HYACINTHOS 28867

Kadir Altintas]:

 

Let ABC be a triangle and DEF the anticevian triangle of a point P wrt triangle ABC.

Let Ga,Gb,Gc be the centroids of triangle BPC, CPA, APB respectively

The liines DGa, EGb, FGc concur at point Q with first barycentrics:

 

Q = x(x^3 + (y+z)x^2 + (-y^2-10 y z-z^2)x - y^3-7 y^2 z-7 y z^2-z^3) : :

 

P=X(1)  ---> Q=X(3646)

P=X(2)  ---> Q=X(2)

P=X(3)  ---> Q=X(15805)

P=X(4)  ---> Q=X(8888)

P=X(6)  ---> Q=?

 

--------------------------------------------------------------------------------------------

 

 

[Ercole Suppa]

 

Let Q = Q(P) the perspector of DEF and GaGbGc

 

*** Q= x(x^3 + (y+z)x^2 + (-y^2-10 y z-z^2)x - y^3-7 y^2 z-7 y z^2-z^3) : :  (barys)

 

*** Some points Q=Q(P):

 

 

--- Q(X(6)) = X(1)X(12329) ∩ X(2)X(159)

 

= a^2 (a^6+a^4 b^2-a^2 b^4-b^6+a^4 c^2-10 a^2 b^2 c^2-7 b^4 c^2-a^2 c^4-7 b^2 c^4-c^6) : : (barys) 

 

= (2 R^2 SB+2 R^2 SC+4 R^2 SW+SW^2)S^2 -SB SC SW^2 : :

 

= lies on the Stammler hyperbola and these lines: {1,12329}, {2,159}, {3,3589}, {6,3917}, {25,2916}, {125,2930}, {140,9937}, {141,16419}, {155,182}, {195,5050}, {206,1498}, {399,12017}, {511,15805}, {610,2297}, {936,22769}, {1351,15047},  {1486,29598}, {1593,16936}, {1609,14096}, {1843,22112}, {2918,6642}, {3313,10601}, {3360,5116}, {3564,13154}, {3618,7485}, {5092,9818}, {5157,19153}, {5898,15694}, {7503,15740}, {7509,11821}, {7866,23333}, {9969,17825}, {11284,20987}, {11479,16656}, {12085,17508}, {15141,26206}

 

= complement of X(15435)

= (6-8-13) search numbers [3.33925477429885877, 2.71915151353364819, 0.216980461323149383]

 

 

Best regards

Ercole Suppa