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[EMHL] Re: Reflections (a sequence of points)

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  • Antreas P. Hatzipolakis
    Dear Peter [APH] ... I think that the 1st (ie for n=1) trilinear perspector is (sin3A/sin4A ::), and, in general, the n-th one is: (sin [2^(n+1) - 1]A / sin
    Message 1 of 8 , Aug 26, 2005
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      Dear Peter

      [APH]
      >>
      >> Let ABC be a triangle, and AoBoCo the orthic triangle of ABC.
      >>
      >> A1 := (Reflection of BoCo in AoBo) /\ (Reflection of BoCo in AoCo).
      >>
      >> A2 := (Reflection of BoCo in A1Bo) /\ (Reflection of BoCo in A1Co)
      >>
      >> .......
      >>
      >> An := (Reflection of BoCo in An-1Bo) /\ (Reflection of BoCo in
      >> An-1Co)
      >> Similarly Bn,Cn
      >>
      >> Are the triangles ABC, AnBnCn perspective?

      [PM]:
      >Yes!. Quite where I don't yet know.


      I think that the 1st (ie for n=1) trilinear perspector is

      (sin3A/sin4A ::), and, in general, the n-th one is:


      (sin [2^(n+1) - 1]A / sin [2^(n+1)]A ::)

      Locus:

      Let ABC be a triangle, P a point and A'B'C' the pedal
      triangle of P.

      A1 := (Reflection of B'C' in A'B') /\ (Reflection of B'C' in A'C').

      Similarly B1,C1.

      Which is the locus of P such that ABC, A1B1C1 are perspective ?

      If my computations are correct, then

      ( ABC, A1B1C1 are perspective ) <==>

      cos(2Cb + Ab)
      ------------------ * CYCLIC = 1
      cos(2Bc + Ac)

      where Ab = angle(PAB), Ac = angle(PAC) etc.

      This leads to a 9th degree equation in terms of x,y,z,
      where (x:y:z) are the trilinears of P.

      (H obviously lies on the locus, since Ab = 90-B deg. etc)


      Antreas









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