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22032Fwd: [EGML] Numerical computation of the coordinates of the points concerning Morley Triangle

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  • Antreas Hatzipolakis
    Jan 9, 2014


      ---------- Forwarded message ----------
      From: César Lozada
      Date: Thu, Jan 9, 2014 at 3:48 PM
      Subject: RE: [EGML] Numerical computation of the coordinates of the points concerning Morley Triangle
      To: Anopolis@yahoogroups.com


       

      Dear Seiichi:

       

      Although I couldn’t understand your constructions, I have found some numerical conjectures about your points:

       

      1) [4.0009748991, 3.1518822374, -0.3880116359]

      On lines: (24,547), (357,1134), (1059,5555), (2052,3407), (3282,5390)

       

      2) [2.408489889, 1.270321463, 1.649600443]

      On lines: (112,298), (604,1354), (1134,3278), (1136,1137), (1215,3143), (1399,4289),(1930,3338), (1987,4564)

       

      3) [3.573704479, 2.53674706, 0.235052911]

      X=Midpoint of: (2,3277)                                                                           

      On lines: (2,3277), (498,3343), (1360,5439), (1411,3351), (2027,3242), (2568,3036)

       

      4) [6.6751430960, 7.0835326970, -4.3441549680]

      On lines: (1134,1136), (3278,5390)

       

      5) [11.02440279, 9.927435954, -8.320361696]

      On lines: (356,357), (1136,3282), (1936,4635)

       

      6) [8.416515372, 7.243447912, -5.258575783]

      X=Midpoint of: (2,3276)                                                                            

      On lines: (2,3276), (570,4787), (1622,2446), (2954,3041)

       

      Maybe you could apply “reverse engineering” in order to prove some or all of these conjectures.

       

      Working precision = 0.00001

       

      Best regards

      César Lozada

       

       


      De: Anopolis@yahoogroups.com [mailto:Anopolis@yahoogroups.com] En nombre de seiichi kirikami
      Enviado el: Jueves, 09 de Enero de 2014 04:47 a.m.
      Para: Anopolis@yahoogroups.com
      Asunto: [EGML] Numerical computation of the coordinates of the points concerning Morley Triangle [2 Attachments]

       

       

      Dear friends,

       

      Given a triangle ABC, we trisect its angles A, B, C and their complementary angles equally. We denote the intersection of equal trisection outside ABC and nearest to BC by Da, the intersection of equal trisection inside the sector of A, outside ABC and nearest to CA by Db and the intersection of equal trisection inside the sector of A, outside ABC and nearest to AB by Dc. Others are defined cyclically. We have 4 equilateral triangles DaDbDc, EaEbEc, FaFbFc and DaEbFc. We denote the the intersections of EaEc and FaFb by G. Others are defined cyclically. We have an equilateral triangle GHI.

      See the attached picture.

       

      We obtain the following results from DaEbFc for the triangle {6,9,13}.

      (1)  Concurrent point of Euler lines AEbFc, BFcDa and CDaEb

      Search kx=4.000974899..

      (2)  Concurrent point of Newton lines AFcDaEb, BDaEbFc and CEbFcDa.

      Search kx=2.408489889..

      (3)  Concurrent point of the lines through centroids of AEbFc and DaBC, BFcDa and EbCA, CDaFb and FcAB.

      Search kx=3.573704479..

      We obtain the following results from GHI for the triangle {6,9,13}.

      (4)  Concurrent point of Euler lines AHI, BIG and CGH

      Search kx=6.675143096..

      (5)  Concurrent point of Newton lines AIGH, BGHI and CHIG.

      Search kx=11.024402788..

      (6)  Concurrent point of the lines through centroids of AHI and GBC, BIG and HCA, CGH and IAB.

      Search kx=8.416515372..

      See the attached MS Excel sheet.

       

      Best regards,

      Seiichi