HYACINTHOS 28809

[Kadir Altintas]:

 

Let ABC be an acute angled triangle with orthocenter H.

Let DEF be the orthic triangle of ABC.

Circles with diameter AH, BH, CH intersects the circumcircle again at A1, B1, C1, resp.

Small circles are externally tangent to these circles and internally tangent to circumcircle at Ta, Tb, Tc, resp.

 

Prove that A1Ta, B1Tb, C1TC concur on the Euler line of ABC

 

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[Ercole Suppa]

 

The lines A1Ta, B1Tb, C1TC concur at point

 

X = ISOGONAL CONJUGATE OF X(28787)

 

= a (a+b) (a+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^3-a^2 b-a b^2+b^3-a^2 c-2 a b c+b^2 c-a c^2+b c^2+c^3) : : (barys) 

 

As a point on the Euler line, X() has Shinagawa coefficients: {(2R + r - p)(p + r + 2 R)(2R + r), p^2 (r + R) - r (r + 2 R) (r + 3 R)} 

 

= on lines X(i)X(j) for these {i,j}: {1,2299}, {2,3}, {9,1474}, {19,5248}, {34,18593}, {37,943}, {72,2203}, {104,1301}, {105,1289}, {107,915}, {112,15344}, {158,2218}, {241,1396}, {1068,8747}, {1104,1870}, {1612,5317}, {1708,1780}, {1848,5259}, {1891,5251}, {1974,10477}, {2164,7040}, {2189,2303}, {2204,5089}, {2332,7719}, {2360,18446}, {2687,22239}, {2752,10423}, {3487,27802}, {5436,7713}, {11517,17776}, {14344,21789}, {20626,26707}

= isogonal conjugate of X(28787)

 

= barycentric product of X(i) and X(j) for these {i,j}: {27,3811}, {28,17776}, {29,1708}, {92,1780}, {286,2911}, {648,15313}, {1289,26217}, {1896,3173}, {2322,4341}, {4567,5521}

 

= trilinear product of X(i) and X(j) for these {i,j}: {4,1780}, {27,2911}, {28,3811}, {162,15313}, {1172,1708}, {1474,17776}, {1896,3215}, {3173,8748}, {4183,4341}, {4570,5521}, {8747,11517}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,186,3651}, {4,3147,6889}, {4,6878,3541}, {4,7505,6829}, {21,28,4227}, {21,4233,28}, {25,405,4}, {25,468,4239}, {25,17520,28}, {28,2074,21}, {28,4183,4}, {29,13739,28}, {405,2915,440}, {440,2915,3651}, {2074,4233,4227}, {3089,6987,4}, {3575,8226,4}, {4248,17515,28}, {5047,7466,5142}, {6846,7487,4}

 

= (6-8-13) search numbers [0.749844467171239750, -0.121653564364109387, 3.37880411854203649]

 

 

Best regards

Ercole Suppa