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5333Four collinear orthopoles

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  • Atul Dixit
    May 1 9:05 AM
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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

      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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