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24438Re: Another NPC center on OI line

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  • Antreas Hatzipolakis
    Sep 20, 2016
       

      [Tran Quang Hung]:

      Let ABC be a triangle.

      A1B1C1 is pedal triangle of incenter I.

      A2,B2,C2 are reflections of A1,B1,C1 through I.

      A3,B3,C3 are reflections of A,B,C through A2,B2,C2, reps.

      Then NPC center of A3B3C3 lies on OI line of ABC.


      *** The NPC center of A3B3C3 is  W =  (3 R-2r) I + (2r-R) O

      W =  ( a (a^5 (b+c)
                     -a^4 (b^2+6 b c+c^2)
                    -a^3 (2 (b^3+c^3)-7 b c(b+c))
                   +2 a^2 (b^4+2 b^3 c-7 b^2 c^2+2 b c^3+c^4)
                   +a (b-c)^2 (b^3-6 b c(b+c)+c^3)
                   -(b-c)^4 (b+c)^2) : ... : ...),
                  
                   with (6-9-13)-search number (0.304863513039357, 0.546310358888081, 3.12174338127447).
                 
      W is the  midpoint of X(i) and X(j) for these {i,j}: {355, 3885}, {1482, 5697}

      W is the reflection of X(i) in X(j) for these {i,j}:  {1385,9957}, {5690,3884}, {5903,6583}
              
           W lies on lines:  {1,3},  {5,2802}, {8,6965}, {140,3898}, {149,355},  {519,5694}, {1483,2800}, .....

      Best regards,
      Angel Montesdeoca



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