Construct (with R&C) the intersection of a line <r>

with a conic (parabola) given by its focus <F> and

directrix <d>.

Dear Hyacinthists,

Dear François,

I will reply directly from the archives.

I have never received this message.

===

>The center of C is one of the intersections sought.

>Otherwise perform a homothecy of C of pole P.

> Please, gimme exactly your construction with your homothecy!

I will suppose the "ideal" position of the data.

Let P=d/\r. With P_0 on r as center draw a circle C

tangent to d. Let P_1 be the contact point.

Draw s=(P,F) and let F_1 and F_2 be the intersections

between <s> and <C>. Draw t=(P_0,F_1) and u=(P_0,F_2).

Draw from <F> t'||t and u'||u. The intersections O=r/\t'

and O'=r/\u' are the intersections of <r> and the parabola.

Your solution seems nice but I still have to

understand it.

Friendly,

Luis

--- In

Hyacinthos@yahoogroups.com, "Francois Rideau"

<francois.rideau@...> wrote:

>

> Dear Luis

> ===

> I think we have a big problem when <d> and <r> are

> near orthogonal

> ===

> I don't think so. In this case one has P=d/\r and with P

> as pole one makes a homothecy. Draw a circle C with center

> in r and tangent do d. If C goes through F fine.

>

> >

> > That's just the problem of this construction: to find circles

tangent to

> > the directrix <d> and through the focus <F> of which the centers

are on line

> > <r>. These centers are the sought points of intersection. These

circles are

> > also on point <F'> symmetric of <F> wrt line <r>. So these

circles are in a

> > pencil and their intersections with <d> are in involution,

(Desargues

> > theorem). For example, if <Gamma> is a circle in this pencil

cutting <d> in

> > <P> and <Q>, then JP.JQ = JF.JF' = power of J wrt <Gamma>,

(product of

> > oriented segments), where J = FF' /\ <d>. In case of a touching

circle in T,

> > we have JT² = JF.JF'. The circle of center J and radius JT is

orthogonal

> > to all circles of the pencil. So draw any circle of the pencil

and draw

> > segments tangents to this circle through J and so on...