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Re: [EMHL] chords

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  • Francois Rideau
    Dear Nikolaos It seems that s true for every conchoid since one adds or substracts the same length on each line through the pole? Have you any idea on my
    Message 1 of 6 , Jul 7, 2005
      Dear Nikolaos
      It seems that's true for every conchoid since one adds or substracts
      the same length on each line through the pole?

      Have you any idea on my question on the ruled quadric?
      I must work on the generators of a ruled quadric given by three
      generators of the same system.
      Calculations are so complicated, I hesitate to begin!
      Friendly
      François

      On 7/7/05, Nikolaos Dergiades <ndergiades@...> wrote:
      > Dear François,
      > I had in mind the cardioide.
      > For example in polar coordinates (p, w)
      > the cardioide with equation p = 1 + cosw
      > has every chord AA' that passes through O
      > equal to 2 since
      > OA = 1 + cosw
      > OA' = 1 + cos(pi + w) = 1-cosw
      > AA' = OA + OA' = 2
      >
      > Best regards
      > Nikolaos Dergiades
      >
      > > -----Original Message-----
      > > From: Hyacinthos@yahoogroups.com
      > > [mailto:Hyacinthos@yahoogroups.com]On Behalf Of Francois Rideau
      > > Sent: Thursday, July 07, 2005 10:39 AM
      > > To: Hyacinthos@yahoogroups.com
      > > Subject: Re: [EMHL] chords
      > >
      > >
      > > Here what the meaning of a curve?
      > > If we begin with a "curve" C and we construct the conchoid C' of C wrt
      > > O for a length l, the reunion of C and C' is a new "curve" C" with
      > > infinitely many chords through O of length l?
      > > We must restrict to some kind of curves, for example irreducible
      > > algebraic curves?
      > > Friendly
      > > François
      > >
      > > On 7/7/05, Nikolaos Dergiades <ndergiades@...> wrote:
      > > > Dear Antreas,
      > > > we had very nice proofs for the case of circle
      > > > but before this you wrote
      > > >
      > > > > If three concurrent chords of a conic are equal, then is
      > > > > the conic a circle (and the chords diameters) ?
      > > >
      > > > and maybe somebody is trying to solve this problem.
      > > > The answer is no because we can construct an ellipse
      > > > that has three equal concurrent chords.
      > > >
      > > > Does anybody can give the equation of a curve (not circle)
      > > > that has infinite equal concurrent chords?
      > > >
      > > > Best regards
      > > > Nikolaos Dergiades
      > > >
      > > >
      > > >
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    • jpehrmfr
      Dear Antreas note that if three lines L1,L2,L3 going through a point M intercept three chords of equal length on a given conic, there exists a 4th line L4
      Message 2 of 6 , Jul 10, 2005
        Dear Antreas
        note that if three lines L1,L2,L3 going through a point M intercept
        three chords of equal length on a given conic, there exists a 4th line
        L4 going through M and intercepting a chord of the same length, namely
        the line L4 going through M such as
        <U,L1+<U,L2+<U,L3+<U,L4 = 0(mod Pi) where U is an axis of the conic
        Friendly. Jean-Pierre
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