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loci of concentric isogonals

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  • rhutson2
    Dear friends, Given two fixed isogonal points, X and X , and two variable isogonal points, P and P , what is the locus of P such that X, X , P, P are
    Message 1 of 3 , Mar 31, 2013
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      Dear friends,

      Given two fixed isogonal points, X and X', and two variable isogonal
      points, P and P', what is the locus of P such that X, X', P, P' are
      concyclic?

      Special cases: X,X' = G,K; O,H; 1st and 2nd Brocard points?

      Best regards,
      Randy
    • Bernard Gibert
      Dear Randy, ... I find a bicircular isogonal circum-sextic passing through the in/excenters, X, X , the intersections of (O) and the line XX , their isogonal
      Message 2 of 3 , Apr 1 1:24 AM
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        Dear Randy,

        > Given two fixed isogonal points, X and X', and two variable isogonal
        > points, P and P', what is the locus of P such that X, X', P, P' are
        > concyclic?
        >
        > Special cases: X,X' = G,K; O,H; 1st and 2nd Brocard points?

        I find a bicircular isogonal circum-sextic passing through the in/excenters, X, X', the intersections of (O) and the line XX', their isogonal conjugates at infinity.
        A, B, C, X, X' are nodes.

        When X lies on (O), the sextic splits into (O), the line at infinity and the pK(X6, X). See

        http://bernard.gibert.pagesperso-orange.fr/Tables/table17.html

        These sextics seem to be not very prolific in ETC centers, in particular your special cases.

        There are 3 bicircular isogonal circum-sextics in CTC but none of them corresponds to this configuration.

        Who's going to find a nice one ?

        Best regards

        Bernard

        [Non-text portions of this message have been removed]
      • rhutson2
        Thanks, Bernard and Francisco! A couple of related problems: 1. What is the locus of the circumcenters? 2. Do ETC centers X(i) and X(j) exist such that X(i),
        Message 3 of 3 , Apr 2 10:21 AM
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          Thanks, Bernard and Francisco!

          A couple of related problems:

          1. What is the locus of the circumcenters?

          2. Do ETC centers X(i) and X(j) exist such that X(i), X(j) and their
          isogonal conjugates (whether in ETC or not) are concyclic?

          Randy


          --- In Hyacinthos@yahoogroups.com, Bernard Gibert <bg42@...> wrote:
          >
          > Dear Randy,
          >
          > > Given two fixed isogonal points, X and X', and two variable isogonal
          > > points, P and P', what is the locus of P such that X, X', P, P' are
          > > concyclic?
          > >
          > > Special cases: X,X' = G,K; O,H; 1st and 2nd Brocard points?
          >
          > I find a bicircular isogonal circum-sextic passing through the
          in/excenters, X, X', the intersections of (O) and the line XX', their
          isogonal conjugates at infinity.
          > A, B, C, X, X' are nodes.
          >
          > When X lies on (O), the sextic splits into (O), the line at infinity
          and the pK(X6, X). See
          >
          > http://bernard.gibert.pagesperso-orange.fr/Tables/table17.html
          >
          > These sextics seem to be not very prolific in ETC centers, in
          particular your special cases.
          >
          > There are 3 bicircular isogonal circum-sextics in CTC but none of them
          corresponds to this configuration.
          >
          > Who's going to find a nice one ?
          >
          > Best regards
          >
          > Bernard
          >
          > [Non-text portions of this message have been removed]
          >
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