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correct Blaschke formula (Re: [EMHL] Area of a quadrilateral)

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  • Sergei Markelov
    Hello Marcello, The formula you are asking for is the following. Let ABCD be (not nesessary convex) quadrilateral with sides AB=k, BC=l, CD=m, DA=n and angles
    Message 1 of 2 , Sep 30, 2002
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      Hello Marcello,

      The formula you are asking for is the following. Let ABCD be (not
      nesessary convex) quadrilateral with sides AB=k, BC=l, CD=m, DA=n and
      angles 2a, 2b, 2c, 2d at vertices A,B,C,D respectively; and let S be the
      area of the quadrilateral.

      Then:

      4*S = (k+l+m+n)^2/(cot(a)+cot(b)+cot(c)+cot(d)) -
      (k-l+m-n)^2/(tan(a)+tan(b)+tan(c)+tan(d))

      As far as I know, this formula was first discovered by german
      mathematition Wilhelm Blaschke. First appeared in 1914 in:

      Jahresbericht Deutsch. Mathem. Vereinigung 23, 1914, page 210-234

      Plain geometry proof and several computer-based proofs of this formula can
      be found in the article:

      "Russian killer No. 2, A challenging geometric theorem with human and
      machine proofs" by Xiaorong Hou, Hongbo Li, Dongming Wang and Lu Yang
      Mathematical Intelligencer, Volume 23, number 1, 2001, pages: 9-15

      It can be downloaded from
      http://groups.yahoo.com/group/Hyacinthos/files/killers.pdf

      Size of file = 1 387 554 bytes, PDF version 1.2 (so you will need Adobe
      Acrobat Reader version 3.0 or later).

      Sergei Markelov
    • Paul Yiu
      Dear Sergei, Since you were the one who sent the Intelligencer article authors the 5 Russian killers, can you post the 5 problems here please? Best regards
      Message 2 of 2 , Sep 30, 2002
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        Dear Sergei,

        Since you were the one who sent the Intelligencer article authors the 5
        Russian killers,
        can you post the 5 problems here please?

        Best regards
        Sincerely,
        Paul

        ==============
        Hyacinthos 2979
        From: yiu@...
        Date: Thu Jun 7, 2001 11:07 am
        Subject: 5 Russian killers


        Dear friends,

        A recent issue of Mathematical Intelligencer* contains an
        article by 4 Chinese authors on Number 2 of 5 Russian killers.
        They
        wrote that ``these theorems have been used to prepare the
        Moscow team
        for the all-Russian school mathematics olympiad, and are called
        killers to analytic ways of geometric problem-solving''. Here is
        Number 2.

        Theorem. Let ABCD be an arbitrary quadrilateral with sides AB
        = k, BC
        = ell, CD = m, DA = n, and internal angles 2a, 2b, 2c, 2d at
        vertices
        A, B, C, D respectively; and let S be the area of the
        quadrilateral.
        Then

        4S = (k+ell+m+n)^2/F - (k-ell+m-n)^2/G,

        where

        F = cot a + cot b + cot c + cot d,

        G = tan a + tan b + tan c + tan d.

        Does any one know what the other killers are?

        Best regards
        Sincerely,
        Paul

        *volume 23 (2001) 9 -- 15.
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