HYACINTHOS 28959

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

L, La, Lb, Lc = the Eiler lines of ABC, IBC, ICA, IAB, resp. (concurrent at X(21))

P, Pa, Pb, Pc = same points on L, La, Lb, Lc, resp.
(ie if OP/OH = t [t = number], then OaPa/OaHa = t, etc).

P1, P2, P3 = the reflections of Pa, Pb, Pc in BC, CA, AB, resp.

The parallels to La, Lb, Lc through P1, P2, P3, resp. are concurrent.

Point of concurrence for some P's (G, O, H, N.........) ?


[Peter Moses]:

Hi Antreas,

If P = X[2] + k X[3], and k = number, then concurrence  = X[2] + k X[191]
= (a+b+c) (a^3-a^2 b-a b^2+b^3-a^2 c-a b c-b^2 c-a c^2-b c^2+c^3)+3 a (a^3+a^2 b-a b^2-b^3+a^2 c-a b c-b^2 c-a c^2-b c^2-c^3) k : :

G, k = 0, ->G.
O, k = inf,  -> X(191).
H, k = -2/3, -> X(14450).
N, k = -1/3 -> X(11263).

X(20), k = -4/3 ->  

X(2)X(191)∩X(4)X(13465)

 

= 3 a^4+4 a^3 b-2 a^2 b^2-4 a b^3-b^4+4 a^3 c-a^2 b c-a b^2 c-2 a^2 c^2-a b c^2+2 b^2 c^2-4 a c^3-c^4 : : 

 

= 3 X[2] - 4 X[191],9 X[2] - 8 X[11263],8 X[21] - 7 X[3622],5 X[21] - 4 X[16137],X[145] - 4 X[3648],5 X[145] - 8 X[5441],3 X[191] - 2 X[11263],4 X[2475] - 5 X[3617],3 X[2475] - 4 X[21677],5 X[3091] - 4 X[16159],5 X[3522] - 4 X[16132],7 X[3523] - 8 X[22937],5 X[3617] - 8 X[11684],5 X[3617] - 2 X[20084],15 X[3617] - 16 X[21677],7 X[3622] - 16 X[3650],35 X[3622] - 32 X[16137],5 X[3623] - 4 X[16126],8 X[3647] - 7 X[15676],5 X[3648] - 2 X[5441],4 X[3649] - 5 X[15674],5 X[3650] - 2 X[16137],7 X[4678] - 4 X[16118],4 X[5441] - 5 X[15680],3 X[5603] - 4 X[22936],3 X[9778] - 2 X[16143],5 X[10595] - 6 X[28453],4 X[11263] - 3 X[14450],8 X[11281] - 9 X[15672],4 X[11544] - 5 X[31254],4 X[11684] - X[20084],3 X[11684] - 2 X[21677],3 X[12535] - 2 X[12682],4 X[12682] - 3 X[12849],3 X[20084] - 8 X[21677]

 

= lies on these lines: {2, 191}, {4, 13465}, {8, 12535}, {20, 2771}, {21, 999}, {30, 12245}, {46, 26792}, {79, 10590}, {144, 1654}, {145, 758}, {390, 17637}, {962, 7701}, {1749, 3086}, {3091, 16159}, {3120, 24898}, {3189, 20066}, {3522, 16132}, {3523, 22937}, {3623, 16126}, {3647, 15676}, {3649, 7288}, {3869, 20067}, {3873, 28646}, {4127, 15228}, {4295, 18259}, {4661, 6361}, {4678, 16118}, {5178, 28534}, {5180, 6763}, {5603, 22936}, {5904, 20095}, {5905, 12913}, {9778, 16143}, {10032, 15677}, {10056, 18244}, {10528, 20214}, {10587, 20059}, {10595, 28453}, {10786, 16116}, {11041, 13100}, {11281, 15672}, {11544, 31254}, {12514, 17483}, {17484, 21077}

 

= anticomplement of X(14450)
= reflection of X(i) and X(j) for these {i,j}: {4, 13465}, {21, 3650}, {145, 15680}, {962, 7701}, {2475, 11684}, {12849, 12535}, {14450, 191}, {15677, 10032}, {15680, 3648}, {16116, 16139}, {20084, 2475}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {191, 14450, 2}, {11684, 20084, 3617}

Best regards,
Peter Moses.