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Re: [EMHL] Locus defined by trilinears' equation

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  • Francois Rideau
    Dear Nikolaos Yes ; In others words, two distinct points have the same tripolar coordinates if they are inverse wrt the circumcircle. So the restriction of f
    Message 1 of 38 , Mar 24, 2005
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      Dear Nikolaos
      Yes ;
      In others words, two distinct points have the same tripolar
      coordinates if they are inverse wrt the circumcircle.

      So the restriction of f or g to the circumcircle and its interior is
      one to one onto the range which is compact since f or g is continuous.

      I just give this riddle so that every one thinks about Ptolemy's
      Theorem as Ptolemy Inequality Theorem and not as Ptolemy Equality
      theorem which is only a cocyclicity
      condition.
      I just have this idea when I read Zbaruh's question.
      Best regards and nice Spring for all Hyacinthos friends
      especially those who live in Tokyo marveling at cherries blossoming!
      François
    • Francois Rideau
      Dear friends I notice the following fact certainly well known for a long time and very easy to prove: Call P any point in the plane of triangle ABC and L the
      Message 38 of 38 , Apr 22, 2005
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        Dear friends
        I notice the following fact certainly well known for a long time and very
        easy to prove:

        Call P any point in the plane of triangle ABC and L the trilinear polar of P
        wrt ABC.
        Call Gamma the inscribed conic in ABC of which P is the perspector or
        Brianchon point, i.e : P is the perspector of ABC and the contact triangle.
        Let M be any point on Gamma and call T the tangent in M at Gamma.
        Then the trilinear pole O of T is on L.
        From this fact can we find :
        1°a simple projective construction of the tangent T of M at Gamma.
        2°If O is any point on L, a simple projective construction of the contact
        point M of the trilinear polar T of O wrt ABC with Gamma.
        Thanks for your swift replies
        François


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