HYACINTHOS 28811

[Kadir Altintas]:

 

Let ABC be a triangle with centroid G

Let DEF be the medial triangle of ABC.

Ga, Gb, Gc are the centroids of GEF, GFD, GDE resp.

Circles passing through Gb, Gc are tangent to circumcircle of ABC at Ta, Taa and their centers are Oa, Oaa

Define Tb,Tbb, Ob, Obb, Tc, Tcc, Oc, Occ cyclically

 

Prove:

1. TaTaa, TbTbb, TcTcc concur on the Euler line of ABC

2. OaOaa, ObObb, OcOcc concur on the Euler line of ABC

 

AoPS

 

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[Ercole Suppa]

 

1.  The lines TaTaa, TbTbb, TcTcc concur at point

 

X = EULER LINE INTERCEPT OF X(32)X(21448)

 

= a^2 (5 a^4-5 b^4+26 b^2 c^2-5 c^4) : : (barys) 

 

= (36 R^2-5 SW)S^2 + 5 SW SB SC : : (barys)

 

As a point on the Euler line, X() has Shinagawa coefficients: {4 E - 5 F, 5 E + 5 F)} 

 

= on lines X(i)X(j) for these {i,j}: {2,3}, {32,21448}, {111,9605}, {373,26864}, {576,3066}, {1351,10545}, {1495,10541}, {3053,8585}, {3167,15019}, {3292,9777}, {5050,5643}, {5544,6800}, {5640,11482}, {5644,9544}, {5651,11477}, {5943,22234}, {8780,11451}, {9306,22330}, {10314,15860}, {11465,14530}, {14924,22112}, {15034,15465}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,5020,16042}, {3,12105,9715}, {3,16042,11284}, {1995,5020,11284}, {1995,11284,25}, {1995,16042,3}

 

= (6-8-13) search numbers [1.31257986180043057, 0.439597320085166473, 2.73052178563291128]

 

 

2. The lines OaOaa, ObObb, OcOcc concur at X(11539)

 

 

Best regards

Ercole Suppa