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Re: [EMHL] Fuss Theorem for Bicentric Quadrilateral

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  • Juan Salazar
    Dear Prof. Nikolaos: Thank you very much for answer me. Perhaps I did not know how to explain the reason of my question. The method that I developed does not
    Message 1 of 2 , Mar 19, 2003
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      Dear Prof. Nikolaos:
      Thank you very much for answer me.
      Perhaps I did not know how to explain the reason of my question.
      The method that I developed does not use trigonometry. I like resolving with
      the traditional methods of the euclidean geometry
      The method that mentions the Prof. Paul Yiu in his e-book Euclidean
      Geometry pag.159-161 also utilizes trigonometry.
      Do I can use the inversion method?
      Do you know another method without use trigonometry?
      I will thank your answer
      Juan Carlos Salazar

      ----- Original Message -----
      From: "Nikolaos Dergiades" <ndergiades@...>
      To: <Hyacinthos@yahoogroups.com>
      Sent: Tuesday, March 18, 2003 5:52 PM
      Subject: Re: [EMHL] Fuss Theorem for Bicentric Quadrilateral


      > Dear Juan,
      > I don't know the who you are asking but
      > I think that it is a simple ralation.
      > Because since A + C = 180
      > then sin(C/2) = cos(A/2) and hence
      > 1/AI² + 1/CI² = sin²(A/2)/r² + sin²(C/2)/r²
      > = [sin²(A/2) + cos²(A/2)]/r² = 1/r²
      > Similarly for the sum 1/BI² + 1/DI² = 1/r²
      >
      > Best regards
      > Nikolaos Dergiades
      >
      > >Dear friends:
      > >I am new member.
      > >I did read the Fuss Theorem to bicentric quadrilateral in
      > the book
      > >Introduction to Modern Geometry by Levi Shively in the year
      > 1975
      > >without proof.
      > >I have developed in the same year this relation:
      > >1/AI^2+1/CI^2=1/BI^2+1/DI^2=1/r^2
      > >(I: incenter and r: inradii,in ABCD bicentric
      > quadrilateral) and
      > >then I did develop the proof to Fuss theorem with
      > utilization of
      > >power of points and Appolonio's theorem.
      > >I don't know one paper on this relation is writed.
      > >I thing this condition "necesary and sufficent" for the
      > >quadrilateral be a bicentric quadrilateral, when the sides
      > be
      > >tangents to one circle.
      > >Please,
      > >Do you send me, Who develop by first time this above
      > relation?
      > >Thanks
      > >Juan Carlos Salazar
      >
      >
      >
      >
      >
      >
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      >
      >
      >
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