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24664Re: Around NPC centers

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  • Antreas Hatzipolakis
    Oct 20, 2016

      [Tran Quang Hung, Anopolis #4049]

      Let ABC be a triangle.

      Denote:

      Ha, Hb, Hc = the orthocenters of NBC, NCA, NAB, resp.
      Oa, Ob, Oc = the circumcenters of NBC, NCA, NAB, resp.

      Nha,Nhb,Nhc = the NPC centers of NHbHc,NHcHa,NHaHb, resp.
      Noa,Nob,Noc = the NPC centers of NObOc,NOcOa,NOaOb, resp.

      The NPC centers of
      1.
      NhaNhbNhc
      2.
      NoaNobNoc

      lie on the Euler line of ABC.

      Tran Quang Hung


      [Peter Moses]:


      Hi Antreas,
       
      1).
      2 a^16-9 a^14 b^2+15 a^12 b^4-9 a^10 b^6-5 a^8 b^8+13 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16-9 a^14 c^2+22 a^12 b^2 c^2-13 a^10 b^4 c^2-15 a^6 b^8 c^2+34 a^4 b^10 c^2-27 a^2 b^12 c^2+8 b^14 c^2+15 a^12 c^4-13 a^10 b^2 c^4+4 a^8 b^4 c^4-7 a^6 b^6 c^4-22 a^4 b^8 c^4+51 a^2 b^10 c^4-28 b^12 c^4-9 a^10 c^6-7 a^6 b^4 c^6-2 a^4 b^6 c^6-29 a^2 b^8 c^6+56 b^10 c^6-5 a^8 c^8-15 a^6 b^2 c^8-22 a^4 b^4 c^8-29 a^2 b^6 c^8-70 b^8 c^8+13 a^6 c^10+34 a^4 b^2 c^10+51 a^2 b^4 c^10+56 b^6 c^10-11 a^4 c^12-27 a^2 b^2 c^12-28 b^4 c^12+5 a^2 c^14+8 b^2 c^14-c^16::
      on lines {{2,3},{54,1263}}.
      Anticomplement X[10126].
      Reflection of X(i) in X(j) for these {i,j}: {{5, 5501}, {10205, 140}}.
      {6,9,13}-Searches: {-6. 92825917464489310823608783454, -7. 77950217490354620974112096771, 12. 2241317605229386828942493884}.
       
      2).
      2 a^22-15 a^20 b^2+50 a^18 b^4-93 a^16 b^6+92 a^14 b^8-14 a^12 b^10-84 a^10 b^12+110 a^8 b^14-62 a^6 b^16+13 a^4 b^18+2 a^2 b^20-b^22-15 a^20 c^2+82 a^18 b^2 c^2-172 a^16 b^4 c^2+139 a^14 b^6 c^2+41 a^12 b^8 c^2-125 a^10 b^10 c^2+3 a^8 b^12 c^2+97 a^6 b^14 c^2-54 a^4 b^16 c^2-a^2 b^18 c^2+5 b^20 c^2+50 a^18 c^4-172 a^16 b^2 c^4+160 a^14 b^4 c^4+52 a^12 b^6 c^4-94 a^10 b^8 c^4-65 a^8 b^10 c^4+58 a^6 b^12 c^4+32 a^4 b^14 c^4-14 a^2 b^16 c^4-7 b^18 c^4-93 a^16 c^6+139 a^14 b^2 c^6+52 a^12 b^4 c^6-72 a^10 b^6 c^6-39 a^8 b^8 c^6-84 a^6 b^10 c^6+98 a^4 b^12 c^6+4 a^2 b^14 c^6-5 b^16 c^6+92 a^14 c^8+41 a^12 b^2 c^8-94 a^10 b^4 c^8-39 a^8 b^6 c^8-18 a^6 b^8 c^8-89 a^4 b^10 c^8+76 a^2 b^12 c^8+22 b^14 c^8-14 a^12 c^10-125 a^10 b^2 c^10-65 a^8 b^4 c^10-84 a^6 b^6 c^10-89 a^4 b^8 c^10-134 a^2 b^10 c^10-14 b^12 c^10-84 a^10 c^12+3 a^8 b^2 c^12+58 a^6 b^4 c^12+98 a^4 b^6 c^12+76 a^2 b^8 c^12-14 b^10 c^12+110 a^8 c^14+97 a^6 b^2 c^14+32 a^4 b^4 c^14+4 a^2 b^6 c^14+22 b^8 c^14-62 a^6 c^16-54 a^4 b^2 c^16-14 a^2 b^4 c^16-5 b^6 c^16+13 a^4 c^18-a^2 b^2 c^18-7 b^4 c^18+2 a^2 c^20+5 b^2 c^20-c^22::
      on line {{2,3}}.
      Midpoint of X[5] and X[5500].
      {6,9,13}-Searches: {-30. 0053674580255540938780238562,- 30. 7957324639432746056338602485, 38. 8094180914184403349624239774}.
       
      X(54) {Nha,Nhb,Nhc} = N ABC.
      X(1141) {Nha,Nhb,Nhc} = H ABC.
      X(1157) {Nha,Nhb,Nhc} = X(1263) ABC.
      X(8254) {Nha,Nhb,Nhc} = X(5501) ABC.
       
      Best regards,
      Peter.