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RE: [EMHL] The orthic limit.

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  • Michael Lambrou
    ... see ... As John pointed out, this part of the argument is erroneous. Sorry for this but I wrote in haste after a five hour(1) Departmental meeting.
    Message 1 of 9 , Jun 5, 2002
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      >
      > One last point: If G is the (common) centre of gravity of the nth
      > circle and O(n) its circumcirle then, as GO(n+1)= (1/2)* GO(n), we
      see
      > the exact rate at which O(n) approaches G. Note also that the sequence
      > O(n), which is on the Euler line, alternates sides around G.
      >

      As John pointed out, this part of the argument is erroneous. Sorry for
      this but I wrote in haste after a five hour(1) Departmental meeting.

      Michael.
    • jpehrmfr
      Dear John [JPE] ... can ... [JHC] ... definition ... that s ... I m just looking at a problem of a French examination (Ecole Centrale : May 2002). The problem
      Message 2 of 9 , Jun 5, 2002
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        Dear John

        [JPE]
        > > If A[n],B[n],C[n] is the n-th orthic triangle (with A[0]=A), we
        can
        > > see, by induction that
        > > B[n]C[n] = R|sin(2^n A)|/2^(n-1)
        > > and A[n]A[n+1] = R|sin(2^n B)sin(2^n C)|/2^(n-1)

        [JHC]

        > I was aware of this behavior, but thought that maybe the
        definition
        > could be extended in some way to circumvent it. I admit that
        that's
        > rather unlikely, but would like to see a proof that the orthic limit
        > isn't analytic.

        I'm just looking at a problem of a French examination (Ecole
        Centrale : May 2002).
        The problem studies the particular case of an isosceles triangle
        A[0]=(0,0); B[0]=(1, cot t); C[0]=(1, -cot t) (in rectangular
        cartesian coordinates) where 0<t<Pi.
        The sequence converges to(x(t),0) where
        x(t) = 1 + sum((-1/2)^n (sin(2^n t)/sin(t))^2, n = 1..infinity)
        or (using Fourier expansion)
        x(t) sin(t)^2 = 1/3 + sum((-1/2)^n cos(2^n t), n=1..infi)
        Then, it is proved that x is continuous on (0,Pi) and admits a
        derivative at no point.
        The whole problem is available at
        http://csmp.ecp.fr/CentraleSupelec/2002/PC/sujets/math1.pdf
        Friendly. Jean-Pierre
      • arnaud.pascal
        Dear friends Owing to google, i just see this reference with the keywords orthic limit pedal triangle www.mathstat.usouthal.edu/~hitt/research/markov.pdf I
        Message 3 of 9 , Jun 5, 2002
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          Dear friends

          Owing to google, i just see this reference with the keywords
          orthic limit pedal triangle
          www.mathstat.usouthal.edu/~hitt/research/markov.pdf
          I haven't the time to analyze now
          Maybe it can help you,

          regards from the rainy south east of the France

          Arnaud PASCAL, teacher in highschool

          [Non-text portions of this message have been removed]
        • Quim Castellsaguer
          I think that this problem is solved in The Sequence of Pedal Triangles , by J.G. Kingston and J.L. Synge, American Mathematical Monthly volume 95 number 7,
          Message 4 of 9 , Jun 5, 2002
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            I think that this problem is solved in "The Sequence of Pedal Triangles", by
            J.G. Kingston and J.L. Synge, American Mathematical Monthly volume 95 number
            7, 1988.

            Quim Castellsaguer
            Barcelona (Spain)

            En/Na John Conway ha escrit:

            > Let A1 B1 C1 be the orthic triangle of A B C, A2 B2 C2 the
            > orthic triangle of A1 B1 C1, and so on. Then I think it's pretty
            > obvious that the triangles An Bn Cn tend to a point, which I'll
            > call the orthic limit point of A B C.
            >
            > Of course this is only one of a number of such limits, but it's
            > about the simplest whose location isn't immediately obvious. May
            > I ask:
            >
            > 1) is it indeed true that the limit exists?
            >
            > 2) is it "algebraic" (ie., are its barycentric or normal coordinates
            > algebraic functions of a,b,c ?
            >
            > John Conway
            >
            >
            >
            > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
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