HYACINTHOS 29575

[Antreas P. Hatzipolakis]:

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

Denote :

A"B"C" = the reflection of A'B'C' in L = IO line
 
A*, B*, C* points on A"Na, B "Nb, C"Nc such that :

A*Na/A "Na = B*Nb/B" Nb = C*Nc/C "Nc = t
 
The orthocenter Ht of A*B*C* lies on L.

 
[Angel Montesdeoca]:


  ***  The envelope of the lines A*Ht is a parabola (Pa).
Define (Pb) and (Pc) cyclically.
 
The parabolae (Pa), (Pb), (Pc) have common focus, F, and their ditrctrices are concurrent at Z.
 
  F = X(1)X(12619) ∩ X(106)X(18976) =  

= (a-b+c) (a+b-c) (a^7 (b+c)+(b^2-c^2)^4+a^6 (b^2-12 b c+c^2)-a^2 (b^2-c^2)^2 (b^2-b c+c^2)-a (b-c)^2 (b+c)^3 (3 b^2-5 b c+3 c^2)+a^5 (-5 b^3+13 b^2 c+13 b c^2-5 c^3)-a^4 (b^4-11 b^3 c+30 b^2 c^2-11 b c^3+c^4)+a^3 (7 b^5-16 b^4 c+10 b^3 c^2+10 b^2 c^3-16 b c^4+7 c^5)) : :
 
= lies on these lines:  {1,12619}, {106,18976}, {11011,23869}
 
 (6 - 9 - 13) - search numbers  of F: (1.02672530231474, 0.972500486415961, 2.49352169793576)
 
 
 Z = a^2 (-a^9 (b-c)^2+a^8 (b^3-2 b^2 c-2 b c^2+c^3)+(b-c)^4 (b+c)^3 (b^4-3 b^3 c+3 b^2 c^2-3 b c^3+c^4)-a (b-c)^4 (b+c)^2 (b^4-2 b^3 c-b^2 c^2-2 b c^3+c^4)+2 a^7 (2 b^4-5 b^3 c+11 b^2 c^2-5 b c^3+2 c^4)-2 a^6 (2 b^5-5 b^4 c+7 b^3 c^2+7 b^2 c^3-5 b c^4+2 c^5)+a^5 (-6 b^6+18 b^5 c-43 b^4 c^2+72 b^3 c^3-43 b^2 c^4+18 b c^5-6 c^6)+2 a^3 (b-c)^2 (2 b^6-3 b^5 c+4 b^4 c^2-17 b^3 c^3+4 b^2 c^4-3 b c^5+2 c^6)-a^2 (b-c)^2 (4 b^7-6 b^6 c+8 b^5 c^2-11 b^4 c^3-11 b^3 c^4+8 b^2 c^5-6 b c^6+4 c^7)+a^4 (6 b^7-18 b^6 c+37 b^5 c^2-26 b^4 c^3-26 b^3 c^4+37 b^2 c^5-18 b c^6+6 c^7)) :  :
 
Z =  (2r^3-24r^2R-R^3+2r s^2) X[65]+ (4r^2(2r-R)) X[1483]

(6 - 9 - 13) - search numbers  of Z:  (0.894824616590753, 1.20332668643293, 2.39459618364277)

Angel Montesdeoca

Parábolas con el mismo foco y directrices concurrentes