Loading ...
Sorry, an error occurred while loading the content.

[EMHL] Re: Parry Locus

Expand Messages
  • peter_mows
    Dear Antreas, ... Another part is the Jerabek hyperbola with perpector on the Euler line. Best regards, Peter.
    Message 1 of 7 , Sep 8, 2005
    • 0 Attachment
      Dear Antreas,

      > [APH]:
      > >> Let ABC be a triangle and P a point.
      > >>
      > >> Which is the locus of P such that the reflections of:
      > >> PA in BC, PB in CA, and PC in AB, are concurrent?
      >
      > Now, let A'B'C' be the medial triangle of ABC.
      >
      > Which is the locus of P such that the reflections of:
      > PA in B'C', PB in C'A', and PC in A'B', are concurrent?
      >
      > It is well known that the line at infinity is part of the
      > locus, since three parallels through A,B,C, reflected on
      > the sidelines of the medial triangle concur at
      > the Nine Point Circle of ABC.
      >

      Another part is the Jerabek hyperbola with perpector on the Euler
      line.

      Best regards,
      Peter.
    • Antreas P. Hatzipolakis
      Dear Peter ... We can replace the medial triangle with the orthic, and ask for the locus etc In general: Let ABC be a triangle, Q = (u:v:w) a fixed point,
      Message 2 of 7 , Sep 8, 2005
      • 0 Attachment
        Dear Peter

        >> [APH]:
        >> >> Let ABC be a triangle and P a point.
        >> >>
        >> >> Which is the locus of P such that the reflections of:
        >> >> PA in BC, PB in CA, and PC in AB, are concurrent?
        >>
        >> Now, let A'B'C' be the medial triangle of ABC.
        >>
        >> Which is the locus of P such that the reflections of:
        >> PA in B'C', PB in C'A', and PC in A'B', are concurrent?
        >>
        >> It is well known that the line at infinity is part of the
        >> locus, since three parallels through A,B,C, reflected on
        >> the sidelines of the medial triangle concur at
        >> the Nine Point Circle of ABC.

        [PM]:
        >Another part is the Jerabek hyperbola with perpector on the Euler
        >line.

        We can replace the medial triangle with the orthic, and ask for
        the locus etc

        In general:

        Let ABC be a triangle, Q = (u:v:w) a fixed point, QaQbQc its pedal
        triangle, and P a variable point.

        Which is the locus of P such that the reflections of:
        PA in QbQc, PB in QcQa, and PC in QaQb, are concurrent?


        Antreas




        --
      • peter_mows
        Dear Antreas, [APH] ... Orthic, 2 cubics one, of which passes through X{3,24,186,1299,2931}. Anticomp, 2 cubics, one of which is Bernards K025. Excentral,
        Message 3 of 7 , Sep 8, 2005
        • 0 Attachment
          Dear Antreas,

          [APH]
          >
          > We can replace the medial triangle with the orthic, and ask for
          > the locus etc
          >
          > In general:
          >
          > Let ABC be a triangle, Q = (u:v:w) a fixed point, QaQbQc its pedal
          > triangle, and P a variable point.
          >
          > Which is the locus of P such that the reflections of:
          > PA in QbQc, PB in QcQa, and PC in QaQb, are concurrent?

          Orthic, 2 cubics one, of which passes through X{3,24,186,1299,2931}.
          Anticomp, 2 cubics, one of which is Bernards K025.
          Excentral, whole plane, perspector the isogonal of P.

          General case looks as if it could be messy.

          Best regards,
          Peter.
        • jpehrmfr
          Dear Antreas ... A circular circumcubic going through the isogonal conjugate of Q. Jean-Pierre
          Message 4 of 7 , Sep 8, 2005
          • 0 Attachment
            Dear Antreas
            > In general:
            >
            > Let ABC be a triangle, Q = (u:v:w) a fixed point, QaQbQc its pedal
            > triangle, and P a variable point.
            >
            > Which is the locus of P such that the reflections of:
            > PA in QbQc, PB in QcQa, and PC in QaQb, are concurrent?

            A circular circumcubic going through the isogonal conjugate of Q.
            Jean-Pierre
          Your message has been successfully submitted and would be delivered to recipients shortly.