Re: intersection of parabola and line
- From: "Philippe 92" <nospam@xxxxxxxxxxxx>
- Date: Thu, 01 Nov 2007 17:01:58 +0100
frank wrote :
On 28 okt, 21:07, "Philippe 92" <nos...@xxxxxxxxxxxx> wrote:
frank wrote :
Hi there,
I am looking for a rule and compass construction of the intersection
points (if they exist) of a line L with a parabola P 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.shtml
http://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 the parabola is 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 at http://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+news@xxxxxxx
site : http://chephip.free.fr/ (recreational mathematics)
.
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