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Re: SIMSON LINES Loci

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  • 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 1 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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