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Re: Ratios in a quadrilateral

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  • Darij Grinberg
    Dear Nikolaos, ... Well, I remember having discovered this relation in 2000, when I was quite young. It was so surprising to me that I called it fundamental
    Message 1 of 4 , Jan 16, 2004
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      Dear Nikolaos,

      In Hyacinthos message #9064, you wrote:

      >> If E,F,G,H are points on the sides AB, BC, CD, DA
      >> of a quadrilateral ABCD such that
      >> AE/EB = DG/GC = m
      >> BF/FC = AH/HD = n
      >> then the intersection K of EG, FH gives
      >> HK/KF = m
      >> EK/KG = n

      Well, I remember having discovered this relation in
      2000, when I was quite young. It was so surprising
      to me that I called it "fundamental theorem of
      quadrilateral geometry". My proof used vectors, just
      as Alexey's one; however, it seems that Alexey and
      me have slightly different conceptions of what is
      an "evident" fact :-)

      Concerning the question whether this fact is new,
      it was used as a lemma in the solution of the
      problem 2 in the 1st Round of the Bundeswettbewerb
      Mathematik 1982 and thus published in the book
      "Bundeswettbewerb Mathematik Aufgaben und Lösungen
      1972-82".

      Sincerely,
      Darij Grinberg
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