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22015Fwd: [EGML] RE: Numerically confirmed points concerning Den Roussel equilateral triangle

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  • Antreas Hatzipolakis
    Nov 12, 2013
    • 0 Attachment


      ---------- Forwarded message ----------
      From:  Seiichi Kirikami
      Date: Tue, Nov 12, 2013 at 12:40 PM
      Subject: [EGML] RE: Numerically confirmed points concerning Den Roussel equilateral triangle
      To: Anopolis@yahoogroups.com


       

      Dear Barry,

       

      If you would see this message, I would like to hear your opinion whether the above described points([0], [1], [2] and [3]) might have such coordinates (sum {hmn <m, n>} : : } as you showed concerning X(5390) in Hyacinthos message #21897.

      A-vertex of Den Roussel triangle= {

      -a^2-2 a c Cos[B/3]-2 a b Cos[C/3]-4 b c Cos[B/3] Cos[C/3]+16 b c Cos[A/3]^2 Cos[B/3] Cos[C/3],

      a b+2 a c Cos[A/3]+4 c^2 Cos[A/3] Cos[B/3]+8 b c Cos[A/3]^2 Cos[B/3]+2 b^2 Cos[C/3]+4 b c Cos[A/3] Cos[C/3],

      a c+2 a b Cos[A/3]+2 c^2 Cos[B/3]+4 b c Cos[A/3] Cos[B/3]+4 b^2 Cos[A/3] Cos[C/3]+8 b c Cos[A/3]^2 Cos[C/3]

      }. Similarly, the others are cyclic permutations of this.

       

      Best regards,

      Seiichi Kirikami, Japan.

       



      ---In Anopolis@yahoogroups.com, <seiichikirikami@...> wrote:

      Dear friends,
       
      The following are numerically confirmed points concerning Den Roussel equilateral triangle.
       
      [0] The center of Den Roussel equilateral triangle:
      Numerical x-coordinate for the triangle {6,9,13} is given in Hyacinthos message #21388.
      Trilinear coordinates are given in Hyacinthos message #21406.
       
      [1] The perspector of Morley triangle(1st) and Den Roussel equilateral triangle:
      Numerical x-coordinate for the triangle {6,9,13} is given in Hyacinthos message #21388.
       
      [2] The perspector of adjunct Morley triangle(1st) and Den Roussel equilateral triangle:
      Numerical x-coordinate for the triangle {6,9,13} is given in Hyacinthos message #21390.
       
      [3] Given a triangle ABC, its Den Roussel equilateral triangle DEF, Newton lines of AFDE, BDEF and CEFD concur in a point. (non-ETC, search x=2.889531296..)
       
      [4] Given a triangle ABC, its Den Roussel equilateral triangle DEF, the lines through the centroids of AFE and BCD, BDF and CAE, CED and ABF concur in a point. (non-ETC, search x=2.609486612..)
       
      Other perspectors are shown without coordinates in Hyacinthos message #13702 and #21390.
       
      Best regards,
      Seiichi.