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Re: [EMHL] Another One Is Sum Of Other Two Cubics

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  • Bernard Gibert
    Dear Quang Tuan Bui, ... Don t you think it s better to say that the three signed distances sum up to zero, the triangle ABC being arbitrarily orientated. ...
    Message 1 of 3 , Jun 27, 2007
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      Dear Quang Tuan Bui,

      > [QTB] Given triangle ABC, one point P with pedal triangle PaPbPc. X
      > is any point with Cevian triangle XaXbXc. We consider to find the
      > locus of X such that three segments XaPa, XbPb, XcPc hold a
      > property: one is sum of other two.

      Don't you think it's better to say that the three signed distances
      sum up to zero, the triangle ABC being arbitrarily orientated.

      > [QTB] The result:
      > If barycentrics of P = (p : q : r), X = (x : y : z) then the locus
      > is one cubic with equation:
      >
      > CyclicSum[a*b*r*((SA*x - SB*y)*(x*y + z^2) + (a - b)*c*x*y*(x + y)
      > - 2*sc*(sb*x^2 - sa*y^2)*z - 2*sc*(a - b)*x*y*z)] = 0
      >
      > If P = incenter then the locus is one well known interesting cubic
      > Z(X(8), X(31)) (by Clark Kimberling, ETC) or
      > K200 (by Bernard Gibert, Cubics In The Triangle Plane)

      You obtain the same cubic K200 for any point P on the line IO = X1X3
      and this is the only case when the cubic is a pK.

      Now, when PaPbPc is the cevian triangle of P (instead of pedal), the
      locus is a cubic again which turns out to be a pK for P on K200 and
      the pK is always K200.
      In other words, for any points P, Q on K200 the cevian triangles
      PaPbPc, QaQbQc of P, Q verify the property above.

      One particular interesting case : the Soddy points X175, X176.

      Best regards

      Bernard



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    • Quang Tuan Bui
      Dear Bernard, Thank you very much for your advices. Yes, I think that it s better to say that the three signed distances sum up to zero, the triangle ABC being
      Message 2 of 3 , Jun 27, 2007
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        Dear Bernard,

        Thank you very much for your advices.
        Yes, I think that it's better to say that the three signed distances sum up to zero, the triangle ABC being arbitrarily orientated. In fact, I use somehow "signed distances" in all my calculations related "one is some of other two" my messages. I keep "one is some of other two" phrase because historical reason only.
        Your remarks about K200 are very interesting for me which I can not see. Thank you for it!

        Best regards,
        Bui Quang Tuan

        Bernard Gibert <bg42@...> wrote: Dear Quang Tuan Bui,

        Don't you think it's better to say that the three signed distances
        sum up to zero, the triangle ABC being arbitrarily orientated.

        You obtain the same cubic K200 for any point P on the line IO = X1X3
        and this is the only case when the cubic is a pK.

        Now, when PaPbPc is the cevian triangle of P (instead of pedal), the
        locus is a cubic again which turns out to be a pK for P on K200 and
        the pK is always K200.
        In other words, for any points P, Q on K200 the cevian triangles
        PaPbPc, QaQbQc of P, Q verify the property above.

        One particular interesting case : the Soddy points X175, X176.

        Best regards

        Bernard



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