Re: intersection of parabola and line

On 1 nov, 17:01, "Philippe 92" <nos...@xxxxxxxxxxxx> wrote:
frank wrote :

On 28 okt, 21:07, "Philippe 92" <nos...@xxxxxxxxxxxx> wrote:
frank wrote :

Hi there,

I am looking for aruleandcompassconstruction of the intersection
points (if they exist) of a line L with aparabolaP with given
directrix and focus.

See figure :http://i21.tinypic.com/v3opxh.gif

Dear Philippe,

Thank you for this magnificent construction. You filled me with awe!

I didn't want to ...

Allow me just to inquire whether you made up this construction from
scratch or you just happened to know it (or knew where to find it) -
in the first case, I would be interested to learn where I could learn
your art, in the latter I would like to know where I could find
similar exercises.

I don't remember from where comes my construction. However I would
not be surprised if I found it by myself !
Sure it is known, and/or can be found as exercices in class books...

It comes from a page that I designed for my web site last year, and
that I didn't have the time to finalize and publish, about a few
constructions related with conic sections, the first chapter is
about parabolas.
This was motivated by a few problems on math forums at that time.

For the way I solved it and for my sources, they come from my old
class books, plus a few books about projective geometry, triangle
geometry etc... (all in french).
And also the "cut-the-knot" web site is a source of inspiring
problems, some with solutions as well.
See also Wolfram and Wikipedia and search through Google, and also
a very interesting site, but in dutch language.
Some abstract of my favorites :

http://www.cut-the-knot.org/content.shtmlhttp://mathworld.wolfram.com/http://www.pandd.demon.nl/

This last one is in dutch language, but I found there simple proofs
of Poncelet porism, Descarte/Soddy formula etc...

http://www.mathcurve.com/
A very complete encyclopedia of curves, unfortunately in french,
but gives a lot of usefull properties of curves.

For the present problem, I think my line of thoughts about it (find
the points) was to find the center of some circle which ... As I know
that theparabolais the locus of centers of circles tangent to
directrix and going at focus.
This is related with Apollonius problem (find circle tangent
to 3 objects, objects being independantly lines/circles/points).
So having the construction method of Apollonius problems in mind,
may be this lend me to the solution of the present problem.

But I really don't remember now what was my exact thinking line
last year when I solved it...

Thank you for motivating me to publish that page on my web site.
It is now athttp://chephip.free.fr/themes_en/parabole1.html
Please be patient when loading the page as it uses dynamic applet
for drawing the figures, so you can drag points to change the
parabolas, lines etc...

Regards.

--
Philippe C., mail : chephip+n...@xxxxxxx
site :http://chephip.free.fr/ (recreational mathematics)

Thanks again. Actually, the construction I asked you about was a
question in a book that I used in my secondary (Latin-Maths)
education, now more than 30 years ago.I remember that we were not
asked to solve it (perhaps our teacher did not know the answer
either). I purchased a second hand version of the very book a few
weeks ago (I like to do some maths for fun), and I had a go at the
problem, but got frustrated after thinking for some two weeks about
it.That is why I posted the problem.The solution you gave (including
its proof), and the way you got at it reassure me of the fact that I
could never ever have come up with it myself. In fact, I am quite
convinced that nobody could have solved the problem using the maths
that we learned at school.For example, we were not familiar with the
relationship that leads to PH^2 = PF.PF' in your proof; I had to look
that up (is http://www.cut-the-knot.org/pythagoras/PPower.shtml the
right reference?).

Thank you for the very useful pointers to websites you gave. The
french and dutch are no problem whatsoever: I am Belgian. Thanks as
well for having added that page to your website. Perhaps you should
add the proofs as well.

I'll leave you now, because I have a battle to conduct. At Hastings.
And it is all your fault.

Sincerely yours,

Frank

.

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