HYACINTHOS 26354

 [Antreas P. Hatzipolakis]:

   
Let ABC be a triangle and A'B'C',IaIbIc the pedal, antipedal triangles of I, resp.

Denote:

A",B",C" = the antipodes of A', B', C', in the incircle, resp.

The circumcircles of IA"Ia, IB"Ib, IC"Ic are coaxial.

2nd intersection (other than I)?

 

[Angel Montesdeoca]:

***  2nd intersection (other than I) of the circumcircles of IA"Ia, IB"Ib, IC"Ic is

W = (a (2 a^9
               -15 a^8 (b+c)
              +10 a^7 (b^2+12 b c+c^2)
              +2 a^6 (21 b^3-61 b^2 c-61 b c^2+21 c^3)
               -2 a^5 (21 b^4+70 b^3 c-62 b^2 c^2+70 b c^3+21 c^4)
                -4 a^4 (9 b^5-67 b^4 c+22 b^3 c^2+22 b^2 c^3-67 b c^4+9 c^5)
               +2 a^3 (b-c)^2 (23 b^4+6 b^3 c-74 b^2 c^2+6 b c^3+23 c^4)
               +2 a^2 (b-c)^2 (3 b^5-49 b^4 c-2 b^3 c^2-2 b^2 c^3-49 b c^4+3 c^5)
               -4 a (b^2-c^2)^2 (4 b^4-25 b^3 c+10 b^2 c^2-25 b c^3+4 c^4)
               3 (b-c)^4 (b+c)^3 (b^2-6 b c+c^2)) : .... : ....).
              
The coaxal axis is X(1)X(5806)
 
 (6 - 9 - 13) - search numbers  of W: (8.16415864716562, 8.77789505718487, -6.20441301022005).
            
              
 *** If Ra is the the radical center of incircle and circumcircle  of IA"Ia,  and define Rb, Rc cyclically.
 
   Ra, Rb , Rc concur in a point
  
   Z = (a (a^5 (b+c)
           -a^4 (b^2+14 b c+c^2)
          +a (b-c)^2 (b^3-9 b^2 c-9 b c^2+c^3)
          -2 a^3 (b^3-5 b^2 c-5 b c^2+c^3)
          +2 a^2 (b^4+4 b^3 c+14 b^2 c^2+4 b c^3+c^4)
           -(b^2-c^2)^2 (b^2-6 b c+c^2)) : ... : ....).
          
 Z lies on the coaxal axis X(1)X(5806) of the circumcircles of IA"Ia, IB"Ib, IC"Ic,  and  on lines X(i)X(j) for these {i, j}:  {1,5806}, {8,3740}, {65,390}, {354,4297}, {950,8581}, {2136,3303}, {2951,11518}, {3057,6738}, {3488,12675}, {3698,10389}, {3893,4423}, {4321,5665}, {5836,8236}, {8275,9957}, {9848,12672}.
 
  (6 - 9 - 13) - search numbers  of Z: (1.98657016904173, 2.01465647989418, 1.32902376396145).
 
  Angel Montesdeoca