HYACINTHOS 28899

[Kadir Altintas] 

 

Let ABC be a triangle and DEF the circumcevian triangle of the incenter I.

Denote:

G1, G2, G3, G4, G5, G6 - the centroids of the triangles EAF, AFB, FBD, BDC, DCE, CEA, resp.

G1a, G1b, G1c = the centroids of the triangles G1G2G6, G2G3G4, G4G5G6, resp.

 

Prove that: the center O' of the circle (G1a,G1b,G1c) lies on Euler line of ABC

 

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[Ercole Suppa]

 

O' = EULER LINE INTERCEPT OF X(1000)X(5424)

 

= 7 a^7-7 a^6 b-15 a^5 b^2+15 a^4 b^3+9 a^3 b^4-9 a^2 b^5-a b^6+b^7-7 a^6 c+a^5 b c+a^4 b^2 c+4 a^3 b^3 c+7 a^2 b^4 c-5 a b^5 c-b^6 c-15 a^5 c^2+a^4 b c^2+14 a^3 b^2 c^2+2 a^2 b^3 c^2+a b^4 c^2-3 b^5 c^2+15 a^4 c^3+4 a^3 b c^3+2 a^2 b^2 c^3+10 a b^3 c^3+3 b^4 c^3+9 a^3 c^4+7 a^2 b c^4+a b^2 c^4+3 b^3 c^4-9 a^2 c^5-5 a b c^5-3 b^2 c^5-a c^6-b c^6+c^7 : : (barys)

 

= (12 a R^2-12 b R^2-4 a SB+4 b SB-4 a SC+4 c SC+a SW+3 b SW)S^2 + 2 R S^3+3 R S SB SC+3 b SB SC^2-3 c SB SC^2-3 b SB SC SW : : (barys)

 

= 5*X[3616]+4*X[22937], 2*X[5426]+X[5657], 11*X[5550]-2*X[16159], 5*X[10595]+4*X[16139], X[12317]+8*X[16164]

 

= lies on these lines: {2,3}, {1000,5424}, {1056,5427}, {3616,22937}, {5426,5657}, {5550,16159}, {10595,16139}, {12317,16164}, {21165,25055}

 

= (6-9-13) search numbers [3.79572217856873888, 2.91618904472331531, -0.130107400701949441]

 

 

Best regards

Ercole Suppa