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Re: [EMHL] A special line in quadrilateral

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  • yakub.aliyev
    Dear friends, Nikolaos Dergiades proposed the following corrected construction. This construction completely resolves the problem. Construction by Nikolaos
    Message 1 of 3 , Mar 10, 2010
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      Dear friends, Nikolaos Dergiades proposed the following corrected construction. This construction completely resolves the problem.

      Construction by Nikolaos Dergiades:
      "So if ABCD is a convex quadrilateral
      with diagonal intersection O and
      OA = a, OB = b, OC = c, OD = d
      you want to construct a line AK (K on AB)
      such that f = sin(KOB)/sin(KOA)
      and f^2 = (1/b^2 + 1/d^2)/(1/a^2 + 1/c^2).
      So my construction is modified as
      Let D' be the rotation of D about O by angle 90.
      Let P be the orthogonal projection of O on BD'.
      Let B' be a point on the cemiline OB such that OB' = OP
      Similarly
      Let C' be the rotation of C about O by angle 90.
      Let Q be the orthogonal projection of O on AC'.
      Let A' be a point on the cemiline OA such that OA' = OQ.
      If S is the mid-point of A'B' then the line OS meets
      AB at the point K."

      Thank you Nikolaos!
      Yakub Aliyev

      --- In Hyacinthos@yahoogroups.com, Nikolaos Dergiades <ndergiades@...> wrote:
      >
      > Dear Yakub,
      > I think that in your study you are
      > interested for f = sin(KOA)/sin(KOB)
      > and not for f = sin(KOB)/sin(KOA).
      > I don't have an explicit geometric meaning
      > but a geometric construction.
      >
      > Let D' be the rotation of D about O by angle 90.
      > Let P be the orthogonal projection of O on BD'.
      > Let A' be a point on the cemiline OA such that OA' = OP
      > Similarly
      > Let C' be the rotation of C about O by angle 90.
      > Let Q be the orthogonal projection of O on AC'.
      > Let B' be a point on the cemiline OB such that OB' = OQ.
      > If S is the mid-point of A'B' then the line OS meets
      > AB at the point K.
      >
      > Best regards
      > Nikos Dergiades
      >
      >
      > > Dear friends, I ask you to find
      > > explicit geometric meaning for the following algebraically
      > > defined construction of special line through intersection
      > > point O of diagonals AC and BD of convex quadrilateral ABCD.
      > > Let K and M be points on sides AB and CD, respectively, so
      > > that KM passes through O. Suppose that
      > > f=sin(angle(KOB))/sin(angle(KOA)). Let
      > > f^2 = (1/BO^2+1/DO^2)/(1/AO^2+1/CO^2). Are there any
      > > explicit geometric construction for this line KM? This line
      > > appeared in my study of some inequality in quadrilateral.
      > > See [1, section 3].
      > >
      > > [1] Y. N. Aliyev, New inequalities on triangle areas,
      > > Journal of Qafqaz University, Number 25, 2009, 
      > > 129-135. (See the link:
      > > http://www.qafqaz.edu.az/journal/20092516nev.pdf )
      >
      >
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