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SIMSON LINES Loci

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  • Antreas
    1. Let ABC be a triangle, P,P* two isogonal conjugate points, and A B C , A B C the pedal triangles of P, P*, resp. (sharing the same pedal circle) Denote:
    Message 1 of 6 , Mar 6, 2013
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      1. Let ABC be a triangle, P,P* two isogonal conjugate points,
      and A'B'C', A"B"C" the pedal triangles of P, P*, resp.
      (sharing the same pedal circle)

      Denote:

      sA' = the Simson line of A' wrt A"B"C"
      sB' = the Simson line of B' wrt A"B"C"
      sC' = the Simson line of C' wrt A"B"C"

      Which is the locus of P such the lines sA',sB',sC' are concurrent?

      O,H are on the locus.

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

      2. Let ABC be a triangle, P,P* two cyclocevian conjugate points,
      and A'B'C', A"B"C" the cevian triangles of P, P*, resp.
      (sharing the same cevian circle)

      Denote:

      sA' = the Simson line of A' wrt A"B"C"
      sB' = the Simson line of B' wrt A"B"C"
      sC' = the Simson line of C' wrt A"B"C"

      Which is the locus of P such the lines sA',sB',sC' are concurrent?

      G,H are on the locus.


      APH
    • rhutson2
      Antreas, For (1.), the locus appears to be the McCay cubic. The concurrence points are, for (P,P*): (X(1),X(1)) - X(65) (X(3),X(4)) - X(389)
      Message 2 of 6 , Mar 6, 2013
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        Antreas,

        For (1.), the locus appears to be the McCay cubic. The concurrence points are, for (P,P*):

        (X(1),X(1)) -> X(65)
        (X(3),X(4)) -> X(389)
        (X(1075),X*(1075)) -> non-ETC 0.540058752563797
        (X(1745),X(3362)) -> non-ETC 1.986126910721852

        What is the locus of the points of concurrence?

        Randy


        --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
        >
        > 1. Let ABC be a triangle, P,P* two isogonal conjugate points,
        > and A'B'C', A"B"C" the pedal triangles of P, P*, resp.
        > (sharing the same pedal circle)
        >
        > Denote:
        >
        > sA' = the Simson line of A' wrt A"B"C"
        > sB' = the Simson line of B' wrt A"B"C"
        > sC' = the Simson line of C' wrt A"B"C"
        >
        > Which is the locus of P such the lines sA',sB',sC' are concurrent?
        >
        > O,H are on the locus.
        >
        > -------------
        >
        > 2. Let ABC be a triangle, P,P* two cyclocevian conjugate points,
        > and A'B'C', A"B"C" the cevian triangles of P, P*, resp.
        > (sharing the same cevian circle)
        >
        > Denote:
        >
        > sA' = the Simson line of A' wrt A"B"C"
        > sB' = the Simson line of B' wrt A"B"C"
        > sC' = the Simson line of C' wrt A"B"C"
        >
        > Which is the locus of P such the lines sA',sB',sC' are concurrent?
        >
        > G,H are on the locus.
        >
        >
        > APH
        >
      • Antreas Hatzipolakis
        Dear Randy I was almost sure it is an isogonal cubic, but I expected a simpler one... (the simple is for the cubic itself, not for its pivot) Thanks APH
        Message 3 of 6 , Mar 6, 2013
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          Dear Randy

          I was almost sure it is an isogonal cubic, but I expected
          a "simpler" one... (the "simple" is for the cubic itself, not for its pivot)

          Thanks

          APH

          On Wed, Mar 6, 2013 at 9:54 PM, rhutson2 <rhutson2@...> wrote:

          > **
          >
          >
          > Antreas,
          >
          > For (1.), the locus appears to be the McCay cubic. The concurrence points
          > are, for (P,P*):
          >
          > (X(1),X(1)) -> X(65)
          > (X(3),X(4)) -> X(389)
          > (X(1075),X*(1075)) -> non-ETC 0.540058752563797
          > (X(1745),X(3362)) -> non-ETC 1.986126910721852
          >
          > What is the locus of the points of concurrence?
          >
          > Randy
          >
          >
          > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
          > >
          > > 1. Let ABC be a triangle, P,P* two isogonal conjugate points,
          > > and A'B'C', A"B"C" the pedal triangles of P, P*, resp.
          > > (sharing the same pedal circle)
          > >
          > > Denote:
          > >
          > > sA' = the Simson line of A' wrt A"B"C"
          > > sB' = the Simson line of B' wrt A"B"C"
          > > sC' = the Simson line of C' wrt A"B"C"
          > >
          > > Which is the locus of P such the lines sA',sB',sC' are concurrent?
          > >
          > > O,H are on the locus.
          > >
          > > -------------
          > >
          > > 2. Let ABC be a triangle, P,P* two cyclocevian conjugate points,
          > > and A'B'C', A"B"C" the cevian triangles of P, P*, resp.
          > > (sharing the same cevian circle)
          > >
          > > Denote:
          > >
          > > sA' = the Simson line of A' wrt A"B"C"
          > > sB' = the Simson line of B' wrt A"B"C"
          > > sC' = the Simson line of C' wrt A"B"C"
          > >
          > > Which is the locus of P such the lines sA',sB',sC' are concurrent?
          > >
          > > G,H are on the locus.
          > >
          > >
          > > APH
          >


          [Non-text portions of this message have been removed]
        • rhutson2
          And for (2.), I expected the locus to be the Lucas cubic, which is self-cyclocevian, as the trivial case (X(7),X(7)) - X(65). However, the pair (X(8),X(189)
          Message 4 of 6 , Mar 6, 2013
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            And for (2.), I expected the locus to be the Lucas cubic, which is self-cyclocevian, as the trivial case (X(7),X(7)) -> X(65). However, the pair (X(8),X(189) does not lie on this locus.

            Randy

            --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
            >
            > Dear Randy
            >
            > I was almost sure it is an isogonal cubic, but I expected
            > a "simpler" one... (the "simple" is for the cubic itself, not for its pivot)
            >
            > Thanks
            >
            > APH
            >
            > On Wed, Mar 6, 2013 at 9:54 PM, rhutson2 <rhutson2@...> wrote:
            >
            > > **
            > >
            > >
            > > Antreas,
            > >
            > > For (1.), the locus appears to be the McCay cubic. The concurrence points
            > > are, for (P,P*):
            > >
            > > (X(1),X(1)) -> X(65)
            > > (X(3),X(4)) -> X(389)
            > > (X(1075),X*(1075)) -> non-ETC 0.540058752563797
            > > (X(1745),X(3362)) -> non-ETC 1.986126910721852
            > >
            > > What is the locus of the points of concurrence?
            > >
            > > Randy
            > >
            > >
            > > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
            > > >
            > > > 1. Let ABC be a triangle, P,P* two isogonal conjugate points,
            > > > and A'B'C', A"B"C" the pedal triangles of P, P*, resp.
            > > > (sharing the same pedal circle)
            > > >
            > > > Denote:
            > > >
            > > > sA' = the Simson line of A' wrt A"B"C"
            > > > sB' = the Simson line of B' wrt A"B"C"
            > > > sC' = the Simson line of C' wrt A"B"C"
            > > >
            > > > Which is the locus of P such the lines sA',sB',sC' are concurrent?
            > > >
            > > > O,H are on the locus.
            > > >
            > > > -------------
            > > >
            > > > 2. Let ABC be a triangle, P,P* two cyclocevian conjugate points,
            > > > and A'B'C', A"B"C" the cevian triangles of P, P*, resp.
            > > > (sharing the same cevian circle)
            > > >
            > > > Denote:
            > > >
            > > > sA' = the Simson line of A' wrt A"B"C"
            > > > sB' = the Simson line of B' wrt A"B"C"
            > > > sC' = the Simson line of C' wrt A"B"C"
            > > >
            > > > Which is the locus of P such the lines sA',sB',sC' are concurrent?
            > > >
            > > > G,H are on the locus.
            > > >
            > > >
            > > > APH
            > >
            >
            >
            > [Non-text portions of this message have been removed]
            >
          • Antreas
            [Antreas] ... [Randy] ... 3. Let P,P* be two isogonal conjugate points, and A B C , A B C the circumcevian triangles of P,P*, resp. Denote: sA = the Simson
            Message 5 of 6 , Mar 7, 2013
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              [Antreas]
              > > 1. Let ABC be a triangle, P,P* two isogonal conjugate points,
              > > and A'B'C', A"B"C" the pedal triangles of P, P*, resp.
              > > (sharing the same pedal circle)
              > >
              > > Denote:
              > >
              > > sA' = the Simson line of A' wrt A"B"C"
              > > sB' = the Simson line of B' wrt A"B"C"
              > > sC' = the Simson line of C' wrt A"B"C"
              > >
              > > Which is the locus of P such the lines sA',sB',sC' are concurrent?
              > >
              > > O,H are on the locus.

              [Randy]
              > For (1.), the locus appears to be the McCay cubic.
              > The concurrence points are, for (P,P*):
              >
              > (X(1),X(1)) -> X(65)
              > (X(3),X(4)) -> X(389)
              > (X(1075),X*(1075)) -> non-ETC 0.540058752563797
              > (X(1745),X(3362)) -> non-ETC 1.986126910721852
              >
              > What is the locus of the points of concurrence?

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

              sA' = the Simson line of A' wrt A"B"C"
              sB' = the Simson line of B' wrt A"B"C"
              sC' = the Simson line of C' wrt A"B"C"

              Which is the locus of P such the lines sA',sB',sC'
              are concurrent?

              The McCay cubic ??

              Antreas
            • Angel
              Dear Antreas For (1.), the locus is the McCay cubic and the circumcircle For (2. Message#321670), the locus is the sextic: CiclicSum[
              Message 6 of 6 , Mar 7, 2013
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                Dear Antreas


                For (1.), the locus is the McCay cubic and the circumcircle


                For (2. Message#321670), the locus is the sextic:
                CiclicSum[ a^2(b^2-c^2)y^3z^3-b^2c^2x^3y(y-z)z) ] =0,

                The vertices of ABC are singular points of multiplicity 3; the vertices of the anticomplementary triangle are inflexion points with tangents the medians. The sextic contains the centroid.


                For (3.), the locus is the McCay cubic, the cubic Kjp = K024 and the circumcircle

                Can be a lucus propertie of K024 not cited in Bernard Gibert Cataloge, CTC?


                Best regards
                Angek Montesdeoca

                --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
                >
                > [Antreas]
                > > > 1. Let ABC be a triangle, P,P* two isogonal conjugate points,
                > > > and A'B'C', A"B"C" the pedal triangles of P, P*, resp.
                > > > (sharing the same pedal circle)
                > > >
                > > > Denote:
                > > >
                > > > sA' = the Simson line of A' wrt A"B"C"
                > > > sB' = the Simson line of B' wrt A"B"C"
                > > > sC' = the Simson line of C' wrt A"B"C"
                > > >
                > > > Which is the locus of P such the lines sA',sB',sC' are concurrent?
                > > >
                > > > O,H are on the locus.
                >
                > [Randy]
                > > For (1.), the locus appears to be the McCay cubic.
                > > The concurrence points are, for (P,P*):
                > >
                > > (X(1),X(1)) -> X(65)
                > > (X(3),X(4)) -> X(389)
                > > (X(1075),X*(1075)) -> non-ETC 0.540058752563797
                > > (X(1745),X(3362)) -> non-ETC 1.986126910721852
                > >
                > > What is the locus of the points of concurrence?
                >
                > 3. Let P,P* be two isogonal conjugate points,
                > and A'B'C', A"B"C" the circumcevian triangles
                > of P,P*, resp.
                > Denote:
                >
                > sA' = the Simson line of A' wrt A"B"C"
                > sB' = the Simson line of B' wrt A"B"C"
                > sC' = the Simson line of C' wrt A"B"C"
                >
                > Which is the locus of P such the lines sA',sB',sC'
                > are concurrent?
                >
                > The McCay cubic ??
                >
                > Antreas
                >
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