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Mantel * Noyer * Droz-Farny * Goormaghtigh

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  • Jean-Louis Ayme
    Dear Hyacinthists, an article concerning “Mantel * Noyer * Droz-Farny * Goormaghtigh  or the Simson-Wallace generalized’’ has been put on my website.
    Message 1 of 5 , Sep 3, 2012
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      Dear Hyacinthists,
      an article concerning “Mantel * Noyer * Droz-Farny * Goormaghtigh  or the Simson-Wallace generalized’’ has been put on my website.


      http://perso.orange.fr/jl.ayme%c2%a0%c2%a0%c2%a0 vol. 12

      Sincerely
      Jean-Louis

      [Non-text portions of this message have been removed]
    • Alexey Zaslavsky
      Dear Jean-Louis! an article concerning “Mantel * Noyer * Droz-Farny * Goormaghtigh or the Simson-Wallace generalized’’ has been put on my website.
      Message 2 of 5 , Sep 3, 2012
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        Dear Jean-Louis!

        an article concerning “Mantel * Noyer * Droz-Farny * Goormaghtigh or the Simson-Wallace generalized’’ has been put on my website.

        http://perso.orange.fr/jl.ayme vol. 12

        Some years ago C.Pohoata and N.Beluhov independntly found a new proof of Goormaghtigh theorem: the line A'B' touches the inconic with center P. Partially if P is the incenter then This line touches the incircle.

        Sincerely Alexey

        [Non-text portions of this message have been removed]
      • Martín Acosta
        Hello Is this result known? Given a quadrilateral ABCD, and the parabola tangent to its sides, any other tangent to the parabola divides the oposit sides of
        Message 3 of 5 , Sep 4, 2012
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          Hello

          Is this result known?

          Given a quadrilateral ABCD, and the parabola tangent to its sides, any
          other tangent to the parabola divides the oposit sides of ABCD in the
          same ratio.

          Thanks

          Martin
        • Francois Rideau
          Dear Martin Yes; it is known for a long long time. Given a conic Gamma and two of its tangent L and L ; a variable tangent T meets L in m and L in m . Then it
          Message 4 of 5 , Sep 5, 2012
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            Dear Martin
            Yes; it is known for a long long time.
            Given a conic Gamma and two of its tangent L and L'; a variable tangent T
            meets L in m and L' in m'.
            Then it is known for long that the correspondance between m and m' is
            projective or homographic.
            In case Gamma is a parabola, the line of infinity is a tangent; hence
            points at infinity of line L and L' are in correspondence or homologuous,
            that is to say the correspondence between m and m' is affine and so
            preserves ratio.
            Friendly
            Francois


            On Tue, Sep 4, 2012 at 2:21 PM, Mart�n Acosta <maedu@...> wrote:

            > **
            >
            >
            > Hello
            >
            > Is this result known?
            >
            > Given a quadrilateral ABCD, and the parabola tangent to its sides, any
            > other tangent to the parabola divides the oposit sides of ABCD in the
            > same ratio.
            >
            > Thanks
            >
            > Martin
            >
            >


            [Non-text portions of this message have been removed]
          • Barry Wolk
            This also follows from some results about cross-ratio. Cross-ratio is defined originally for 4 collinear points, and then for 4 concurrent lines, then for 4
            Message 5 of 5 , Sep 6, 2012
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              This also follows from some results about cross-ratio. Cross-ratio is defined
              originally for 4 collinear points, and then for 4 concurrent lines, then for 4 points on
              a conic, and then for 4 tangents to a conic. This last case gives the following lemma:
               
              Let lines a,c,e,f,x be 5 tangents to a conic. The cross-ratio R(a,c;e,f) is
                   (a.x - e.x)*(c.x - f.x)/((a.x - f.x)*(c.x - e.x))
              where p.q means the point of intersection of lines p and q, and P - Q means
              the directed distance from point Q to point P. The lemma is that changing x
              to any tangent of this conic does not change the value of R.
               
              If the conic is a parabola, then the line at infinity is a tangent line.
              Put f = the line at infinity, so R simplifies to R' = (a.x - e.x)/(c.x - e.x).
              Then -R' is the ratio in which e.x divides the segment of x between a.x and c.x
               
              Let your quadrilateral have sides a,b,c,d in cyclic order, and let e be the other tangent
              line. Then x=b and x=d both give the same value of -R', which is just your result.
              --
              Barry Wolk

               
              > From: Francois Rideau <francois.rideau@...>
              >
              > Dear Martin
              > Yes; it is known for a long long time.
              > Given a conic Gamma and two of its tangent L and L'; a variable tangent T
              > meets L in m and L' in m'.
              > Then it is known for long that the correspondance between m and m' is
              > projective or homographic.
              > In case Gamma is a parabola, the line of infinity is a tangent; hence
              > points at infinity of line L and L' are in correspondence or homologuous,
              > that is to say the correspondence between m and m' is affine and so
              > preserves ratio.
              > Friendly
              > Francois
              >
              >>
              >> Hello
              >>
              >> Is this result known?
              >>
              >> Given a quadrilateral ABCD, and the parabola tangent to its sides, any
              >> other tangent to the parabola divides the oposit sides of ABCD in the
              >> same ratio.
              >>
              >> Thanks
              >>
              >> Martin
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