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14678Θέμα: [EMHL] Re: Locus of Simson line intersection point

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  • Nikolaos Dergiades
    Jan 1, 2007
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      Dear Jean-Pierre,

      Thank you very much.

      > > many thanks for your nice mails about this
      > problem.
      > > You've probably noticed that when the angle of the
      > Simson lines is
      > > Pi/3, we get a nice trifolium inscribed in the
      > circle(N,R) N = NP-
      > center
      > > With N as pole and a tangent at a cusp of the
      > Steiner deltoid as
      > polar
      > > axis, the trifolium has polar equation rho =
      > R.cos(3.theta)
      > Some remarks about this trifolium :
      > The vertices - contact points with the circle(N,R) -
      > are the vertices
      > of the equilateral triangle bounded by the Simson
      > lines of the
      > vertices of the circumnormal triangle; these Simson
      > lines are the
      > common tangents of the Steiner deltoid and the
      > NP-circle at the
      > points where they touch each other.

      These points are found by drawing from N parallells
      to the bisectors of Morley's triangle.

      > These vertices
      > are homothetic of
      > the cusps of the Steiner deltoid in (N,2/3)
      > The rectangular circumhyperbola through N intersects
      > the circle (N,R)
      > at 4 points; one of them is the antipode of N on the
      > hyperbola; the
      > three other ones are the vertices above.

      Very interesting.

      Happy New Year to you and all Hyacinthists.
      Best regards
      Nikos Dergiades.

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