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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