HYACINTHOS 26956

[Tran Quang Hung]:

Let N be NPC center of ABC.

A'B'C' is circumcevian triangle of N.

A''B''C'' is anticevian triangle of N wrt A'B'C'.

Then the isogonal conjugate of N wrt A''B''C'' is on Euler line of ABC.

Which is this point ?


[Angel Montesdeoca]:


Dear Tran Quang Hung,
 
  The  isogonal conjugate of N wrt A''B''C'' is the point of intersection od Euler line and the line X(11584)X(12254):

W =  a^2 (-a^20
    +8 a^18 (b^2+c^2)    
    -a^16 (27 b^4+41 b^2 c^2+27 c^4)
    +16 a^14 (3 b^6+5 b^4 c^2+5 b^2 c^4+3 c^6)    
    -a^12 (42 b^8+64 b^6 c^2+67 b^4 c^4+64 b^2 c^6+42 c^8)
    +4 a^10 b^2 c^2 (b^2-c^2)^2 (b^2+c^2)
    +a^8 (42 b^12+2 b^10 c^2+14 b^8 c^4+19 b^6 c^6+14 b^4 c^8+2 b^2 c^10+42 c^12)
    -2 a^6 (b^2-c^2)^2 (24 b^10+20 b^8 c^2+23 b^6 c^4+23 b^4 c^6+20 b^2 c^8+24 c^10)
    +a^4 (b^2-c^2)^2 (27 b^12-26 b^10 c^2-9 b^6 c^6-26 b^2 c^10+27 c^12)
    -2 a^2 (b^2-c^2)^4 (4 b^10-6 b^8 c^2-b^6 c^4-b^4 c^6-6 b^2 c^8+4 c^10)
    +(b^2-c^2)^6 (b^8-3 b^6 c^2-3 b^2 c^6+c^8)) : .... : ... ,

On the lines: {2,3},{11584, 12254}

with  (6 - 9 - 13) - search numbers 0.240104080495344, -0.630049243029177, 3.96603515146827

Best regards,
Angel Montesdeoca