HYACINTHOS 24008

 

 

• Antreas P. Hatzipolakis

   

  
  

  [APH]:

  
  

  Let ABC be a triangle and A'B'C' the pedal triangle of H.
  
  

  Denote:
  
  

  Ab, Ac = the reflections of A' in BO, CO, resp.
  
  

  Bc, Ba = the reflections of B' in CO, AO, resp.
  
  

  Ca, Cb = the reflections of C' in AO, BO, resp.
  
  

  The NPCs of A'AbAc, B'BcBa, C'CaCb are concurrent.
  
  

  Let Na, Nb, Nc be the NPC centers of A'AbAc, B'BcBa, C'CaCb, resp.
  
  

  ABC, NaNbNc are orthologic.

   

  APH
  
  
  

  [Peter Moses]:
  
  

  Hi Antreas,

   

  >The NPCs of A'AbAc, B'BcBa, C'CaCb are concurrent.
  At 

  (a^2-b^2-c^2) (2 a^14-6 a^12 b^2+5 a^10 b^4+a^8 b^6-4 a^6 b^8+4 a^4 b^10-3 a^2 b^12+b^14-6 a^12 c^2+12 a^10 b^2 c^2-9 a^8 b^4 c^2+8 a^6 b^6 c^2-12 a^4 b^8 c^2+12 a^2 b^10 c^2-5 b^12 c^2+5 a^10 c^4-9 a^8 b^2 c^4-4 a^6 b^4 c^4+8 a^4 b^6 c^4-21 a^2 b^8 c^4+9 b^10 c^4+a^8 c^6+8 a^6 b^2 c^6+8 a^4 b^4 c^6+24 a^2 b^6 c^6-5 b^8 c^6-4 a^6 c^8-12 a^4 b^2 c^8-21 a^2 b^4 c^8-5 b^6 c^8+4 a^4 c^10+12 a^2 b^2 c^10+9 b^4 c^10-3 a^2 c^12-5 b^2 c^12+c^14):: on lines {{6,13},{49,5972},{125,1147},{ 539,6699},...}

   

  >ABC, NaNbNc are orthologic.

  At X(265) & X(6102).

   

  Also, NaNbNc is perspective to the medial triangle at X(1147) and orthologic to it at X(1511) and X(6102).

   

  Orthic & NaNbNc are orthologic at X(113) & {{4,542},{5,539},...}.

  Tangential & NaNbNc orthologic at X(2931) & {{4,542},{5,539},...}.

  Euler & NaNbNc orthlogic at {{4,94},{5,1511},{30,125},{ 74,382},{110,381},{113,137},.. .} & X(6102).

   

  It is orthologic to plenty of other triangles too, even the Hatzipolakis-Moses triangle at {{6,13},{54,125},...} & {{52,54},{140,389},...} !

   

  Orthic & NaNbNc are paralogic at X(125) & {{2,9705},{3,539},{5,542},...} .

  Tangential & NaNbNc are paralogic at {{3,113},{6,1112},{22,110},{ 23,1503},{24,64},{25,125},{26, 1498},...} & {{2,9705},{3,539},{5,542},...} .

  Intangents & NaNbNc are paralogic at {{1,2777},{33,125},{56,2935},{ 65,74},{73,9627},{110,3100},{ 113,1062},...} & {{2,9705},{3,539},{5,542},...} .

  Extangents & NaNbNc are paralogic at {{19,125},{40,2777},{65,2906}, {74,6197},{110,3101},{113, 8251},...}, & {{2,9705},{3,539},{5,542},...} .

  ....

   

  Best regards,

  Peter Moses.