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Reflections in the sidelines of the circumcevian triangle

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  • Antreas Hatzipolakis
    For the paper on the 20th Anniversary of Hyacinthos....... :-) Let ABC be a triangle, P,P* two isogonal conjugate points and A B C the circumcevian triangle
    Message 1 of 2 , Mar 7, 2013
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      For the paper on the 20th Anniversary of Hyacinthos....... :-)

      Let ABC be a triangle, P,P* two isogonal conjugate points
      and A'B'C' the circumcevian triangle of P.

      Which is the locus of P such that the reflections of

      1. AP, BP, CP

      2. AP*, BP*, CP*

      in the respective sidelines B'C', C'A', A'B' of A'B'C' are concurrent?

      APH
    • Francisco Javier
      This time I include the Mathematica instructions. We can find the loci: 1. circumcircle + nonic 2. circuncircle + McCay cubic ... I think that the code is neat
      Message 2 of 2 , Mar 7, 2013
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        This time I include the Mathematica instructions. We can find the loci:

        1. circumcircle + nonic
        2. circuncircle + McCay cubic

        -----

        I think that the code is neat enough:

        Det - Determinant of a matrix (included in Mathematica)

        ConjugadoIsogonal - Isogonal conjugate of a point
        TrianguloCircunceviano - Circuncevian triangle of a point
        Recta - Line through two points
        SimetriaAxial - Reflection of a point on the line through two points

        ----------

        << Baricentricas`;

        ptP = {x, y, z};
        ptQ = ConjugadoIsogonal[ptP];
        {ptA1, ptB1, ptC1} = TrianguloCircunceviano[ptP];

        Factor[Det[{
        Recta[SimetriaAxial[ptA, ptB1, ptC1],
        SimetriaAxial[ptP, ptB1, ptC1]],
        Recta[SimetriaAxial[ptB, ptC1, ptA1],
        SimetriaAxial[ptP, ptC1, ptA1]],
        Recta[SimetriaAxial[ptC, ptA1, ptB1],
        SimetriaAxial[ptP, ptA1, ptB1]]}]]

        Factor[Det[{
        Recta[SimetriaAxial[ptA, ptB1, ptC1],
        SimetriaAxial[ptQ, ptB1, ptC1]],
        Recta[SimetriaAxial[ptB, ptC1, ptA1],
        SimetriaAxial[ptQ, ptC1, ptA1]],
        Recta[SimetriaAxial[ptC, ptA1, ptB1],
        SimetriaAxial[ptQ, ptA1, ptB1]]}]]
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