
      of ABC, then your theorem is equivalent to say that if Gamma is a
      hyperbola through ABC and its center of gravity G, then the tangent to
      Gamma at G meets both asymptots of Gamma in two points situated on the ABC
      Steiner inellipse.
      I note that the center O of Gamma and the symmetric T of O wrt G are also
      on this ellipse.
      Friendly
      Francois

      >
      >

      [Non-text portions of this message have been removed]






      [Non-text portions of this message have been removed]
    • Francois Rideau
      Dear Alexey To sum up your sketch, we start with a triangle ABC and a point P. is the ABC -inconic with perspector P. The tripolar line L of P wrt ABC
      Message 2 of 3 , Nov 1, 2008
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        Dear Alexey
        To sum up your sketch, we start with a triangle ABC and a point P.
        <Gamma> is the ABC -inconic with perspector P.
        The tripolar line L of P wrt ABC is also the polar line of P wrt <Gamma>.
        We look at the pencil (F) of conics through ABCP and at the Desargues
        involution determined by this pencil on L.
        If <gamma> is a conic in (F) cutting line L in Q and R, we get a triangle
        PQR autopolar wrt <Gamma>.
        Then the triangle made up of the tangents at conic <gamma> in P, Q, R is
        inscribed in conic <Gamma>.
        Friendly
        Francois

        On Sat, Nov 1, 2008 at 8:07 AM, Alexey.A.Zaslavsky <zasl@...>wrote:

        > Dear Francois!
        > You are right, I proved the theorem by this way.
        >
        > Sincerely Alexey
        >
        >
        > Dear Alexey
        > That reminds me something already met here but I don't remember who and
        > when!
        > Maybe Jean-Pierre ?
        > If we choose a projective frame ABCP in which P is the unit point that is
        > to
        > say an affine chart of the projective plane for which P is center of
        > gravity
        > of ABC, then your theorem is equivalent to say that if Gamma is a
        > hyperbola through ABC and its center of gravity G, then the tangent to
        > Gamma at G meets both asymptots of Gamma in two points situated on the ABC
        > Steiner inellipse.
        > I note that the center O of Gamma and the symmetric T of O wrt G are also
        > on this ellipse.
        > Friendly
        > Francois
        >
        > >
        > >
        >
        > [Non-text portions of this message have been removed]
        >
        > [Non-text portions of this message have been removed]
        >
        >
        >


        [Non-text portions of this message have been removed]
      • pamfilos
        ... Dear Alexey, Last year I have handled this subject and several other similar questions in my Gallery http://www.math.uoc.gr/~pamfilos/eGallery/Gallery.html
        Message 3 of 3 , Nov 3, 2008
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          Alexey.A.Zaslavsky wrote:
          >
          > Dear Francois!
          > You are right, I proved the theorem by this way.
          >
          > Sincerely Alexey
          >
          > Dear Alexey
          > That reminds me something already met here but I don't remember who and
          > when!
          > Maybe Jean-Pierre ?
          > If we choose a projective frame ABCP in which P is the unit point that
          > is to
          > say an affine chart of the projective plane for which P is center of
          > gravity
          > of ABC, then your theorem is equivalent to say that if Gamma is a
          > hyperbola through ABC and its center of gravity G, then the tangent to
          > Gamma at G meets both asymptots of Gamma in two points situated on the ABC
          > Steiner inellipse.
          > I note that the center O of Gamma and the symmetric T of O wrt G are also
          > on this ellipse.
          > Friendly
          > Francois
          >
          > >
          > >
          >
          > [Non-text portions of this message have been removed]
          >
          > [Non-text portions of this message have been removed]
          >
          >
          > ------------------------------------------------------------------------
          >
          > Internal Virus Database is out-of-date.
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          Dear Alexey,
          Last year I have handled this subject and several other similar
          questions in my Gallery
          http://www.math.uoc.gr/~pamfilos/eGallery/Gallery.html
          It may interest you to have a look there
          I like very much projective geometry
          and handle from time to time subjects
          whose proofs are missing or not accessible to me.
          Best regards
          Paris Pamfilos
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