HYACINTHOS 25778

 

[Antreas P. Hatzipolakis]:

 

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

    Denote:

    A"B"C" = the pedal triangle of O wrt triangle A'B'C'.

    Which is the locus of P such that ABC, A"B"C" are orthologic?

    I lies on the locus. Orthologic centers?

 

[Angel Montesdeoca]:

 

***** The locus of P such that ABC, A"B"C" are orthologic is the McCay cubic.

If U is the orthologic center (ABC, A"B"C"), pairs {P,U}: {1,90}, {3,3}, {4,68}.

If V is the orthologic center (A"B"C",ABC), pairs {P,V}: {3,3}, {4,6146}.
X(6146) =  HATZIPOLAKIS-MOSES-TRIANGLE- TO-ABC ORTHOLOGY CENTER

***** Case  P=X(1)

 Let A'B'C' be the intouch triangle of ABC; i.e., the pedal triangle of the incenter, I.

    Denote:     A"B"C" = the pedal triangle of O wrt triangle A'B'C'.

   The triangles ABC, A"B"C" are orthologic.
   The orthologic center (ABC, A"B"C") is X(90).
  The orthologic center (A"B"C",ABC) is

 V = (a (a^5 (b+c)-(b^2-c^2)^2 (b^2+c^2)-a^4 (b^2-4 b c+c^2)-2 a^3 (b^3+c^3)+2 a^2 (b^4-2 b^3 c-2 b c^3+c^4)+a (b^5-b^4 c-b c^4+c^5)) : ... : ...),

  with (6-9-13)-search numbers  (7.33950856404248, 7.42420457803130, -4.88663494782613)
 
  V= Midpoint of X(i) and X(j) for these {i,j}: {3,1071},{355,12680},{3555, 12702},{4297,5884},{5787, 12671},{9943,12675},{10202, 11220}
 
   V =Reflection of X(i) in X(j) for these {i,j}: {5,9940},{3627,5806},{5777, 140},{7686,5885},{9856,5901}
  
   V lies on lines: {1,1406}, {3,63}, {4,10202}, {5,142}, {7,6851}, {30,553}, {36,1858},{40,4880}, {57,6985,10399},{65,4299}, {84,3560},......
  
   Angel Montesdeoca