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Antipedal triangles of P,P*

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  • Antreas
    Let ABC be a triangle, A B C , A B C the antipedal tiangles of H, O [antimedial, tangential, resp.]. The triangles: 1. ABC, A B C 2. ABC, A BC 3. ABC,
    Message 1 of 8 , Mar 18 4:57 AM
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      Let ABC be a triangle, A'B'C', A"B"C" the antipedal tiangles
      of H, O [antimedial, tangential, resp.].

      The triangles:

      1. ABC, A'B'C'

      2. ABC, A"BC"

      3. ABC, Triangle bounded by the lines (A'A",B'B",C'C")

      are perspectives.

      [the 3rd perspector is on the circumcircle(ABC) = NPC(A'B'C') =
      Incircle(A"B"C")]

      Generalization:

      Let P,P* be two isogonal conjugate points and A'B'C',A"B"C"
      the antipedal triangles of P,P*, resp.

      Which is the locus of P such that the triangles:

      1. ABC, A'B'C'

      2. ABC, A"BC"

      3. ABC, Triangle bounded by the lines (A'A",B'B",C'C")

      are perspectives ?

      APH
    • Antreas
      Let ABC be a triangle, A B C , A B C the antipedal tiangles of H, O [antimedial, tangential, resp.]. The triangles: 1. ABC, A B C 2. ABC, A BC 3. ABC,
      Message 2 of 8 , Mar 18 5:43 AM
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        Let ABC be a triangle, A'B'C', A"B"C" the antipedal tiangles
        of H, O [antimedial, tangential, resp.].

        The triangles:

        1. ABC, A'B'C'

        2. ABC, A"BC"

        3. ABC, Triangle bounded by the lines (A'A",B'B",C'C")

        are perspective.

        [the 3rd perspector is on the circumcircle(ABC) = NPC(A'B'C') =
        Incircle(A"B"C")]

        Generalization:

        Let P,P* be two isogonal conjugate points and A'B'C',A"B"C"
        the antipedal triangles of P,P*, resp.

        Which is the locus of P such that the triangles:

        1. A"B"C", Triangle bounded by (AA',BB',CC')

        2. A'B'C', Triangle bounded by (AA",BB",CC")

        3. ABC, Triangle bounded by (A'A",B'B",C'C")

        are perspective ?

        APH
      • Francisco Javier
        [APH] Generalization: Let P,P* be two isogonal conjugate points and A B C ,A B C the antipedal triangles of P,P*, resp. Which is the locus of P such that the
        Message 3 of 8 , Mar 18 12:31 PM
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          [APH]

          Generalization:
          Let P,P* be two isogonal conjugate points and A'B'C',A"B"C"
          the antipedal triangles of P,P*, resp.
          Which is the locus of P such that the triangles:
          1. A"B"C", Triangle bounded by (AA',BB',CC')
          2. A'B'C', Triangle bounded by (AA",BB",CC")
          3. ABC, Triangle bounded by (A'A",B'B",C'C")
          are perspective ?

          ------------------------


          The loci in 1. and 2 are the same:

          line at infinity + circumcircle + K004 + three cubics, each one relative to one of the vertices.

          The cubic relative to A has equation
          2 b^2 c^2 x^2 y + a^2 c^2 x y^2 + b^2 c^2 x y^2 + c^4 x y^2 +
          2 b^2 c^2 x^2 z + 4 b^2 c^2 x y z + 2 a^2 c^2 y^2 z + a^2 b^2 x z^2 +
          b^4 x z^2 + b^2 c^2 x z^2 + 2 a^2 b^2 y z^2 =0.

          It intersects the line at infinity at the infinite points of the bisectors of angle A and at the infinite point of the A-altitude.

          It is its isogonal conjugate, then it intersects the circumcircle at the intersections of the circumcircle and the bisectors of angle A and at the antipode of A.

          With respect to 3., the triangle bounded by (A'A", B' B ",C'C")
          is ALWAYS perspective with ABC.

          If P=(x:y:z), then the perspector S is complicated: the coordinates of its isotomic conjugate, that is infinite when P lies on K004, are:

          {2 a^4 b^2 c^2 x^2 y - 4 a^2 b^4 c^2 x^2 y + 2 b^6 c^2 x^2 y +
          4 a^2 b^2 c^4 x^2 y + 4 b^4 c^4 x^2 y - 6 b^2 c^6 x^2 y +
          a^6 c^2 x y^2 - a^4 b^2 c^2 x y^2 - a^2 b^4 c^2 x y^2 +
          b^6 c^2 x y^2 - a^4 c^4 x y^2 + 10 a^2 b^2 c^4 x y^2 -
          b^4 c^4 x y^2 - a^2 c^6 x y^2 - b^2 c^6 x y^2 + c^8 x y^2 -
          2 a^4 b^2 c^2 x^2 z - 4 a^2 b^4 c^2 x^2 z + 6 b^6 c^2 x^2 z +
          4 a^2 b^2 c^4 x^2 z - 4 b^4 c^4 x^2 z - 2 b^2 c^6 x^2 z +
          2 a^6 c^2 y^2 z + 4 a^4 b^2 c^2 y^2 z - 6 a^2 b^4 c^2 y^2 z -
          4 a^4 c^4 y^2 z + 4 a^2 b^2 c^4 y^2 z + 2 a^2 c^6 y^2 z -
          a^6 b^2 x z^2 + a^4 b^4 x z^2 + a^2 b^6 x z^2 - b^8 x z^2 +
          a^4 b^2 c^2 x z^2 - 10 a^2 b^4 c^2 x z^2 + b^6 c^2 x z^2 +
          a^2 b^2 c^4 x z^2 + b^4 c^4 x z^2 - b^2 c^6 x z^2 -
          2 a^6 b^2 y z^2 + 4 a^4 b^4 y z^2 - 2 a^2 b^6 y z^2 -
          4 a^4 b^2 c^2 y z^2 - 4 a^2 b^4 c^2 y z^2 +
          6 a^2 b^2 c^4 y z^2, -a^6 c^2 x^2 y + a^4 b^2 c^2 x^2 y +
          a^2 b^4 c^2 x^2 y - b^6 c^2 x^2 y + a^4 c^4 x^2 y -
          10 a^2 b^2 c^4 x^2 y + b^4 c^4 x^2 y + a^2 c^6 x^2 y +
          b^2 c^6 x^2 y - c^8 x^2 y - 2 a^6 c^2 x y^2 + 4 a^4 b^2 c^2 x y^2 -
          2 a^2 b^4 c^2 x y^2 - 4 a^4 c^4 x y^2 - 4 a^2 b^2 c^4 x y^2 +
          6 a^2 c^6 x y^2 + 6 a^4 b^2 c^2 x^2 z - 4 a^2 b^4 c^2 x^2 z -
          2 b^6 c^2 x^2 z - 4 a^2 b^2 c^4 x^2 z + 4 b^4 c^4 x^2 z -
          2 b^2 c^6 x^2 z - 6 a^6 c^2 y^2 z + 4 a^4 b^2 c^2 y^2 z +
          2 a^2 b^4 c^2 y^2 z + 4 a^4 c^4 y^2 z - 4 a^2 b^2 c^4 y^2 z +
          2 a^2 c^6 y^2 z + 2 a^6 b^2 x z^2 - 4 a^4 b^4 x z^2 +
          2 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 + 4 a^2 b^4 c^2 x z^2 -
          6 a^2 b^2 c^4 x z^2 + a^8 y z^2 - a^6 b^2 y z^2 - a^4 b^4 y z^2 +
          a^2 b^6 y z^2 - a^6 c^2 y z^2 + 10 a^4 b^2 c^2 y z^2 -
          a^2 b^4 c^2 y z^2 - a^4 c^4 y z^2 - a^2 b^2 c^4 y z^2 +
          a^2 c^6 y z^2, -6 a^4 b^2 c^2 x^2 y + 4 a^2 b^4 c^2 x^2 y +
          2 b^6 c^2 x^2 y + 4 a^2 b^2 c^4 x^2 y - 4 b^4 c^4 x^2 y +
          2 b^2 c^6 x^2 y - 2 a^6 c^2 x y^2 - 4 a^4 b^2 c^2 x y^2 +
          6 a^2 b^4 c^2 x y^2 + 4 a^4 c^4 x y^2 - 4 a^2 b^2 c^4 x y^2 -
          2 a^2 c^6 x y^2 + a^6 b^2 x^2 z - a^4 b^4 x^2 z - a^2 b^6 x^2 z +
          b^8 x^2 z - a^4 b^2 c^2 x^2 z + 10 a^2 b^4 c^2 x^2 z -
          b^6 c^2 x^2 z - a^2 b^2 c^4 x^2 z - b^4 c^4 x^2 z + b^2 c^6 x^2 z -
          a^8 y^2 z + a^6 b^2 y^2 z + a^4 b^4 y^2 z - a^2 b^6 y^2 z +
          a^6 c^2 y^2 z - 10 a^4 b^2 c^2 y^2 z + a^2 b^4 c^2 y^2 z +
          a^4 c^4 y^2 z + a^2 b^2 c^4 y^2 z - a^2 c^6 y^2 z +
          2 a^6 b^2 x z^2 + 4 a^4 b^4 x z^2 - 6 a^2 b^6 x z^2 -
          4 a^4 b^2 c^2 x z^2 + 4 a^2 b^4 c^2 x z^2 + 2 a^2 b^2 c^4 x z^2 +
          6 a^6 b^2 y z^2 - 4 a^4 b^4 y z^2 - 2 a^2 b^6 y z^2 -
          4 a^4 b^2 c^2 y z^2 + 4 a^2 b^4 c^2 y z^2 - 2 a^2 b^2 c^4 y z^2}
        • Antreas
          [APH] ... Dear Francisco, Very good!! Thanks!! For the simple case of 1 & 2 (ABC, A B C or A B C perspective ie ABC, antipedal triangle of P are perspective)
          Message 4 of 8 , Mar 18 1:13 PM
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            [APH]
            >
            > Generalization:
            > Let P,P* be two isogonal conjugate points and A'B'C',A"B"C"
            > the antipedal triangles of P,P*, resp.
            > Which is the locus of P such that the triangles:
            > 1. A"B"C", Triangle bounded by (AA',BB',CC')
            > 2. A'B'C', Triangle bounded by (AA",BB",CC")
            > 3. ABC, Triangle bounded by (A'A",B'B",C'C")
            > are perspective ?

            [Francisco]:

            > The loci in 1. and 2 are the same:
            >
            > line at infinity + circumcircle + K004 + three cubics, each one relative to one of the vertices.
            >
            > The cubic relative to A has equation
            > 2 b^2 c^2 x^2 y + a^2 c^2 x y^2 + b^2 c^2 x y^2 + c^4 x y^2 +
            > 2 b^2 c^2 x^2 z + 4 b^2 c^2 x y z + 2 a^2 c^2 y^2 z + a^2 b^2 x z^2 +
            > b^4 x z^2 + b^2 c^2 x z^2 + 2 a^2 b^2 y z^2 =0.
            >
            > It intersects the line at infinity at the infinite points of the bisectors of angle A and at the infinite point of the A-altitude.
            >
            > It is its isogonal conjugate, then it intersects the circumcircle at the intersections of the circumcircle and the bisectors of angle A and at the antipode of A.
            >
            > With respect to 3., the triangle bounded by (A'A", B' B ",C'C")
            > is ALWAYS perspective with ABC.
            >
            > If P=(x:y:z), then the perspector S is complicated: the coordinates of its isotomic conjugate, that is infinite when P lies on K004, are:
            >
            > {2 a^4 b^2 c^2 x^2 y - 4 a^2 b^4 c^2 x^2 y + 2 b^6 c^2 x^2 y +
            > 4 a^2 b^2 c^4 x^2 y + 4 b^4 c^4 x^2 y - 6 b^2 c^6 x^2 y +
            > a^6 c^2 x y^2 - a^4 b^2 c^2 x y^2 - a^2 b^4 c^2 x y^2 +
            > b^6 c^2 x y^2 - a^4 c^4 x y^2 + 10 a^2 b^2 c^4 x y^2 -
            > b^4 c^4 x y^2 - a^2 c^6 x y^2 - b^2 c^6 x y^2 + c^8 x y^2 -
            > 2 a^4 b^2 c^2 x^2 z - 4 a^2 b^4 c^2 x^2 z + 6 b^6 c^2 x^2 z +
            > 4 a^2 b^2 c^4 x^2 z - 4 b^4 c^4 x^2 z - 2 b^2 c^6 x^2 z +
            > 2 a^6 c^2 y^2 z + 4 a^4 b^2 c^2 y^2 z - 6 a^2 b^4 c^2 y^2 z -
            > 4 a^4 c^4 y^2 z + 4 a^2 b^2 c^4 y^2 z + 2 a^2 c^6 y^2 z -
            > a^6 b^2 x z^2 + a^4 b^4 x z^2 + a^2 b^6 x z^2 - b^8 x z^2 +
            > a^4 b^2 c^2 x z^2 - 10 a^2 b^4 c^2 x z^2 + b^6 c^2 x z^2 +
            > a^2 b^2 c^4 x z^2 + b^4 c^4 x z^2 - b^2 c^6 x z^2 -
            > 2 a^6 b^2 y z^2 + 4 a^4 b^4 y z^2 - 2 a^2 b^6 y z^2 -
            > 4 a^4 b^2 c^2 y z^2 - 4 a^2 b^4 c^2 y z^2 +
            > 6 a^2 b^2 c^4 y z^2, -a^6 c^2 x^2 y + a^4 b^2 c^2 x^2 y +
            > a^2 b^4 c^2 x^2 y - b^6 c^2 x^2 y + a^4 c^4 x^2 y -
            > 10 a^2 b^2 c^4 x^2 y + b^4 c^4 x^2 y + a^2 c^6 x^2 y +
            > b^2 c^6 x^2 y - c^8 x^2 y - 2 a^6 c^2 x y^2 + 4 a^4 b^2 c^2 x y^2 -
            > 2 a^2 b^4 c^2 x y^2 - 4 a^4 c^4 x y^2 - 4 a^2 b^2 c^4 x y^2 +
            > 6 a^2 c^6 x y^2 + 6 a^4 b^2 c^2 x^2 z - 4 a^2 b^4 c^2 x^2 z -
            > 2 b^6 c^2 x^2 z - 4 a^2 b^2 c^4 x^2 z + 4 b^4 c^4 x^2 z -
            > 2 b^2 c^6 x^2 z - 6 a^6 c^2 y^2 z + 4 a^4 b^2 c^2 y^2 z +
            > 2 a^2 b^4 c^2 y^2 z + 4 a^4 c^4 y^2 z - 4 a^2 b^2 c^4 y^2 z +
            > 2 a^2 c^6 y^2 z + 2 a^6 b^2 x z^2 - 4 a^4 b^4 x z^2 +
            > 2 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 + 4 a^2 b^4 c^2 x z^2 -
            > 6 a^2 b^2 c^4 x z^2 + a^8 y z^2 - a^6 b^2 y z^2 - a^4 b^4 y z^2 +
            > a^2 b^6 y z^2 - a^6 c^2 y z^2 + 10 a^4 b^2 c^2 y z^2 -
            > a^2 b^4 c^2 y z^2 - a^4 c^4 y z^2 - a^2 b^2 c^4 y z^2 +
            > a^2 c^6 y z^2, -6 a^4 b^2 c^2 x^2 y + 4 a^2 b^4 c^2 x^2 y +
            > 2 b^6 c^2 x^2 y + 4 a^2 b^2 c^4 x^2 y - 4 b^4 c^4 x^2 y +
            > 2 b^2 c^6 x^2 y - 2 a^6 c^2 x y^2 - 4 a^4 b^2 c^2 x y^2 +
            > 6 a^2 b^4 c^2 x y^2 + 4 a^4 c^4 x y^2 - 4 a^2 b^2 c^4 x y^2 -
            > 2 a^2 c^6 x y^2 + a^6 b^2 x^2 z - a^4 b^4 x^2 z - a^2 b^6 x^2 z +
            > b^8 x^2 z - a^4 b^2 c^2 x^2 z + 10 a^2 b^4 c^2 x^2 z -
            > b^6 c^2 x^2 z - a^2 b^2 c^4 x^2 z - b^4 c^4 x^2 z + b^2 c^6 x^2 z -
            > a^8 y^2 z + a^6 b^2 y^2 z + a^4 b^4 y^2 z - a^2 b^6 y^2 z +
            > a^6 c^2 y^2 z - 10 a^4 b^2 c^2 y^2 z + a^2 b^4 c^2 y^2 z +
            > a^4 c^4 y^2 z + a^2 b^2 c^4 y^2 z - a^2 c^6 y^2 z +
            > 2 a^6 b^2 x z^2 + 4 a^4 b^4 x z^2 - 6 a^2 b^6 x z^2 -
            > 4 a^4 b^2 c^2 x z^2 + 4 a^2 b^4 c^2 x z^2 + 2 a^2 b^2 c^4 x z^2 +
            > 6 a^6 b^2 y z^2 - 4 a^4 b^4 y z^2 - 2 a^2 b^6 y z^2 -
            > 4 a^4 b^2 c^2 y z^2 + 4 a^2 b^4 c^2 y z^2 - 2 a^2 b^2 c^4 y z^2}
            >

            Dear Francisco,

            Very good!! Thanks!!

            For the simple case of 1 & 2 (ABC, A'B'C' or A"B"C" perspective
            ie ABC, antipedal triangle of P are perspective) the locus is K004
            (= Darboux cubic)+(O) + Linf, and most likely we have seen it before.
            But is it listed in Bernard's properties of K004 ?

            The perspector of the 3rd case is complicated indeed!!

            And I am wondering if for the simple pairs of isogonal
            conjugate points are the perspectors in ETC.
            That is, the perspectors for (P,P*) = (O,H), (G,K), (N, X54)

            APH
          • Antreas
            Do we have one more perspectivity??? ... Now, denote: Abc = A B / A C , Acb = A C / A B Bca = B C / B A , Bac = B A / B C Cab = C A / C B , Cba =
            Message 5 of 8 , Mar 19 3:51 AM
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              Do we have one more perspectivity???

              > [APH]:
              > Generalization:
              > Let P,P* be two isogonal conjugate points and A'B'C',A"B"C"
              > the antipedal triangles of P,P*, resp.
              > Which is the locus of P such that the triangles:
              > 3. ABC, Triangle bounded by (A'A",B'B",C'C")
              > are perspective ?

              [Francisco]:
              > With respect to 3., the triangle bounded by (A'A", B' B ",C'C")
              > is ALWAYS perspective with ABC.

              Now, denote:

              Abc = A'B' /\ A"C", Acb = A'C' /\ A"B"

              Bca = B'C' /\ B"A", Bac = B'A' /\ B"C"

              Cab = C'A' /\ C"B", Cba = C'B' /\ C"A"

              Which is the locus of P such that the triangles
              AbcBcaCab, AcbBacCba are perspective?

              Figure:
              http://anthrakitis.blogspot.gr/2013/03/perspectivity-antipedal-triangles-of-pp_19.html

              APH
            • Francisco Javier
              They are again ALWAYS perspective. The coordinates of the perspector are shown below and it is infinite when P lies on K003. {a^2 (-2 a^4 b^2 c^2 x^2 y + 4 a^2
              Message 6 of 8 , Mar 19 6:41 AM
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                They are again ALWAYS perspective. The coordinates of the perspector are shown below and it is infinite when P lies on K003.

                {a^2 (-2 a^4 b^2 c^2 x^2 y + 4 a^2 b^4 c^2 x^2 y - 2 b^6 c^2 x^2 y -
                4 a^2 b^2 c^4 x^2 y - 4 b^4 c^4 x^2 y + 6 b^2 c^6 x^2 y -
                a^6 c^2 x y^2 + a^4 b^2 c^2 x y^2 + a^2 b^4 c^2 x y^2 -
                b^6 c^2 x y^2 + a^4 c^4 x y^2 - 10 a^2 b^2 c^4 x y^2 +
                b^4 c^4 x y^2 + a^2 c^6 x y^2 + b^2 c^6 x y^2 - c^8 x y^2 +
                2 a^4 b^2 c^2 x^2 z + 4 a^2 b^4 c^2 x^2 z - 6 b^6 c^2 x^2 z -
                4 a^2 b^2 c^4 x^2 z + 4 b^4 c^4 x^2 z + 2 b^2 c^6 x^2 z -
                2 a^6 c^2 y^2 z - 4 a^4 b^2 c^2 y^2 z + 6 a^2 b^4 c^2 y^2 z +
                4 a^4 c^4 y^2 z - 4 a^2 b^2 c^4 y^2 z - 2 a^2 c^6 y^2 z +
                a^6 b^2 x z^2 - a^4 b^4 x z^2 - a^2 b^6 x z^2 + b^8 x z^2 -
                a^4 b^2 c^2 x z^2 + 10 a^2 b^4 c^2 x z^2 - b^6 c^2 x z^2 -
                a^2 b^2 c^4 x z^2 - b^4 c^4 x z^2 + b^2 c^6 x z^2 +
                2 a^6 b^2 y z^2 - 4 a^4 b^4 y z^2 + 2 a^2 b^6 y z^2 +
                4 a^4 b^2 c^2 y z^2 + 4 a^2 b^4 c^2 y z^2 -
                6 a^2 b^2 c^4 y z^2), -b^2 (-a^6 c^2 x^2 y + a^4 b^2 c^2 x^2 y +
                a^2 b^4 c^2 x^2 y - b^6 c^2 x^2 y + a^4 c^4 x^2 y -
                10 a^2 b^2 c^4 x^2 y + b^4 c^4 x^2 y + a^2 c^6 x^2 y +
                b^2 c^6 x^2 y - c^8 x^2 y - 2 a^6 c^2 x y^2 +
                4 a^4 b^2 c^2 x y^2 - 2 a^2 b^4 c^2 x y^2 - 4 a^4 c^4 x y^2 -
                4 a^2 b^2 c^4 x y^2 + 6 a^2 c^6 x y^2 + 6 a^4 b^2 c^2 x^2 z -
                4 a^2 b^4 c^2 x^2 z - 2 b^6 c^2 x^2 z - 4 a^2 b^2 c^4 x^2 z +
                4 b^4 c^4 x^2 z - 2 b^2 c^6 x^2 z - 6 a^6 c^2 y^2 z +
                4 a^4 b^2 c^2 y^2 z + 2 a^2 b^4 c^2 y^2 z + 4 a^4 c^4 y^2 z -
                4 a^2 b^2 c^4 y^2 z + 2 a^2 c^6 y^2 z + 2 a^6 b^2 x z^2 -
                4 a^4 b^4 x z^2 + 2 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 +
                4 a^2 b^4 c^2 x z^2 - 6 a^2 b^2 c^4 x z^2 + a^8 y z^2 -
                a^6 b^2 y z^2 - a^4 b^4 y z^2 + a^2 b^6 y z^2 - a^6 c^2 y z^2 +
                10 a^4 b^2 c^2 y z^2 - a^2 b^4 c^2 y z^2 - a^4 c^4 y z^2 -
                a^2 b^2 c^4 y z^2 + a^2 c^6 y z^2),
                c^2 (6 a^4 b^2 c^2 x^2 y - 4 a^2 b^4 c^2 x^2 y - 2 b^6 c^2 x^2 y -
                4 a^2 b^2 c^4 x^2 y + 4 b^4 c^4 x^2 y - 2 b^2 c^6 x^2 y +
                2 a^6 c^2 x y^2 + 4 a^4 b^2 c^2 x y^2 - 6 a^2 b^4 c^2 x y^2 -
                4 a^4 c^4 x y^2 + 4 a^2 b^2 c^4 x y^2 + 2 a^2 c^6 x y^2 -
                a^6 b^2 x^2 z + a^4 b^4 x^2 z + a^2 b^6 x^2 z - b^8 x^2 z +
                a^4 b^2 c^2 x^2 z - 10 a^2 b^4 c^2 x^2 z + b^6 c^2 x^2 z +
                a^2 b^2 c^4 x^2 z + b^4 c^4 x^2 z - b^2 c^6 x^2 z + a^8 y^2 z -
                a^6 b^2 y^2 z - a^4 b^4 y^2 z + a^2 b^6 y^2 z - a^6 c^2 y^2 z +
                10 a^4 b^2 c^2 y^2 z - a^2 b^4 c^2 y^2 z - a^4 c^4 y^2 z -
                a^2 b^2 c^4 y^2 z + a^2 c^6 y^2 z - 2 a^6 b^2 x z^2 -
                4 a^4 b^4 x z^2 + 6 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 -
                4 a^2 b^4 c^2 x z^2 - 2 a^2 b^2 c^4 x z^2 - 6 a^6 b^2 y z^2 +
                4 a^4 b^4 y z^2 + 2 a^2 b^6 y z^2 + 4 a^4 b^2 c^2 y z^2 -
                4 a^2 b^4 c^2 y z^2 + 2 a^2 b^2 c^4 y z^2)}


                --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
                >
                > Do we have one more perspectivity???
                >
                > > [APH]:
                > > Generalization:
                > > Let P,P* be two isogonal conjugate points and A'B'C',A"B"C"
                > > the antipedal triangles of P,P*, resp.
                > > Which is the locus of P such that the triangles:
                > > 3. ABC, Triangle bounded by (A'A",B'B",C'C")
                > > are perspective ?
                >
                > [Francisco]:
                > > With respect to 3., the triangle bounded by (A'A", B' B ",C'C")
                > > is ALWAYS perspective with ABC.
                >
                > Now, denote:
                >
                > Abc = A'B' /\ A"C", Acb = A'C' /\ A"B"
                >
                > Bca = B'C' /\ B"A", Bac = B'A' /\ B"C"
                >
                > Cab = C'A' /\ C"B", Cba = C'B' /\ C"A"
                >
                > Which is the locus of P such that the triangles
                > AbcBcaCab, AcbBacCba are perspective?
                >
                > Figure:
                > http://anthrakitis.blogspot.gr/2013/03/perspectivity-antipedal-triangles-of-pp_19.html
                >
                > APH
                >
              • Angel
                Dear Fancisco Javier. Also: If P lies on Darboux cubic the perspector of the triangles AbcBcaCab, AcbBacCba coincides with the isotomic conjugate of the
                Message 7 of 8 , Mar 19 7:03 AM
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                  Dear Fancisco Javier.

                  Also:

                  If P lies on Darboux cubic the perspector of the triangles AbcBcaCab, AcbBacCba coincides with the isotomic conjugate of the perspector of ABC and triangle bounded by (A'A", B'B ",C'C").

                  Does this happen only if P is in the Darboux cubic?


                  Some pairs of isogonal conjugate points (on Darboux cubic), and their corresponding perspector of the triangles AbcBcaCab, AcbBacCba:

                  (X3,X4): X512; (X20,X64): X647; (X40, X84): X663; .....


                  Angel M.



                  --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
                  >
                  > They are again ALWAYS perspective. The coordinates of the perspector are shown below and it is infinite when P lies on K003.
                  >
                  > {a^2 (-2 a^4 b^2 c^2 x^2 y + 4 a^2 b^4 c^2 x^2 y - 2 b^6 c^2 x^2 y -
                  > 4 a^2 b^2 c^4 x^2 y - 4 b^4 c^4 x^2 y + 6 b^2 c^6 x^2 y -
                  > a^6 c^2 x y^2 + a^4 b^2 c^2 x y^2 + a^2 b^4 c^2 x y^2 -
                  > b^6 c^2 x y^2 + a^4 c^4 x y^2 - 10 a^2 b^2 c^4 x y^2 +
                  > b^4 c^4 x y^2 + a^2 c^6 x y^2 + b^2 c^6 x y^2 - c^8 x y^2 +
                  > 2 a^4 b^2 c^2 x^2 z + 4 a^2 b^4 c^2 x^2 z - 6 b^6 c^2 x^2 z -
                  > 4 a^2 b^2 c^4 x^2 z + 4 b^4 c^4 x^2 z + 2 b^2 c^6 x^2 z -
                  > 2 a^6 c^2 y^2 z - 4 a^4 b^2 c^2 y^2 z + 6 a^2 b^4 c^2 y^2 z +
                  > 4 a^4 c^4 y^2 z - 4 a^2 b^2 c^4 y^2 z - 2 a^2 c^6 y^2 z +
                  > a^6 b^2 x z^2 - a^4 b^4 x z^2 - a^2 b^6 x z^2 + b^8 x z^2 -
                  > a^4 b^2 c^2 x z^2 + 10 a^2 b^4 c^2 x z^2 - b^6 c^2 x z^2 -
                  > a^2 b^2 c^4 x z^2 - b^4 c^4 x z^2 + b^2 c^6 x z^2 +
                  > 2 a^6 b^2 y z^2 - 4 a^4 b^4 y z^2 + 2 a^2 b^6 y z^2 +
                  > 4 a^4 b^2 c^2 y z^2 + 4 a^2 b^4 c^2 y z^2 -
                  > 6 a^2 b^2 c^4 y z^2), -b^2 (-a^6 c^2 x^2 y + a^4 b^2 c^2 x^2 y +
                  > a^2 b^4 c^2 x^2 y - b^6 c^2 x^2 y + a^4 c^4 x^2 y -
                  > 10 a^2 b^2 c^4 x^2 y + b^4 c^4 x^2 y + a^2 c^6 x^2 y +
                  > b^2 c^6 x^2 y - c^8 x^2 y - 2 a^6 c^2 x y^2 +
                  > 4 a^4 b^2 c^2 x y^2 - 2 a^2 b^4 c^2 x y^2 - 4 a^4 c^4 x y^2 -
                  > 4 a^2 b^2 c^4 x y^2 + 6 a^2 c^6 x y^2 + 6 a^4 b^2 c^2 x^2 z -
                  > 4 a^2 b^4 c^2 x^2 z - 2 b^6 c^2 x^2 z - 4 a^2 b^2 c^4 x^2 z +
                  > 4 b^4 c^4 x^2 z - 2 b^2 c^6 x^2 z - 6 a^6 c^2 y^2 z +
                  > 4 a^4 b^2 c^2 y^2 z + 2 a^2 b^4 c^2 y^2 z + 4 a^4 c^4 y^2 z -
                  > 4 a^2 b^2 c^4 y^2 z + 2 a^2 c^6 y^2 z + 2 a^6 b^2 x z^2 -
                  > 4 a^4 b^4 x z^2 + 2 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 +
                  > 4 a^2 b^4 c^2 x z^2 - 6 a^2 b^2 c^4 x z^2 + a^8 y z^2 -
                  > a^6 b^2 y z^2 - a^4 b^4 y z^2 + a^2 b^6 y z^2 - a^6 c^2 y z^2 +
                  > 10 a^4 b^2 c^2 y z^2 - a^2 b^4 c^2 y z^2 - a^4 c^4 y z^2 -
                  > a^2 b^2 c^4 y z^2 + a^2 c^6 y z^2),
                  > c^2 (6 a^4 b^2 c^2 x^2 y - 4 a^2 b^4 c^2 x^2 y - 2 b^6 c^2 x^2 y -
                  > 4 a^2 b^2 c^4 x^2 y + 4 b^4 c^4 x^2 y - 2 b^2 c^6 x^2 y +
                  > 2 a^6 c^2 x y^2 + 4 a^4 b^2 c^2 x y^2 - 6 a^2 b^4 c^2 x y^2 -
                  > 4 a^4 c^4 x y^2 + 4 a^2 b^2 c^4 x y^2 + 2 a^2 c^6 x y^2 -
                  > a^6 b^2 x^2 z + a^4 b^4 x^2 z + a^2 b^6 x^2 z - b^8 x^2 z +
                  > a^4 b^2 c^2 x^2 z - 10 a^2 b^4 c^2 x^2 z + b^6 c^2 x^2 z +
                  > a^2 b^2 c^4 x^2 z + b^4 c^4 x^2 z - b^2 c^6 x^2 z + a^8 y^2 z -
                  > a^6 b^2 y^2 z - a^4 b^4 y^2 z + a^2 b^6 y^2 z - a^6 c^2 y^2 z +
                  > 10 a^4 b^2 c^2 y^2 z - a^2 b^4 c^2 y^2 z - a^4 c^4 y^2 z -
                  > a^2 b^2 c^4 y^2 z + a^2 c^6 y^2 z - 2 a^6 b^2 x z^2 -
                  > 4 a^4 b^4 x z^2 + 6 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 -
                  > 4 a^2 b^4 c^2 x z^2 - 2 a^2 b^2 c^4 x z^2 - 6 a^6 b^2 y z^2 +
                  > 4 a^4 b^4 y z^2 + 2 a^2 b^6 y z^2 + 4 a^4 b^2 c^2 y z^2 -
                  > 4 a^2 b^4 c^2 y z^2 + 2 a^2 b^2 c^4 y z^2)}
                  >
                  >
                  > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:
                  > >
                  > > Do we have one more perspectivity???
                  > >
                  > > > [APH]:
                  > > > Generalization:
                  > > > Let P,P* be two isogonal conjugate points and A'B'C',A"B"C"
                  > > > the antipedal triangles of P,P*, resp.
                  > > > Which is the locus of P such that the triangles:
                  > > > 3. ABC, Triangle bounded by (A'A",B'B",C'C")
                  > > > are perspective ?
                  > >
                  > > [Francisco]:
                  > > > With respect to 3., the triangle bounded by (A'A", B' B ",C'C")
                  > > > is ALWAYS perspective with ABC.
                  > >
                  > > Now, denote:
                  > >
                  > > Abc = A'B' /\ A"C", Acb = A'C' /\ A"B"
                  > >
                  > > Bca = B'C' /\ B"A", Bac = B'A' /\ B"C"
                  > >
                  > > Cab = C'A' /\ C"B", Cba = C'B' /\ C"A"
                  > >
                  > > Which is the locus of P such that the triangles
                  > > AbcBcaCab, AcbBacCba are perspective?
                  > >
                  > > Figure:
                  > > http://anthrakitis.blogspot.gr/2013/03/perspectivity-antipedal-triangles-of-pp_19.html
                  > >
                  > > APH
                  > >
                  >
                • Francisco Javier
                  [FJGC] If P=(x:y:z), then the perspector S is complicated: the coordinates of its isotomic conjugate, that is infinite when P lies on K004, are... ... I was
                  Message 8 of 8 , Mar 21 10:29 PM
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                    [FJGC]

                    If P=(x:y:z), then the perspector S is complicated: the coordinates of its isotomic conjugate, that is infinite when P lies on K004, are...

                    -------------------

                    I was curious about the locus of the isotomic conjugate of S when P moves along Euler line. This lead to a elimination problem and gives a cubic.

                    You can see the calculations and the results at

                    http://garciacapitan.blogspot.com.es/2013/03/an-elimination-problem.html



                    --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
                    >
                    > [APH]
                    >
                    > Generalization:
                    > Let P,P* be two isogonal conjugate points and A'B'C',A"B"C"
                    > the antipedal triangles of P,P*, resp.
                    > Which is the locus of P such that the triangles:
                    > 1. A"B"C", Triangle bounded by (AA',BB',CC')
                    > 2. A'B'C', Triangle bounded by (AA",BB",CC")
                    > 3. ABC, Triangle bounded by (A'A",B'B",C'C")
                    > are perspective ?
                    >
                    > ------------------------
                    >
                    >
                    > The loci in 1. and 2 are the same:
                    >
                    > line at infinity + circumcircle + K004 + three cubics, each one relative to one of the vertices.
                    >
                    > The cubic relative to A has equation
                    > 2 b^2 c^2 x^2 y + a^2 c^2 x y^2 + b^2 c^2 x y^2 + c^4 x y^2 +
                    > 2 b^2 c^2 x^2 z + 4 b^2 c^2 x y z + 2 a^2 c^2 y^2 z + a^2 b^2 x z^2 +
                    > b^4 x z^2 + b^2 c^2 x z^2 + 2 a^2 b^2 y z^2 =0.
                    >
                    > It intersects the line at infinity at the infinite points of the bisectors of angle A and at the infinite point of the A-altitude.
                    >
                    > It is its isogonal conjugate, then it intersects the circumcircle at the intersections of the circumcircle and the bisectors of angle A and at the antipode of A.
                    >
                    > With respect to 3., the triangle bounded by (A'A", B' B ",C'C")
                    > is ALWAYS perspective with ABC.
                    >
                    > If P=(x:y:z), then the perspector S is complicated: the coordinates of its isotomic conjugate, that is infinite when P lies on K004, are:
                    >
                    > {2 a^4 b^2 c^2 x^2 y - 4 a^2 b^4 c^2 x^2 y + 2 b^6 c^2 x^2 y +
                    > 4 a^2 b^2 c^4 x^2 y + 4 b^4 c^4 x^2 y - 6 b^2 c^6 x^2 y +
                    > a^6 c^2 x y^2 - a^4 b^2 c^2 x y^2 - a^2 b^4 c^2 x y^2 +
                    > b^6 c^2 x y^2 - a^4 c^4 x y^2 + 10 a^2 b^2 c^4 x y^2 -
                    > b^4 c^4 x y^2 - a^2 c^6 x y^2 - b^2 c^6 x y^2 + c^8 x y^2 -
                    > 2 a^4 b^2 c^2 x^2 z - 4 a^2 b^4 c^2 x^2 z + 6 b^6 c^2 x^2 z +
                    > 4 a^2 b^2 c^4 x^2 z - 4 b^4 c^4 x^2 z - 2 b^2 c^6 x^2 z +
                    > 2 a^6 c^2 y^2 z + 4 a^4 b^2 c^2 y^2 z - 6 a^2 b^4 c^2 y^2 z -
                    > 4 a^4 c^4 y^2 z + 4 a^2 b^2 c^4 y^2 z + 2 a^2 c^6 y^2 z -
                    > a^6 b^2 x z^2 + a^4 b^4 x z^2 + a^2 b^6 x z^2 - b^8 x z^2 +
                    > a^4 b^2 c^2 x z^2 - 10 a^2 b^4 c^2 x z^2 + b^6 c^2 x z^2 +
                    > a^2 b^2 c^4 x z^2 + b^4 c^4 x z^2 - b^2 c^6 x z^2 -
                    > 2 a^6 b^2 y z^2 + 4 a^4 b^4 y z^2 - 2 a^2 b^6 y z^2 -
                    > 4 a^4 b^2 c^2 y z^2 - 4 a^2 b^4 c^2 y z^2 +
                    > 6 a^2 b^2 c^4 y z^2, -a^6 c^2 x^2 y + a^4 b^2 c^2 x^2 y +
                    > a^2 b^4 c^2 x^2 y - b^6 c^2 x^2 y + a^4 c^4 x^2 y -
                    > 10 a^2 b^2 c^4 x^2 y + b^4 c^4 x^2 y + a^2 c^6 x^2 y +
                    > b^2 c^6 x^2 y - c^8 x^2 y - 2 a^6 c^2 x y^2 + 4 a^4 b^2 c^2 x y^2 -
                    > 2 a^2 b^4 c^2 x y^2 - 4 a^4 c^4 x y^2 - 4 a^2 b^2 c^4 x y^2 +
                    > 6 a^2 c^6 x y^2 + 6 a^4 b^2 c^2 x^2 z - 4 a^2 b^4 c^2 x^2 z -
                    > 2 b^6 c^2 x^2 z - 4 a^2 b^2 c^4 x^2 z + 4 b^4 c^4 x^2 z -
                    > 2 b^2 c^6 x^2 z - 6 a^6 c^2 y^2 z + 4 a^4 b^2 c^2 y^2 z +
                    > 2 a^2 b^4 c^2 y^2 z + 4 a^4 c^4 y^2 z - 4 a^2 b^2 c^4 y^2 z +
                    > 2 a^2 c^6 y^2 z + 2 a^6 b^2 x z^2 - 4 a^4 b^4 x z^2 +
                    > 2 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 + 4 a^2 b^4 c^2 x z^2 -
                    > 6 a^2 b^2 c^4 x z^2 + a^8 y z^2 - a^6 b^2 y z^2 - a^4 b^4 y z^2 +
                    > a^2 b^6 y z^2 - a^6 c^2 y z^2 + 10 a^4 b^2 c^2 y z^2 -
                    > a^2 b^4 c^2 y z^2 - a^4 c^4 y z^2 - a^2 b^2 c^4 y z^2 +
                    > a^2 c^6 y z^2, -6 a^4 b^2 c^2 x^2 y + 4 a^2 b^4 c^2 x^2 y +
                    > 2 b^6 c^2 x^2 y + 4 a^2 b^2 c^4 x^2 y - 4 b^4 c^4 x^2 y +
                    > 2 b^2 c^6 x^2 y - 2 a^6 c^2 x y^2 - 4 a^4 b^2 c^2 x y^2 +
                    > 6 a^2 b^4 c^2 x y^2 + 4 a^4 c^4 x y^2 - 4 a^2 b^2 c^4 x y^2 -
                    > 2 a^2 c^6 x y^2 + a^6 b^2 x^2 z - a^4 b^4 x^2 z - a^2 b^6 x^2 z +
                    > b^8 x^2 z - a^4 b^2 c^2 x^2 z + 10 a^2 b^4 c^2 x^2 z -
                    > b^6 c^2 x^2 z - a^2 b^2 c^4 x^2 z - b^4 c^4 x^2 z + b^2 c^6 x^2 z -
                    > a^8 y^2 z + a^6 b^2 y^2 z + a^4 b^4 y^2 z - a^2 b^6 y^2 z +
                    > a^6 c^2 y^2 z - 10 a^4 b^2 c^2 y^2 z + a^2 b^4 c^2 y^2 z +
                    > a^4 c^4 y^2 z + a^2 b^2 c^4 y^2 z - a^2 c^6 y^2 z +
                    > 2 a^6 b^2 x z^2 + 4 a^4 b^4 x z^2 - 6 a^2 b^6 x z^2 -
                    > 4 a^4 b^2 c^2 x z^2 + 4 a^2 b^4 c^2 x z^2 + 2 a^2 b^2 c^4 x z^2 +
                    > 6 a^6 b^2 y z^2 - 4 a^4 b^4 y z^2 - 2 a^2 b^6 y z^2 -
                    > 4 a^4 b^2 c^2 y z^2 + 4 a^2 b^4 c^2 y z^2 - 2 a^2 b^2 c^4 y z^2}
                    >
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