HYACINTHOS 25871

 [Antreas P. Hatzipolakis]:

 

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

    Denote:

    Ba, Ca = the orthogonal projections of B', C' on BA, CA, resp.

    A* = (Perpendicular from Ba to PB') intersection (Perpendicular from Ca to PC')

    Similarly B*, C*.
   
     Which is the locus of P such that ABC, A*B*C* are
    1. orthologic
    2. perspective
     Orthologic centers?    Perspector ?

 

     [Angel Montesdeoca]:
    
    
     *** The locus of P such that ABC, A*B*C* are  orthologic is Euler line.
    
     The  orthology center U of ABC with respect to A*B*C*  lies on the circum-hyperbola that passes through  {3, 185, 235, 800, 1093, 9307}.
    
     Pairs {P,U}:  {3,3}, {20,1093}, {376,9307}.
    
      The  orthology center V of A*B*C* with respect to ABC  lies on line  {4, 51, 185, 389, 1075, 1093, 1896, 1899, 2052, 3168, 3567, 5878, 5890, 6000, 6225, 6241, 6524, 6761, 8887, 9781, 10110, 10379, 10380, 11381, 11433, 11455, 11457, 11550, 11572, 12290, 12324}.
     
      Pairs {P,V}:  {2, 5890}, {3, 389}, {4, 185}, {20, 4}, {376, 51}, {381, 11457}, {550, 10110}, {3146, 6241}, {3522, 3567}, {3529, 11381}, {5059, 12290}
    
    
     *** The locus of P such that ABC, A*B*C* are perspective is the Darboux cubic,  the perspector Q being on the Thomson cubic.
     Pairs {P,Q} : {1, 1}, {3, 6}, {4, 2}, {20, 4}, {40, 57}, {64, 3}, {84, 9}, {1490, 282}, {1498, 1073}, {2130, 3349}, {2131, 3350}, {3182, 3342}, {3183, 3344}, {3345, 223}, {3346, 1249}, {3347, 3341}, {3348, 3343}, {3353, 3352}, {3354, 3351}, {3355, 3356}.
   
   
    *** For  {X(3), X(4), X(20)} = (Euler line) /\ (Darboux cibic) the triangles  ABC and A*B*C* are bilogic (orthologic and perspective).  
   
    If P=X(4) then   the  orthology center  of ABC with respect to A*B*C*  is:
  
   U4 = ( 1/(a^8-2 b^2 c^2 (b^2-c^2)^2-3 a^6 (b^2+c^2)-a^2 (b^2+c^2)^3+a^4 (3 b^4+8 b^2 c^2+3 c^4)): ... : ...),
   
    with (6.9,13)-search numbers (-56.8088277846917, -69.1675816949657, 77.7453723252030) 

and lies on lines: {2, 185}, {3, 801}, {4, 800}, {76, 6823}, {83, 11479}, {96, 11456}, {98, 1498}, ...
    
     *** If P=X(5), the triangles ABC and A*B*C* are orthologic.
    
       The  orthology center  of ABC with respect to A*B*C*  is  U5 = X(140)X(185) /\ X(800)X(6748):

  U5 =    (1/(a^8-3 a^6 (b^2+c^2)+3 a^4 (b^4+3 b^2 c^2+c^4)-a^2 (b^2+c^2)^3-3 b^2 c^2 (b^2-c^2)^2):...:...),
 
            with (6,9,13)-search numbers (25.1312469231680, 28.9406525228175, -27.9942089215053).
    
     The  orthology center  of A*B*C* with respect to ABC  is  V5 = X(185) + X(389) = X(4)X(51) /\  X(6)X(3357)
 
     V5 =  (a^2 (3 a^6 (b^2+c^2)+a^4 (-9 b^4+4 b^2 c^2-9 c^4)+9 a^2 (b^2-c^2)^2 (b^2+c^2)-(b^2-c^2)^2 (3 b^4+4 b^2 c^2+3 c^4)) : .. . : ....).
    
     with (6,9,13)-search numbers  (7.36718182475004, 7.81259491018046, -5.16829283656365).
   V5 is reflection of X(i) in X(j) for these {i,j}:  {5907,11695}, {10110,389}, {11793,9729}.
  
  
     Angel Montesdeoca