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Re: locus related to Darboux cubic

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  • Francisco Javier
    Hello, Randy, it is a cubic, but it doesn t go through the vertices. One of the three points where the cubic intersects the line BC is the point X1={0, 3 a^2 -
    Message 1 of 5 , Mar 6, 2013
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      Hello, Randy,

      it is a cubic, but it doesn't go through the vertices.

      One of the three points where the cubic intersects the line BC is the point X1={0, 3 a^2 - b^2 + c^2, 3 a^2 + b^2 - c^2} that satisfies DX:XM=-4, where D is the foot of the A-altitude and M is the midpoint of BC.

      Equation of the cubic:
      -a^4 b^2 x^3 - 2 a^2 b^4 x^3 + 3 b^6 x^3 + a^4 c^2 x^3 -
      13 b^4 c^2 x^3 + 2 a^2 c^4 x^3 + 13 b^2 c^4 x^3 - 3 c^6 x^3 -
      3 a^6 x^2 y - 3 a^2 b^4 x^2 y + 6 b^6 x^2 y - 21 a^4 c^2 x^2 y +
      9 b^4 c^2 x^2 y - 45 a^2 c^4 x^2 y - 48 b^2 c^4 x^2 y -
      3 c^6 x^2 y - 6 a^6 x y^2 + 3 a^4 b^2 x y^2 + 3 b^6 x y^2 -
      9 a^4 c^2 x y^2 + 21 b^4 c^2 x y^2 + 48 a^2 c^4 x y^2 +
      45 b^2 c^4 x y^2 + 3 c^6 x y^2 - 3 a^6 y^3 + 2 a^4 b^2 y^3 +
      a^2 b^4 y^3 + 13 a^4 c^2 y^3 - b^4 c^2 y^3 - 13 a^2 c^4 y^3 -
      2 b^2 c^4 y^3 + 3 c^6 y^3 + 3 a^6 x^2 z + 21 a^4 b^2 x^2 z +
      45 a^2 b^4 x^2 z + 3 b^6 x^2 z + 48 b^4 c^2 x^2 z +
      3 a^2 c^4 x^2 z - 9 b^2 c^4 x^2 z - 6 c^6 x^2 z - 24 a^4 b^2 x y z +
      24 a^2 b^4 x y z + 24 a^4 c^2 x y z - 24 b^4 c^2 x y z -
      24 a^2 c^4 x y z + 24 b^2 c^4 x y z - 3 a^6 y^2 z -
      45 a^4 b^2 y^2 z - 21 a^2 b^4 y^2 z - 3 b^6 y^2 z -
      48 a^4 c^2 y^2 z + 9 a^2 c^4 y^2 z - 3 b^2 c^4 y^2 z + 6 c^6 y^2 z +
      6 a^6 x z^2 + 9 a^4 b^2 x z^2 - 48 a^2 b^4 x z^2 - 3 b^6 x z^2 -
      3 a^4 c^2 x z^2 - 45 b^4 c^2 x z^2 - 21 b^2 c^4 x z^2 -
      3 c^6 x z^2 + 3 a^6 y z^2 + 48 a^4 b^2 y z^2 - 9 a^2 b^4 y z^2 -
      6 b^6 y z^2 + 45 a^4 c^2 y z^2 + 3 b^4 c^2 y z^2 +
      21 a^2 c^4 y z^2 + 3 c^6 y z^2 + 3 a^6 z^3 - 13 a^4 b^2 z^3 +
      13 a^2 b^4 z^3 - 3 b^6 z^3 - 2 a^4 c^2 z^3 + 2 b^4 c^2 z^3 -
      a^2 c^4 z^3 + b^2 c^4 z^3 = 0.

      --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
      >
      > Friends,
      >
      > Let P be a point on the Darboux cubic. What is the locus of the centroid of the pedal triangle of P? It would include x(2), X(51), X(154), X(210), X(354), X(1853), and X(3917).
      >
      > Randy
      >
    • rhutson2
      Thanks, Francisco! Anything interesting about the triangle formed by X1 and the corresponding points on sides CA and AB? Do the six remaining intersections
      Message 2 of 5 , Mar 6, 2013
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        Thanks, Francisco!

        Anything interesting about the triangle formed by X1 and the corresponding points on sides CA and AB? Do the six remaining intersections lie on a common conic?

        Randy

        --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
        >
        > Hello, Randy,
        >
        > it is a cubic, but it doesn't go through the vertices.
        >
        > One of the three points where the cubic intersects the line BC is the point X1={0, 3 a^2 - b^2 + c^2, 3 a^2 + b^2 - c^2} that satisfies DX:XM=-4, where D is the foot of the A-altitude and M is the midpoint of BC.
        >
        > Equation of the cubic:
        > -a^4 b^2 x^3 - 2 a^2 b^4 x^3 + 3 b^6 x^3 + a^4 c^2 x^3 -
        > 13 b^4 c^2 x^3 + 2 a^2 c^4 x^3 + 13 b^2 c^4 x^3 - 3 c^6 x^3 -
        > 3 a^6 x^2 y - 3 a^2 b^4 x^2 y + 6 b^6 x^2 y - 21 a^4 c^2 x^2 y +
        > 9 b^4 c^2 x^2 y - 45 a^2 c^4 x^2 y - 48 b^2 c^4 x^2 y -
        > 3 c^6 x^2 y - 6 a^6 x y^2 + 3 a^4 b^2 x y^2 + 3 b^6 x y^2 -
        > 9 a^4 c^2 x y^2 + 21 b^4 c^2 x y^2 + 48 a^2 c^4 x y^2 +
        > 45 b^2 c^4 x y^2 + 3 c^6 x y^2 - 3 a^6 y^3 + 2 a^4 b^2 y^3 +
        > a^2 b^4 y^3 + 13 a^4 c^2 y^3 - b^4 c^2 y^3 - 13 a^2 c^4 y^3 -
        > 2 b^2 c^4 y^3 + 3 c^6 y^3 + 3 a^6 x^2 z + 21 a^4 b^2 x^2 z +
        > 45 a^2 b^4 x^2 z + 3 b^6 x^2 z + 48 b^4 c^2 x^2 z +
        > 3 a^2 c^4 x^2 z - 9 b^2 c^4 x^2 z - 6 c^6 x^2 z - 24 a^4 b^2 x y z +
        > 24 a^2 b^4 x y z + 24 a^4 c^2 x y z - 24 b^4 c^2 x y z -
        > 24 a^2 c^4 x y z + 24 b^2 c^4 x y z - 3 a^6 y^2 z -
        > 45 a^4 b^2 y^2 z - 21 a^2 b^4 y^2 z - 3 b^6 y^2 z -
        > 48 a^4 c^2 y^2 z + 9 a^2 c^4 y^2 z - 3 b^2 c^4 y^2 z + 6 c^6 y^2 z +
        > 6 a^6 x z^2 + 9 a^4 b^2 x z^2 - 48 a^2 b^4 x z^2 - 3 b^6 x z^2 -
        > 3 a^4 c^2 x z^2 - 45 b^4 c^2 x z^2 - 21 b^2 c^4 x z^2 -
        > 3 c^6 x z^2 + 3 a^6 y z^2 + 48 a^4 b^2 y z^2 - 9 a^2 b^4 y z^2 -
        > 6 b^6 y z^2 + 45 a^4 c^2 y z^2 + 3 b^4 c^2 y z^2 +
        > 21 a^2 c^4 y z^2 + 3 c^6 y z^2 + 3 a^6 z^3 - 13 a^4 b^2 z^3 +
        > 13 a^2 b^4 z^3 - 3 b^6 z^3 - 2 a^4 c^2 z^3 + 2 b^4 c^2 z^3 -
        > a^2 c^4 z^3 + b^2 c^4 z^3 = 0.
        >
        > --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@> wrote:
        > >
        > > Friends,
        > >
        > > Let P be a point on the Darboux cubic. What is the locus of the centroid of the pedal triangle of P? It would include x(2), X(51), X(154), X(210), X(354), X(1853), and X(3917).
        > >
        > > Randy
        > >
        >
      • Francisco Javier
        Dear Randy: If Y1, Z1 are the corresponding points of X1 on CA, AB, then X1Y1Z1 is the pedal triangle of X376. The other six points doesn t lie on the same
        Message 3 of 5 , Mar 6, 2013
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          Dear Randy:

          If Y1, Z1 are the corresponding points of X1 on CA, AB, then X1Y1Z1 is the pedal triangle of X376.

          The other six points doesn't lie on the same conic.

          Best regards,
          Francisco Javier.

          --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
          >
          > Thanks, Francisco!
          >
          > Anything interesting about the triangle formed by X1 and the corresponding points on sides CA and AB? Do the six remaining intersections lie on a common conic?
          >
          > Randy
          >
          > --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@> wrote:
          > >
          > > Hello, Randy,
          > >
          > > it is a cubic, but it doesn't go through the vertices.
          > >
          > > One of the three points where the cubic intersects the line BC is the point X1={0, 3 a^2 - b^2 + c^2, 3 a^2 + b^2 - c^2} that satisfies DX:XM=-4, where D is the foot of the A-altitude and M is the midpoint of BC.
          > >
          > > Equation of the cubic:
          > > -a^4 b^2 x^3 - 2 a^2 b^4 x^3 + 3 b^6 x^3 + a^4 c^2 x^3 -
          > > 13 b^4 c^2 x^3 + 2 a^2 c^4 x^3 + 13 b^2 c^4 x^3 - 3 c^6 x^3 -
          > > 3 a^6 x^2 y - 3 a^2 b^4 x^2 y + 6 b^6 x^2 y - 21 a^4 c^2 x^2 y +
          > > 9 b^4 c^2 x^2 y - 45 a^2 c^4 x^2 y - 48 b^2 c^4 x^2 y -
          > > 3 c^6 x^2 y - 6 a^6 x y^2 + 3 a^4 b^2 x y^2 + 3 b^6 x y^2 -
          > > 9 a^4 c^2 x y^2 + 21 b^4 c^2 x y^2 + 48 a^2 c^4 x y^2 +
          > > 45 b^2 c^4 x y^2 + 3 c^6 x y^2 - 3 a^6 y^3 + 2 a^4 b^2 y^3 +
          > > a^2 b^4 y^3 + 13 a^4 c^2 y^3 - b^4 c^2 y^3 - 13 a^2 c^4 y^3 -
          > > 2 b^2 c^4 y^3 + 3 c^6 y^3 + 3 a^6 x^2 z + 21 a^4 b^2 x^2 z +
          > > 45 a^2 b^4 x^2 z + 3 b^6 x^2 z + 48 b^4 c^2 x^2 z +
          > > 3 a^2 c^4 x^2 z - 9 b^2 c^4 x^2 z - 6 c^6 x^2 z - 24 a^4 b^2 x y z +
          > > 24 a^2 b^4 x y z + 24 a^4 c^2 x y z - 24 b^4 c^2 x y z -
          > > 24 a^2 c^4 x y z + 24 b^2 c^4 x y z - 3 a^6 y^2 z -
          > > 45 a^4 b^2 y^2 z - 21 a^2 b^4 y^2 z - 3 b^6 y^2 z -
          > > 48 a^4 c^2 y^2 z + 9 a^2 c^4 y^2 z - 3 b^2 c^4 y^2 z + 6 c^6 y^2 z +
          > > 6 a^6 x z^2 + 9 a^4 b^2 x z^2 - 48 a^2 b^4 x z^2 - 3 b^6 x z^2 -
          > > 3 a^4 c^2 x z^2 - 45 b^4 c^2 x z^2 - 21 b^2 c^4 x z^2 -
          > > 3 c^6 x z^2 + 3 a^6 y z^2 + 48 a^4 b^2 y z^2 - 9 a^2 b^4 y z^2 -
          > > 6 b^6 y z^2 + 45 a^4 c^2 y z^2 + 3 b^4 c^2 y z^2 +
          > > 21 a^2 c^4 y z^2 + 3 c^6 y z^2 + 3 a^6 z^3 - 13 a^4 b^2 z^3 +
          > > 13 a^2 b^4 z^3 - 3 b^6 z^3 - 2 a^4 c^2 z^3 + 2 b^4 c^2 z^3 -
          > > a^2 c^4 z^3 + b^2 c^4 z^3 = 0.
          > >
          > > --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@> wrote:
          > > >
          > > > Friends,
          > > >
          > > > Let P be a point on the Darboux cubic. What is the locus of the centroid of the pedal triangle of P? It would include x(2), X(51), X(154), X(210), X(354), X(1853), and X(3917).
          > > >
          > > > Randy
          > > >
          > >
          >
        • rhutson2
          A related problem: what is the locus of the centroid of the antipedal triangle of P, for P on the Darboux cubic? It would include X(2), X(154), X(165),
          Message 4 of 5 , Mar 18, 2013
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            A related problem: what is the locus of the centroid of the antipedal triangle of P, for P on the Darboux cubic? It would include X(2), X(154), X(165), X(3158), and X(3167).

            What points are in common with the first locus, besides X(2) and X(154)?

            Randy

            --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
            >
            > Hello, Randy,
            >
            > it is a cubic, but it doesn't go through the vertices.
            >
            > One of the three points where the cubic intersects the line BC is the point X1={0, 3 a^2 - b^2 + c^2, 3 a^2 + b^2 - c^2} that satisfies DX:XM=-4, where D is the foot of the A-altitude and M is the midpoint of BC.
            >
            > Equation of the cubic:
            > -a^4 b^2 x^3 - 2 a^2 b^4 x^3 + 3 b^6 x^3 + a^4 c^2 x^3 -
            > 13 b^4 c^2 x^3 + 2 a^2 c^4 x^3 + 13 b^2 c^4 x^3 - 3 c^6 x^3 -
            > 3 a^6 x^2 y - 3 a^2 b^4 x^2 y + 6 b^6 x^2 y - 21 a^4 c^2 x^2 y +
            > 9 b^4 c^2 x^2 y - 45 a^2 c^4 x^2 y - 48 b^2 c^4 x^2 y -
            > 3 c^6 x^2 y - 6 a^6 x y^2 + 3 a^4 b^2 x y^2 + 3 b^6 x y^2 -
            > 9 a^4 c^2 x y^2 + 21 b^4 c^2 x y^2 + 48 a^2 c^4 x y^2 +
            > 45 b^2 c^4 x y^2 + 3 c^6 x y^2 - 3 a^6 y^3 + 2 a^4 b^2 y^3 +
            > a^2 b^4 y^3 + 13 a^4 c^2 y^3 - b^4 c^2 y^3 - 13 a^2 c^4 y^3 -
            > 2 b^2 c^4 y^3 + 3 c^6 y^3 + 3 a^6 x^2 z + 21 a^4 b^2 x^2 z +
            > 45 a^2 b^4 x^2 z + 3 b^6 x^2 z + 48 b^4 c^2 x^2 z +
            > 3 a^2 c^4 x^2 z - 9 b^2 c^4 x^2 z - 6 c^6 x^2 z - 24 a^4 b^2 x y z +
            > 24 a^2 b^4 x y z + 24 a^4 c^2 x y z - 24 b^4 c^2 x y z -
            > 24 a^2 c^4 x y z + 24 b^2 c^4 x y z - 3 a^6 y^2 z -
            > 45 a^4 b^2 y^2 z - 21 a^2 b^4 y^2 z - 3 b^6 y^2 z -
            > 48 a^4 c^2 y^2 z + 9 a^2 c^4 y^2 z - 3 b^2 c^4 y^2 z + 6 c^6 y^2 z +
            > 6 a^6 x z^2 + 9 a^4 b^2 x z^2 - 48 a^2 b^4 x z^2 - 3 b^6 x z^2 -
            > 3 a^4 c^2 x z^2 - 45 b^4 c^2 x z^2 - 21 b^2 c^4 x z^2 -
            > 3 c^6 x z^2 + 3 a^6 y z^2 + 48 a^4 b^2 y z^2 - 9 a^2 b^4 y z^2 -
            > 6 b^6 y z^2 + 45 a^4 c^2 y z^2 + 3 b^4 c^2 y z^2 +
            > 21 a^2 c^4 y z^2 + 3 c^6 y z^2 + 3 a^6 z^3 - 13 a^4 b^2 z^3 +
            > 13 a^2 b^4 z^3 - 3 b^6 z^3 - 2 a^4 c^2 z^3 + 2 b^4 c^2 z^3 -
            > a^2 c^4 z^3 + b^2 c^4 z^3 = 0.
            >
            > --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@> wrote:
            > >
            > > Friends,
            > >
            > > Let P be a point on the Darboux cubic. What is the locus of the centroid of the pedal triangle of P? It would include x(2), X(51), X(154), X(210), X(354), X(1853), and X(3917).
            > >
            > > Randy
            > >
            >
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