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Re: [EMHL] Brocard & Isodynamic points

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  • Alexey.A.Zaslavsky
    Dear Bernard! ... My proof isn t purely synthetic, but the calculations are minimal. We know that the inversion wrt the circumcicle of X(15) is X(16) and this
    Message 1 of 38 , Apr 3, 2005
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      Dear Bernard!
      >
      >let W1, W2 be the Brocard points and X15, X16 the isodynamic points.
      >
      >P1 = W1 X15 /\ W2 X16
      >P2 = W1 X16 /\ W2 X15
      >
      >KP1P2 is an equilateral triangle inscribed in the Brocard circle. (K =
      >X6)
      >
      >any synthetic proof for this fact ?
      >
      My proof isn't purely synthetic, but the calculations are minimal. We know
      that the inversion wrt the circumcicle of X(15) is X(16) and this inversion
      of X(6) is the midpoint of X(15)X(16). Also it is known that the Brocard
      points are in the circle with diameter OX(6) and the angles WiOX(6) are
      equal to the Brocard angle $\phi$ (i=1,2). So using the expressions for
      distance OX(6) and $\phi$ it is easy to obtain that KP1=P1P2.

      Sincerely Alexey




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    • 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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