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Re: [EMHL] loci of concentric isogonals

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  • 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 1 of 3 , Apr 1, 2013
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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 2 of 3 , Apr 2, 2013
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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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