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5334Re: [EMHL] Four collinear orthopoles

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  • Richard Guy
    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.


      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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