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ex-extra perspectors

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  • Fred Lang
    Dear Hyacinthers, A question: How can I construct the triangle (extra-operation) PaPbPc from P =Po ? A remark: Rouché-Comberousse (Note III sur la géométrie
    Message 1 of 2 , Sep 6, 2000
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      Dear Hyacinthers,
      A question: How can I construct the triangle (extra-operation) PaPbPc
      from P =Po ?
      A remark: Rouché-Comberousse (Note III sur la géométrie récente du
      triangle, traité de géométrie 1912)
      speak about "points associés" instead of "precevians points" and "points
      adjoints" instead of "extra-points".

      Regards
      Fred Lang
      Switzerland
    • John Conway
      ... You can t construct Pa Pb Pc from Po, because they aren t functions of Po. Rather, they are the algebraic conjugates of the functions that were used
      Message 2 of 2 , Sep 9, 2000
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        On Wed, 6 Sep 2000, Fred Lang wrote:

        > A question: How can I construct the triangle (extra-operation) PaPbPc
        > from P =Po ?
        > A remark: Rouché-Comberousse (Note III sur la géométrie récente du
        > triangle, traité de géométrie 1912)
        > speak about "points associés" instead of "precevians points"
        > and "points adjoints" instead of "extra-points".

        You can't construct Pa Pb Pc from Po, because they
        aren't functions of Po. Rather, they are the algebraic
        conjugates of the functions that were used to define Po.

        Let's consider, for instance,

        the incenter Io = (a:b:c),
        and
        the Spieker point So = (b+c:c+a:a+b).

        For these, b-extraversion yields

        Ib = (a:-b:c) and Sb = (c-b:c+a:a-b).

        Now for an equilateral triangle both Io and So are
        the center, but we have

        Ib = (1:-1:1) , Sb = (0:2:0),

        which are different.

        The extraversion operations are not defined on points,
        but rather, on constructions for points.

        I'm glad to hear your remark, which confirms that the
        standard term for A^P, B^P, C^P is "harmonic associates"
        (of P). I have not previously seen a term for extraversion
        in any published work, and so am glad to hear of "points
        adjoints".

        My guess is that it doesn't mean precisely the same thing,
        because there are several closely related notions here, from
        which I chose the particular one I call "extraversion" only
        after considerable thought.

        John Conway
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