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

[EMHL] Re: Parry Locus

Expand Messages
  • 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 1 of 7 , Sep 8 6:26 AM
    • 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 2 of 7 , Sep 8 7:21 AM
      • 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 3 of 7 , Sep 8 8:25 AM
        • 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.