HYACINTHOS 25111

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point on the Neuberg cubic.

Denote:

L1, L2, L3 = the Euler lines of PBC, PCA, PAB, resp. (concurrent on the Euler line of ABC)

L11, L12, L13 = the reflections of L1 in BC, CA, AB, resp.
A1B1C1 = the triangle bounded by L11, L12, L13
(O1) = the incircle or an excircle of A1B1C1 whose the center lies on the circumcircle of ABC
[reflection point of L1]

L21, L22, L23 = the reflections of L2 in BC, CA, AB, resp.
A2B2C2 = the triangle bounded by L21, L22, L23
(O2) = the incircle or an excircle of A2B2C2 whose the center lies on the circumcircle of ABC.
[reflection point of L2]

L31, L32, L33 = the reflections of L3 in BC, CA, AB, resp.
A3B3C3 = the triangle bounded by L31, L32, L33
(O3) = the incircle or an excircle of A3B3C3 whose the center lies on the circumcircle of ABC
[reflection point of L3]

The locus of the radical center of (O1), (O2), (O3), as P moves on the Neuberg cubic, is the line X(3)X(74).

[Angel Montesdeoca];

The locus of the radical center W of (O1), (O2), (O3), as P moves on the Neuberg cubic, is the line X(3)X(74).

Some pairs {P, W}, with P on the Neuberg cubic and W on the line X(3)X(74):

* P=X(1),
  W = ( a^2(-(b-c)^4 (b+c)^3 (b^4+3 b^2 c^2+c^4)+(b-c)^2 (b+c)^4 (b^4+3 b^2 c^2+c^4) a
           +2 (2 b^9-b^8 c-2 b^7 c^2-b^6 c^3-8 b^5 c^4-8 b^4 c^5-b^3 c^6-2 b^2 c^7-b c^8+2 c^9) a^2
           -2 (2 b^8+3 b^7 c+5 b^6 c^2+6 b^5 c^3+6 b^4 c^4+6 b^3 c^5+5 b^2 c^6+3 b c^7+2 c^8) a^3
            +(-6 b^7+3 b^5 c^2+b^4 c^3+b^3 c^4+3 b^2 c^5-6 c^7) a^4
           +(6 b^6+6 b^5 c+11 b^4 c^2+12 b^3 c^3+11 b^2 c^4+6 b c^5+6 c^6) a^5
             +2 (2 b^5+b^4 c+b^3 c^2+b^2 c^3+b c^4+2 c^5)a^6
            -2 (2 b^4+b^3 c+2 b^2 c^2+b c^3+2 c^4) a^7
            +(-b^3-b^2 c-b c^2-c^3) a^8
           +(b^2+c^2) a^9) : ... : ...),

with (6-9-13)-search number (3.96388625672962,  2.63467073054016,  -0.0128242961878620).

* P = X(3),  W =   X(143) - 2  X(5)

   W =  ( a^2 (a^6 (b^2+c^2) 
           -a^4 (3 b^4+2 b^2 c^2+3 c^4)
         +a^2 (3 b^6+2 b^4 c^2+2 b^2 c^4+3 c^6)
          -(b^2-c^2)^2 (b^4+3 b^2 c^2+c^4) ) : ... : ... ),
  
   with (6-9-13)-search number (1.20124098002602,-0. 561026393574956,3. 47464845590888).
   W is the midpoint of X(i) and X(j) for these {i,j}: {3,5876}, {4,6101}, {5,5562}, {1216,5907}, {1511,7723}, {3627,10625}.
   W is the reflection of X(i) in X(j) for these {i,j}: {52,10095}, {143,5}, {389,3628}, {548,5447}, {5446,3850}, {10627,1216}.
   W lies on lines:  {2, 6102}, {3, 74}, {4, 2889}, {5, 51}, {30, 1216}, {49, 10610}, {140, 9729}, {155, 7514}, {185, 549}, {265, 2888}, {381, 10263}, {382, 2979}, {389, 3628}, {394, 7526}, {511, 546}, {547, 5462}, {548, 5447}, {550, 3917}, {567, 1493}, {568, 3090}, {632, 9730}, {1181, 7516}, {1352, 9973}, {1656, 5889}, {1658, 9306}, {1660, 3357}, {2070, 7691}, {3060, 3851}, {3091, 6243}, {3153, 6288}, {3526, 5890}, {3530, 3819}, {3544, 11002}, {3567, 5055}, {3627, 10625}, {3850, 5446}, {5054, 10574}, {5066, 10110}, {5072, 9781}, {5079, 5640}, {5972, 10125}, {6746, 7577}, {6759, 7525}, {7502, 10539}, {7512, 10540}, {8703, 10575}, {10272, 10628}
  
   * P = X(4),  W = X(3530) + X(5447)
  
     W = ( a^2 (a^6 (b^2+c^2)
                     -a^4 (3 b^4+14 b^2 c^2+3 c^4)
                     +a^2 (3 b^6+14 b^4 c^2+14 b^2 c^4+3 c^6)
                      -(b^2-c^2)^2 (b^4+3 b^2 c^2+c^4) ) : ... : ... ),
                    
     with (6-9-13)-search number (5.38708241565376, 4.28095659665481, -1.80942043069377)
    
       W is the midpoint of X(3530) and X(5447).
        W lies on lines: {3,74}, {30,11017}, {52,549}, {140,5446}, {143,631}, {548,3819}, {1154,3530}, {3523,6101}, {3524,6102}, {3627,5650}, {5054,10263}, {10110,10124}.
       
   * P = X(13) or X(14), W = X(7998) = anticomplement of X(373) ,  X(373) =centroid of the pedal triangle of the centroid.
  
   * P = X(1276) or X(7325),  W = X(3).
  
   Angel Montesdeoca