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Re: SIMSON LINES Loci

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  • 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 1 of 6 , Mar 7 2:13 PM
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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 2 of 6 , Mar 7 4:37 PM
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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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