HYACINTHOS 29270

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

A', B',  C' = the orthogonal projections of A, B, C on NbNc, NcNa, NaNb, resp.

Ab, Ac = the orthogonal projections of A' on NaNb, NaNc, resp,

Bc, Ba = the orthogonal projections of B' on NbNc, NbNa, resp,
Ca, Cb = the orthogonal projections of C' on NcNa, NcNb, resp.

 

(Oa) = the circumcircle of A'AbAc
(Ob) = the circumcircle of B'BcBa
(Oc) = the circumcircle of C'CaCb

For P = I: 

ABC, OaObOc are orthologic. 

The orthologic center (ABC, OaObOc) lies on the Euler line (it is X(21))
The other one?

 

 

[Petre Moses]:


Hi Antreas,

(OaObOc, ABC) =

= X(2)X(5541)∩X(10)X(1320)

= 2*a^4 - a^3*b - 5*a^2*b^2 + a*b^3 + 3*b^4 - a^3*c + 8*a^2*b*c - 5*a^2*c^2 - 6*b^2*c^2 + a*c^3 + 3*c^4 : :

= X[1] - 9 X[32558],9 X[2] - X[5541],15 X[2] + X[9802],3 X[2] + X[21630],3 X[10] + X[1320],X[10] + 3 X[16173],X[10] - 5 X[31272],3 X[11] + X[214],7 X[11] + X[10609],9 X[11] - X[12690],X[11] + 3 X[32557],X[80] + 3 X[551],5 X[80] + 3 X[10031],X[100] - 5 X[19862],X[104] + 3 X[3817],X[119] - 3 X[10171],X[149] + 7 X[3624],X[153] - 9 X[7988],X[214] - 3 X[1125],7 X[214] - 3 X[10609],3 X[214] + X[12690],X[214] - 9 X[32557],5 X[551] - X[10031],3 X[946] + X[12515],7 X[1125] - X[10609],9 X[1125] + X[12690],X[1125] - 3 X[32557],X[1145] - 3 X[3828],X[1320] - 9 X[16173],X[1320] + 15 X[31272],3 X[1387] + X[3036],X[1484] + 3 X[11230],X[3036] - 3 X[6702],X[3065] + 3 X[11263],7 X[3090] + X[6264],X[3218] - 9 X[3582],X[3218] + 3 X[11813],3 X[3582] + X[11813],15 X[3616] + X[20085],5 X[3616] - X[33337],7 X[3622] + X[9897],3 X[3898] + X[17636],3 X[4669] + X[26726],5 X[5541] + 3 X[9802],X[5,541] + 3 X[21630],11 X[5550] - 3 X[15015],3 X[5883] + X[17638],3 X[5886] + X[10265],X[6224] - 9 X[25055],5 X[8227] - X[21635],7 X[9624] + X[12247],7 X[9780] + X[12653],X[9802] - 5 X[21630],3 X[10164] + X[14217],3 X[10165] + X[10738],3 X[10175] + X[12737],9 X[10609] + 7 X[12690],X[10609] - 21 X[32557],X[10707] + 3 X[19883],X[11715] + 3 X[23513],X[12512] - 3 X[21154],X[12690] + 27 X[32557],3 X[16173] + 5 X[31272],X[19925] - 3 X[23513],X[20085] + 3 X[33337]

= lies on these lines: {1, 32558}, {2, 5541}, {10, 1320}, {11, 214}, {80, 551}, {100, 19862}, {104, 3817}, {119, 10171}, {149, 3624}, {153, 7988}, {516, 6713}, {519, 1387}, {946, 12515}, {952, 3636}, {1145, 3828}, {1484, 11230}, {1749, 3218}, {2771, 12009}, {2802, 3634}, {2829, 12571}, {3035, 19878}, {3065, 10266}, {3090, 6264}, {3306, 10199}, {3616, 20085}, {3622, 9897}, {3635, 15863}, {3884, 6797}, {3898, 17636}, {4669, 26726}, {4691, 5854}, {4701, 25416}, {4997, 21087}, {5533, 13411}, {5550, 15015}, {5883, 17638}, {5886, 10265}, {6224, 25055}, {6681, 7743}, {6715, 25351}, {8227, 21635}, {9624, 12247}, {9780, 12653}, {10164, 14217}, {10165, 10738}, {10175, 12737}, {10200, 11024}, {10707, 19883}, {11715, 19925}, {12053, 20107}, {12512, 21154}, {12619, 13464}, {14028, 24222}, {17719, 23869}

= midpoint of X(i) and X(j) for these {i,j}: {11, 1125}, {1387, 6702}, {3635, 15863}, {3884, 6797}, {4701, 25416}, {6681, 7743}, {6713, 16174}, {11715, 19925}, {12619, 13464}

= reflection of X(i) in X(j) for these {i,j}: {3035, 19878}, {3634, 6667}

 

= X(i)-complementary conjugate of X(j) for these (i,j): {3065, 121}, {19302, 16594}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 32557, 1125}, {11715, 23513, 19925}, {16173, 31272, 10}


Best regards
Peter Moses.