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Re: [EMHL] Circumconics

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  • Alexey.A.Zaslavsky
    Dear Paris! The Emelyanov s paper was published (in Russsian) in Matematicheskoe prosveschenije , 2002, N 6. May be it also was published in FG, but I amn t
    Message 1 of 29 , Feb 17, 2008
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      Dear Paris!
      The Emelyanov's paper was published (in Russsian) in "Matematicheskoe prosveschenije", 2002, N 6. May be it also was published in FG, but I amn't sure.

      Sincerely Alexey

      Dear Alexey,
      1) Could I have some concrete reference on Lev and Tatyana's
      presentation?
      2) The theorem you proved with Arsenij is the old and classical way of
      "Maclaurin's generation of conics"
      and have already pictures in my gallery (HYPERLINK
      "http://www.math.uoc.gr/~pamfilos/eGallery/problems/Maclaurin.html"http:
      //www.math.uoc.gr/~pamfilos/eGallery/problems/Maclaurin.html)
      3) This gives of course a direct proof of the proposition I posted.
      4) I prefer though the reduction to parabola and the stress of this
      property of anticomplementary
      and more general the precevian triangle. They clarify greatly I think
      the structure of the
      circumconics tangent to a given line, and especially the circumparabolas
      of a triangle.
      Sincerely Paris



      -----Original Message-----
      From: Hyacinthos@yahoogroups.com [mailto:Hyacinthos@yahoogroups.com] On
      Behalf Of Alexey.A.Zaslavsky
      Sent: Tuesday, February 05, 2008 7:29 AM
      To: Hyacinthos@yahoogroups.com
      Subject: [SPAM] Re: [EMHL] Circumconics

      Dear Paris and Francois!
      It seems that these problems were considered by Lev and Tatyana
      Emelyanoff. They presented the problems devoted to trypolars at Summer
      conference of Towns tournament-2005.
      Also in our with Arsenij Akopyan book next theorem is proved. Given the
      triangle ABC and the point C0. An arbitrary line passing through C0
      intersect AC and BC in B' and A' resp. P is the common point of AA' and
      BB'. Then
      1. The locus of P is the circumconic touching CC0.
      2. If CP intersect AB in C' then all lines A'C' have the common point B0
      and BB0 touches the circumconic.

      Sincerely Alexey

      Dear Paris

      This projective configuration of a conic through 3 points A, B, C and
      tangent at Q to a line (L) is very beautiful!

      I am sure there is a nice proof in the general case using Desargues and
      Plücker
      involution theorems known for a long time that we can find as an
      exercise in
      some old book.

      But the idea to send line (L) at infinity and to prove the theorem in
      this
      special case is very educational.

      Another (educational?-) idea should be to send both points B and C at
      infinity that is to say looking at the hyperbola through point A with
      given
      asymptotic directions and tangent to a line (L) at Q. I am sure that
      elementary affine properties of hyperbola can also do the job.

      Friendly
      Francois

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    • Paris Pamfilos
      Dear Francois thank you for the nice remarks on the alternative description of circumparabolas. Dear Alexei thank you for the reference. Is Matematicheskoe
      Message 2 of 29 , Feb 18, 2008
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        Dear Francois
        thank you for the nice remarks on the alternative description
        of circumparabolas.
        Dear Alexei
        thank you for the reference.
        Is "Matematicheskoe prosveschenije" electronically accessible?
        Do you know?
        Best regards to all
        Paris

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      • Alexey.A.Zaslavsky
        Dear Paris! Yes in www.mccme.ru you can find the numbers of MP but I think that there is only russian text. Sincerely
        Message 3 of 29 , Feb 19, 2008
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          Dear Paris!
          Yes in www.mccme.ru you can find the numbers of "MP" but I think that there is only russian text.

          Sincerely Alexey

          Dear Francois
          thank you for the nice remarks on the alternative description
          of circumparabolas.
          Dear Alexei
          thank you for the reference.
          Is "Matematicheskoe prosveschenije" electronically accessible?
          Do you know?
          Best regards to all
          Paris

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