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Re: More on the Jerabek points

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  • Jean-Pierre.EHRMANN
    ... De : Steve Sigur À : Envoyé : mardi 1 février 2000 05:53 Objet : Re: [EMHL] More on the Jerabek points
    Message 1 of 7 , Feb 1, 2000
      ----- Message d'origine -----
      De : "Steve Sigur" <ssigur@...>
      � : <Hyacinthos@onelist.com>
      Envoy� : mardi 1 f�vrier 2000 05:53
      Objet : Re: [EMHL] More on the Jerabek points

      Dear Steve and all,
      Steve wrote

      Jean- Pierre wrote
      "There is an easy way to generalize the Brocard points and the first
      > >Brocard triangle :
      > >Take any point M on the line GK; the parallel lines from M to BC, CA, AB
      > >intersect the medians AA1, BB1, CC1 at Pa, Pb, Pc.
      > >Let Ma, Mb, Mc the reflections of M with respect to Pa, Pb, Pc.
      > >Then the triangle MaMbMc is triple perspective and similar to ABC. "

      > >> I think this result is significant because the line GK is a strong
      line,
      > and I suspect from their construction that Ma, Mb, and Mc are strong
      > points.
      >
      > In playing with this on GSP I have found that as M moves, Ma Mb and Mc
      > all move on a straight line through G and the perspectors dance some more
      > complicated curve that goes twice through each vertex and 3 times through
      > G. I do not recognize this curve, but you might.
      >
      > In fact, an homothecy with center G maps the first Brocard triangle to
      MaMbMc and your curve is the union of three hyperbolas through A, B, C, G,
      one of them being Kiepert hyperbola.
      I know some tiny results, almost nothing; for instance :
      - if you reflect ABC with respect to the line
      U X + V Y + W Z = 0, the new triangle is triple perspective with ABC iff
      a^4 U^4 - 2 b^2 c^2 V^2 W^2 + 4 (SA^2 - (a^2 - b^2)(a^2 - c^2)] U^2 V W +
      ...(similar terms) = 0, but it seems to be the tangential equation of a
      quite complicated curve.
      - if you rotate ABC around a point of the circumcircle, for particular
      values of the angle, the new triangle is triple perspective with ABC
      Friendly from France.
      Jean-Pierre
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