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## 5334Re: [EMHL] Four collinear orthopoles

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• May 1 12:18 PM
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)
>
>
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