- Dear Garcia, Nikolaos, and François,

here is an observation that might lead to a more simple construction of

the focal points of the two parabolas circumscribing a (convex)

quadrilateral:

Note that for every triangle self-polar w.r.t. a parabola the three

lines connecting the midpoints of its sides are tangent lines to the

parabola. This means that the focus of a parabola lies on the nine-point

circle of any triangle which is self-polar with respect to that

parabola. Since the diagonal triangle of any quadrilateral is

self-polar w.r.t. any conic section circumscribing the quadrilateral, it

follows that the focal points of both circumscribing parabolas are on

the nine-point circle of the diagonal triangle. Moreover, in addition to

the four points on the parabola, we now have four lines tangent to the

parabola (the three lines connecting the midpoints of the diagonal

triangle and the line at infinity). Perhaps it is possible to now find

the focal points as the points of intersection of this nine-point circle

with a straight line.

best,

Eisso

--

========================================

Eisso J. Atzema, Ph.D.

Department of Mathematics & Statistics

University of Maine

Orono, ME 04469

Tel.: (207) 581-3928 (office)

(207) 866-3871 (home)

Fax.: (207) 581-3902

E-mail: atzema@...

======================================== - Dear François,

I hoped the drawing that I uploaded would clear that up. Anyway, what I

mean is that if the line through the midpoints of EFG intersects AD in

F*, BC in F**, AB in G* and CD in G** then EF* is parallel to BC, EF**

is parallel to AD, EG* is parallel to CD and G** is parallel to AB.

Eisso

Francois Rideau wrote:>

--

> >

> > Dear Eisso

> >

> > Moreover, the lines connecting the points of intersection of this lines

> > with the sides of ABCD with E are each parallel to the side opposite the

> > side they are on.

> >

>

> I don't see exactly what you mean here.Can you be more explicit?

> >

>

> As for the sides of the medial triangle of the diagonal triangle of

> ABCD are

> > tangent to the parabola, it is just a consequence of the fact that for a

> > point M having polar line L wrt a parabola <P>, the line homothetic

> of L in

> > the dilation of center M and ratio 1/2 is tangent to the parabola.

> >

>

> .

> > Friendly

> >

>

> Francois

>

> >

>

> [Non-text portions of this message have been removed]

>

>

========================================

Eisso J. Atzema, Ph.D.

Department of Mathematics & Statistics

University of Maine

Orono, ME 04469

Tel.: (207) 581-3928 (office)

(207) 866-3871 (home)

Fax.: (207) 581-3902

E-mail: atzema@...

========================================