Four collinear orthopoles

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• Dear all While playing around with 4 lines,I came upon this result which I feel is interesting.Is this known? The orthopoles of the four triangles formed by
Message 1 of 3 , May 1, 2002
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Dear all
While playing around with 4 lines,I came upon this result which I
feel is interesting.Is this known?

The orthopoles of the four triangles formed by any four arbitrary
lines,taken 3 at a time,with respect to the other 4th line respectively,are
collinear.(Sorry for not being able to quote it properly)

In other words,consider any 4 arbitrary lines,l1,l2,l3,l4.
Let P1 be the orthopole of the triangle formed by l2,l3,l4 w.r.t line
l1.Similarly define P2,P3 and P4.Then Pi(i=1,2,3,4) are collinear.

Does the same hold for extended orthopole also?

Yours faithfully
Atul.A.Dixit

P.S(to Paul Yiu)-The another property of 4 collinear orthopoles which I had
sent earlier is same as that of the one (by Goormaghtigh) which you
stated.It was my carelessness,that I didn't look at it properly.Sorry for
the confusion.But I feel,that the only additional thing is that it works
also when the fourth vertex is inside the triangle formed by other 3.
(I'm sorry if this mail is sent again;but didn't receive this mail in
my inbox from quite a long time)

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• Do the following theorems help? There s a unique parabola which touches 4 lines (and the line at infinity). Three tangents to a parabola form a triangle whose
Message 2 of 3 , May 1, 2002
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Do the following theorems help?

There's a unique parabola which touches
4 lines (and the line at infinity).

Three tangents to a parabola form a
triangle whose circumcircle passes
through the focus and whose orthocentre
lies on the directrix.

R.

On Wed, 1 May 2002, Atul Dixit wrote:

> Dear all
> While playing around with 4 lines,I came upon this result which I
> feel is interesting.Is this known?
>
> The orthopoles of the four triangles formed by any four arbitrary
> lines,taken 3 at a time,with respect to the other 4th line respectively,are
> collinear.(Sorry for not being able to quote it properly)
>
> In other words,consider any 4 arbitrary lines,l1,l2,l3,l4.
> Let P1 be the orthopole of the triangle formed by l2,l3,l4 w.r.t line
> l1.Similarly define P2,P3 and P4.Then Pi(i=1,2,3,4) are collinear.
>
> Does the same hold for extended orthopole also?
>
> Yours faithfully
> Atul.A.Dixit
>
> P.S(to Paul Yiu)-The another property of 4 collinear orthopoles which I had
> sent earlier is same as that of the one (by Goormaghtigh) which you
> stated.It was my carelessness,that I didn't look at it properly.Sorry for
> the confusion.But I feel,that the only additional thing is that it works
> also when the fourth vertex is inside the triangle formed by other 3.
> (I'm sorry if this mail is sent again;but didn't receive this mail in
> my inbox from quite a long time)
>
>
> _________________________________________________________________
> Join the worlds largest e-mail service with MSN Hotmail.
> http://www.hotmail.com
>
>
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>
• Dear Atul and Richard ... If we add that the orthopole wrt ABC of a line L lies on the directrix of the inscribed parabola tangent to L, I think that
Message 3 of 3 , May 1, 2002
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Dear Atul and Richard
> Do the following theorems help?
>
> There's a unique parabola which touches
> 4 lines (and the line at infinity).
>
> Three tangents to a parabola form a
> triangle whose circumcircle passes
> through the focus and whose orthocentre
> lies on the directrix.
>
> R.

If we add that the orthopole wrt ABC of a line L lies on the
directrix of the inscribed parabola tangent to L, I think that
everything is clear now.
Friendly. Jean-Pierre

> > Dear all
> > While playing around with 4 lines,I came upon this result
which I
> > feel is interesting.Is this known?
> >
> > The orthopoles of the four triangles formed by any four arbitrary
> > lines,taken 3 at a time,with respect to the other 4th line
respectively,are
> > collinear.(Sorry for not being able to quote it properly)
> >
> > In other words,consider any 4 arbitrary lines,l1,l2,l3,l4.
> > Let P1 be the orthopole of the triangle formed by l2,l3,l4 w.r.t
line
> > l1.Similarly define P2,P3 and P4.Then Pi(i=1,2,3,4) are collinear.
> >
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