## Re: [EMHL] chords

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• 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

> 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
>
> > -----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
> >
> > > 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
> > >
> > >
> > >
> > > Yahoo! Groups Links
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• 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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