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Re: Re: Three Concurrent Lines in a Circle.

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  • xpolakis@xxxxxx.xxxxxxxxxxxxx.xxxxxxxxxx
    ... First Paragraph: The late Leon Bankoff (he died in 1997) was a Beverly Hills, California, dentist who also was a world expert on plane geometry....In 1979
    Message 1 of 6 , Dec 31, 1969
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      Julio Gonzalez Cabillon wrote:

      >At 04:49 PM 28/12/1999 -0700, Richard Guy wrote:
      >>
      >> See also Martin Gardner's article on the Asymmetric Propellor,
      >> somewhere in this year's Coll. Math. J. R.
      >>
      >
      >See:
      >
      >Gardner, Martin:
      >"The Asymmetric Propeller", _College Math Journal_, vol 30 (1999),
      >no 1, pp 18-22.

      First Paragraph:
      The late Leon Bankoff (he died in 1997) was a Beverly Hills, California,
      dentist who also was a world expert on plane geometry....In 1979 he told me
      about a series of fascinating discoveries he had made about what he called
      the asymmetric propeller theorem. He intended to discuss them in an article,
      but never got around to it. This is a summary of what he told me.

      Also:
      Bankoff, Leon - Erdos, Paul - Klamkin, Murray: The Asymmetric Propeller.
      Mathematics Magazine 46 (1973) 270-272.

      Antreas
    • xpolakis@xxxxxx.xxxxxxxxxxxxx.xxxxxxxxxx
      ... The Asymmetric Propeller Martin Gardner A theorem, seventy years old at least and of unknown origin, says that if three congruent equilateral triangles are
      Message 2 of 6 , Dec 31, 1969
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        Richard Guy wrote:

        >See also Martin Gardner's article on the Asymmetric Propellor,
        >somewhere in this year's Coll. Math. J. R.

        The Asymmetric Propeller
        Martin Gardner

        A theorem, seventy years old at least and of unknown origin,
        says that if three congruent equilateral triangles are have
        corners meeting, the midpoints of the lines joining the other
        two vertices of the triangles are vertices of an equilateral
        triangle. The late Leon Bankoff discovered that the triangles
        don't have be congruent and don't have to meet at a point.
        Martin Gardner describes the results, and conjectures that the
        triangles don't have to be triangular--squares seem to work
        as well.
        The College Mathematics Journal, January 1999.

        Antreas
      • xpolakis@xxxxxx.xxxxxxxxxxxxx.xxxxxxxxxx
        ... H. S. M. Coxeter: Given six consecutive points A, B, C, D, E and F on a circle, prove that if (AB)(CD)(EF) = (BC)(DE)(FA), then AD, BE and CF are
        Message 3 of 6 , Dec 31, 1969
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          Den Roussel wrote:

          >When Cevians are drawn from the vertices of a triangle ABC, they
          >intersect the opposite sides at A'B'C' and the circumcircle at
          >A''B''C''.
          >
          >Ceva's theorem shows that the product of the ratios of the segments of
          >the sides is equal to 1.
          >
          >(AB'/B'C) (CA'/A'B) (BC'/C'A) = 1
          >
          >In like manner, the product of the ratios of the consecutive sides of
          >the inscribed hexagon is equal to 1.
          >
          >(AB''/B''C) (CA''/A''B) (BC''/C''A) = 1
          >
          >In other words, when three concurrent lines are drawn in a circle, then
          >the hexagon they form has the property that the product of three
          >non-adjacent sides is equal to the product of the other three sides.
          >
          >I'm not sure if this is known or not.


          H. S. M. Coxeter:

          Given six consecutive points A, B, C, D, E and F on a circle, prove
          that if (AB)(CD)(EF) = (BC)(DE)(FA), then AD, BE and CF are concurrent.
          _The Mathematics Student Journal_ 27:5 (1980) 3.


          Antreas
        • Julio Gonzalez Cabillon
          - ... See: Gardner, Martin: The Asymmetric Propeller , _College Math Journal_, vol 30 (1999), no 1, pp 18-22. Regards, JGC -
          Message 4 of 6 , Dec 28, 1999
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            -


            At 04:49 PM 28/12/1999 -0700, Richard Guy wrote:
            >
            > See also Martin Gardner's article on the Asymmetric Propellor,
            > somewhere in this year's Coll. Math. J. R.
            >

            See:

            Gardner, Martin:
            "The Asymmetric Propeller", _College Math Journal_, vol 30 (1999),
            no 1, pp 18-22.

            Regards, JGC













            -
          • Julio Gonzalez Cabillon
            ... Which according to ZfM, the paper seems to provide two proofs of the theorem: If OAB, OCD and OEF are equilateral triangles of the same orientation then
            Message 5 of 6 , Dec 28, 1999
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              Antreas quoted:

              > Bankoff, Leon - Erdos, Paul - Klamkin, Murray: The Asymmetric Propeller.
              > Mathematics Magazine 46 (1973) 270-272.
              >

              Which according to ZfM, the paper seems to provide two proofs of the
              theorem:

              If OAB, OCD and OEF are equilateral triangles of the same
              orientation then the midpoints of BC, DE and FA are the
              vertices of an equilateral triangle.

              Regards, JGC
            • Richard Guy
              See also Martin Gardner s article on the Asymmetric Propellor, somewhere in this year s Coll. Math. J. R.
              Message 6 of 6 , Dec 28, 1999
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                See also Martin Gardner's article on the Asymmetric Propellor,
                somewhere in this year's Coll. Math. J. R.
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