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Re: [EMHL] Please help me with a reference... (Third)

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  • Darij Grinberg
    Dear Gilles, ... Yes, this is the proof of Third s theorem. By the way, if I remember correctly, Third formulated his theorem with the help of antiparallels.
    Message 1 of 6 , May 17, 2004
      Dear Gilles,

      In Hyacinthos message #9794, you wrote:

      >> Precisely, the three pairwise radical axis of the 3
      >> circles , i.e. the sidelines of ABC, are concurrant,
      >> this fact is impossible for ABC non-degenerate.

      Yes, this is the proof of Third's theorem.

      By the way, if I remember correctly, Third formulated his
      theorem with the help of antiparallels.

      Sincerely,
      Darij Grinberg
    • Ken Pledger
      ... Was Third s reference earlier than 1949? I remember striking this problem as part of a longer proof of an old text-book exercise: E. A. Maxwell, Geometry
      Message 2 of 6 , Jun 1 2:54 PM
        >Dear Hyacinthians,
        >
        >The following theorem is an example of a
        >non-standard triangle geometry result because
        >the proof (at least, the proof I know) uses a
        >reductio ad absurdum with the help of the
        >fact that the three pairwise radical axes of
        >three circles always concur:
        >
        > If ABC is a non-degenerate triangle, X and X'
        > are two points on the line BC, Y and Y' are
        > two points on the line CA, and Z and Z' are
        > two points on the line AB, such that the
        > points Y, Y', Z and Z' lie on one circle, the
        > points Z, Z', X and X' lie on one circle, and
        > the points X, X', Y and Y' lie on one circle,
        > then all six points X, X', Y, Y', Z and Z'
        > lie on one circle.
        >
        >I remember having seen this result in a paper
        >by Third in the Edinburgh Math. Proceedings.
        >Can anybody give an *exact* reference with
        >year, page nos, etc.? ....


        Was Third's reference earlier than 1949? I remember striking
        this problem as part of a longer proof of an old text-book exercise:
        E. A. Maxwell, "Geometry for Advanced Pupils," (1949), p.170, ex.14.
        The first stage of the proof I know is to show that certain specially
        defined points like your Y, Y', Z, Z' are concyclic; which prepares
        the way for the radical axes argument which you mention. Maxwell
        ascribes his problem to the Oxford and Cambridge Schools Examination
        Board, presumably some time before 1949.

        Ken Pledger.
      • Darij Grinberg
        Dear Ken Pledger, Thanks for the information in Hyacinthos message #9836. Lately, I have succeeded to localise the Third reference: J. A. Third, Systems of
        Message 3 of 6 , Jun 2 2:29 AM
          Dear Ken Pledger,

          Thanks for the information in Hyacinthos message
          #9836. Lately, I have succeeded to localise the
          Third reference:

          J. A. Third, "Systems of circles analogous to
          Tucker circles", Proc. Edinburgh Math. Soc.,
          17 (1898) p. 70-99.

          In fact, the fact I called "Third theorem" is
          mentioned and proved on the very first page of
          the above article. Third states it using the
          notion of antiparallels (instead of saying that
          the points Y, Y', Z and Z' lie on one circle,
          he says that the lines YZ' and Y'Z are
          antiparallel to each other with respect to the
          angle CAB).

          Thanks to all for the replies to my inquiry.

          Sincerely,
          Darij Grinberg
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