HYACINTHOS 29452

[Kadir Altintas]

Let ABC be a triangle with incenter I.

Denote:

DEF = the circumcevian triangle of I

Ga = the centroid of AFE;
Define Gb,Gc cyclically

Gaa = the centroid of BDC; 

Define Gbb,Gcb cyclically

Ga' = the centroid of GaGbbGcc; 

Define Gb',Gc' cyclically

Prove:  

1) the circumcenter of Ga'Gb'Gc' lies on Euler line of ABC

2) what happens if Na,Nb,Nc,Naa,Nbb,Ncc,Na',Nb',Nc' are NPC of their respective triangles?

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[Ercole Suppa]
 
1) The circumcenter of Ga'Gb'Gc' is X(31669) which lies on Euler line of ABC

2) The circumcenter of Na'Nb'Nc' is the point:

 
X = MIDPOINT OF  X(946) AND X(5506)

= 2 a^7-a^6 b-6 a^5 b^2+3 a^4 b^3+6 a^3 b^4-3 a^2 b^5-2 a b^6+b^7-a^6 c-6 a^5 b c+14 a^4 b^2 c+29 a^3 b^3 c-12 a^2 b^4 c-23 a b^5 c-b^6 c-6 a^5 c^2+14 a^4 b c^2-22 a^3 b^2 c^2+15 a^2 b^3 c^2+2 a b^4 c^2-3 b^5 c^2+3 a^4 c^3+29 a^3 b c^3+15 a^2 b^2 c^3+46 a b^3 c^3+3 b^4 c^3+6 a^3 c^4-12 a^2 b c^4+2 a b^2 c^4+3 b^3 c^4-3 a^2 c^5-23 a b c^5-3 b^2 c^5-2 a c^6-b c^6+c^7 : : (barys)

= (52 a R^2-52 b R^2+8 a SB-4 b SB-13 c SB+8 a SC-13 b SC-4 c SC-8 a SW+13 b SW)S^2 -58 R S^3-34 R S SB SC+13 b SB SC^2-13 c SB SC^2-13 b SB SC SW : : (barys)

= X[946]+X[5506] 

 
= lies on these lines: {946,5506}, {1158,3306}, {3628,6684}, {19919,33592}

= midpoint of X(946) and X(5506)

= (6-9-13) search numbers [0.5455756327077551887, -0.2045972182321856280, 3.5305045717415436815]


Best regards,
Ercole Suppa