HYACINTHOS 28880

[Kadir Altintas]:

 

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

 

Let Xa = X(546) of AFE ( = midpoint of its orthocenter and NPC center). Define Xb, Xc cyclically.

 

(1) Prove that the center X of the circle through Xa, Xb, Xc lies on Euler line of ABC with first barycentrics

 

X = a (-2 a^6+2 a^5 (b+c)+3 b c (b^2-c^2)^2+4 a^4 (b^2-b c+c^2)-4 a^3 (b^3+c^3)+a^2 (-2 b^4+b^3 c-4 b^2 c^2+b c^3-2 c^4)+2 a (b^5-b^4 c-b c^4+c^5))::

 

Which is this point?

 

(2) Conjecture: if Xa, Xb, Xc are the same centers lying on Euler lines of AFE, BFD, DCE then the center X of circle through Xa, Xb, Xc lies on Euler line of ABC

 

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[Ercole Suppa]

 

(1)

 

*** X = EULER LINE INTERCEPT OF  X(1)X(3652)

 

= a (-2 a^6+2 a^5 (b+c)+3 b c (b^2-c^2)^2+4 a^4 (b^2-b c+c^2)-4 a^3 (b^3+c^3)+a^2 (-2 b^4+b^3 c-4 b^2 c^2+b c^3-2 c^4)+2 a (b^5-b^4 c-b c^4+c^5)) : : (barys)

 

= R S^3 + (-8 a R^2+8 b R^2+2 a SB-2 b SB+2 a SC-2 c SC-2 b SW)S^2 -7 R S SB SC-2 b SB SC^2+2 c SB SC^2+2 b SB SC SW : :

 

=  X[1]+X[3652], 3*X[191]+X[7982], X[355]+X[5441], X[1385]+X[26202], X[1482]+X[11684], X[3065]+X[6265], 5*X[3616]-X[16116], X[3648]+3*X[5603], 3*X[5426]+X[7701], X[5690]-2*X[18253], X[5887]+X[17637], 2*X[6701]-3*X[11230], X[7991]-3*X[16139], 5*X[8227]-X[16118], 2*X[10122]-X[24475], X[12699]+X[16113], X[16132]+X[16138], X[16150]-5*X[18493]

As a point on the Euler line, X has Shinagawa coefficients: {4 r + 3 R, -4 r + 3 R} 

 

= lies on these on lines: {1,3652}, {2,3}, {35,18357}, {56,11544}, {79,5427}, {191,7982}, {355,5441}, {515,22798}, {517,3647}, {758,10222}, {952,3746}, {993,22791}, {1125,12611}, {1385,26202}, {1482,11684}, {1483,3303}, {1621,26321}, {2771,5609}, {2975,3650}, {3065,6265}, {3304,10283}, {3579,3918}, {3616,16116}, {3648,5603}, {3649,5563}, {4265,18358}, {4653,5453}, {5267,9955}, {5298,27197}, {5426,7701}, {5690,18253}, {5887,17637}, {6246,9956}, {6701,11230}, {7173,14792}, {7991,16139}, {8227,16118}, {10058,11698}, {10122,24475}, {10175,26086}, {10593,14793}, {10902,28186}, {11375,16152}, {11376,16153}, {11518,24467}, {11545,14882}, {12047,18977}, {12699,16113}, {13391,22076}, {15446,15950}, {16132,16138}, {16150,18493}, {17768,20330}, {22938,25639}

 

= midpoint of X(i) and X(j) for these {i,j}: {1,3652}, {3,21669}, {21,13743}, {355,5441}, {381,15678}, {1385,26202}, {1482,11684}, {3065,6265}, {5887,17637}, {16132,16138}, {28453,28461}

 

= reflection of X(i) in X(j) for these {i,j}: {3,12104}, {5,16617}, {442,10021}, {549,15673}, {1483,15174}, {3649,5901}, {5428,21}, {5499,6675}, {5690,18253}, {16125,9955}, {19919,22936}, {24475,10122}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,21,12104}, {3,1656,17572}, {3,5047,140}, {3,6912,546}, {3,6920,3628}, {3,7489,5047}, {3,12104,5428}, {3,13743,21669}, {21,405,15673}, {21,3560,16617}, {21,3651,28443}, {21,5428,28463}, {21,15678,4189}, {21,21669,3}, {21,28461,13743}, {1012,19526,3}, {3560,6914,5}, {5047,6906,3}, {6906,7489,140}, {6913,6924,5}, {6914,16617,5428}, {6950,17572,3}, {13743,28453,21}

 

= (6-8-13) search numbers [1.37456884337425377, 0.501422773160396588, 2.65910924970060280]

 

 

Best regards

Ercole Suppa