HYACINTHOS 29037

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

A', B', C' = the midpoints of AI, BI, CI, resp.
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

 

Ab, Ac = the reflections of A' in INb, INc, resp.

Bc, Ba = the reflections of B' in INc, INa. resp.

Ca, Cb = the reflections of C' in INa, INb, resp.
(they are collinear lying on the perpendicular bisector of IFe = (X)1)X(11))

 

N1, N2, N3 = the NPC centers of A'AbAc, B'BcBa, C'CaCb, resp.

La, Lb, Lc = the Euler lines of A'AbAc, B'BcBa, C'CaCb, resp
(concurrent at their common circumcenter I)

1. ABC, N1N2N3 are homothetic.
Homothetic center (on X(1)X(11)) ?

2. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent (on X(1)X(11))

[Peter Moses]:

Hi Antreas,

>1. ABC, N1N2N3 are homothetic.

X(16173).

So too with the medial triangle at:

 

 = X(1)X(6702)∩X(2)X(2802)

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

= X[1] + 2 X[6702],5 X[1] + X[12531],2 X[1] + X[15863],X[1] + 5 X[31272],X[3] + 2 X[16174],2 X[5] + X[11715],X[10] + 2 X[1387],X[10] - 4 X[6667],2 X[11] + X[214],X[11] + 2 X[1125],5 X[11] + X[10609],7 X[11] - X[12690],X[80] + 5 X[3616],X[100] - 7 X[3624],X[104] + 5 X[8227],X[124] + 2 X[29008],X[149] + 11 X[5550],X[214] - 4 X[1125],5 X[214] - 2 X[10609],7 X[214] + 2 X[12690],4 X[551] - X[11274],5 X[631] + X[14217],X[946] + 2 X[6713],10 X[1125] - X[10609],14 X[1125] + X[12690],X[1145] - 4 X[3634],X[1317] - 4 X[3636],X[1320] + 5 X[1698],2 X[1385] + X[6246],X[1387] + 2 X[6667],5 X[1656] + X[12737],2 X[3035] - 5 X[19862],2 X[3035] + X[21630],2 X[3036] + X[3244],7 X[3090] - X[12751],5 X[3617] + X[26726],7 X[3622] - X[7972],2 X[3626] + X[25416],5 X[3698] + X[17652],2 X[3754] + X[12758],X[3874] - 4 X[18240],X[3874] + 2 X[18254],X[3878] + 2 X[12736],X[4973] + 2 X[11813],X[4973] - 4 X[15325],5 X[5439] + X[17638],2 X[5901] + X[12619],4 X[5901] - X[25485],2 X[6681] + X[30384],10 X[6702] - X[12531],4 X[6702] - X[15863],2 X[6702] - 5 X[31272],5 X[7987] + X[10724],7 X[9624] - X[10698],X[10265] + 2 X[11729],7 X[10609] + 5 X[12690],X[11813] + 2 X[15325],2 X[12019] + 7 X[15808],X[12515] + 5 X[18493],2 X[12531] - 5 X[15863],X[12531] - 25 X[31272],X[12532] + 5 X[18398],2 X[12619] + X[25485],2 X[13624] + X[22938],X[13996] - 10 X[31253],2 X[15528] + X[31803],X[15863] - 10 X[31272],2 X[18240] + X[18254],2 X[18857] + X[24042],5 X[19862] + X[21630],8 X[19878] - 5 X[31235],2 X[20418] + X[21635].

= lies on these lines: {1, 6702}, {2, 2802}, {3, 16174}, {5, 11715}, {10, 1387}, {11, 214}, {80, 3616}, {100, 3624}, {104, 8227}, {124, 29008}, {149, 5550}, {499, 3878}, {515, 23513}, {516, 21154}, {528, 19883}, {547, 551}, {631, 14217}, {758, 3582}, {946, 6713}, {1145, 3634}, {1317, 3636}, {1320, 1698}, {1385, 6246}, {1656, 12737}, {2771, 3742}, {2800, 5883}, {2801, 5817}, {2829, 3817}, {3035, 14150}, {3036, 3244}, {3057, 20107}, {3086, 3874}, {3090, 12751}, {3617, 26726}, {3622, 7972}, {3626, 25416}, {3698, 17652}, {3754, 12758}, {3825, 8068}, {3892, 10072}, {4973, 11813}, {4996, 5259}, {5083, 11375}, {5248, 10090}, {5439, 17638}, {5443, 11570}, {5840, 10165}, {5901, 12619}, {6265, 30143}, {6681, 30384}, {7968, 8988}, {7969, 13976}, {7987, 10724}, {9624, 10698}, {10057, 10584}, {10058, 23708}, {10265, 11729}, {10527, 14740}, {10707, 15015}, {11108, 22560}, {11263, 12611}, {11376, 15558}, {11717, 24222}, {12019, 15808}, {12515, 18493}, {12532, 18398}, {12740, 30147}, {13205, 16408}, {13624, 22938}, {13902, 19077}, {13959, 19078}, {13996, 31253}, {15528, 31803}, {15950, 20118}, {17636, 20104}, {17724, 23869}, {18857, 24042}, {19878, 31235}, {20418, 21635}.

= midpoint of X(i) and X(j) for these {i,j}: {2, 16173}, {10707, 15015}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6702, 15863}, {1, 31272, 6702}, {11, 1125, 214}, {1387, 6667, 10}, {5886, 10199, 5883}, {5901, 12619, 25485}, {11813, 15325, 4973}, {18240, 18254, 3874}, {19862, 21630, 3035}

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and anticomplementary triangle at:

 

= X(2)X(2802)∩X(8)X(1387)

= 3*a^4 - 2*a^3*b - 6*a^2*b^2 + 2*a*b^3 + 3*b^4 - 2*a^3*c + 11*a^2*b*c - a*b^2*c - 6*a^2*c^2 - a*b*c^2 - 6*b^2*c^2 + 2*a*c^3 + 3*c^4 : :

= X[2] + 2 X[16173],X[8] + 8 X[1387],5 X[8] + 4 X[25416],X[8] - 10 X[31272],4 X[11] + 5 X[3616],8 X[11] + X[6224],X[20] + 8 X[16174],2 X[80] + 7 X[3622],2 X[100] - 11 X[5550],X[145] + 8 X[6702],X[149] + 8 X[1125],X[149] + 2 X[15015],X[153] - 10 X[8227],X[962] + 8 X[6713],4 X[1125] - X[15015],4 X[1145] - 13 X[19877],X[1320] + 8 X[6667],2 X[1320] + 7 X[9780],10 X[1387] - X[25416],4 X[1387] + 5 X[31272],8 X[3035] + X[9802],8 X[3036] + X[20050],7 X[3090] + 2 X[12737],5 X[3091] + 4 X[11715],7 X[3523] + 2 X[14217],10 X[3616] - X[6224],5 X[3623] + 4 X[15863],7 X[3624] + 2 X[21630],8 X[3634] + X[12653],8 X[3636] + X[9897],5 X[3890] + 4 X[6797],7 X[4678] + 2 X[26726],11 X[5056] - 2 X[12751],X[5180] + 8 X[15325],X[5541] - 10 X[19862],8 X[5901] + X[12247],16 X[6667] - 7 X[9780],7 X[9624] + 2 X[10265],X[9778] - 4 X[21154],X[9803] + 8 X[11729],X[9809] + 8 X[20418],8 X[9955] + X[12248],5 X[10595] + 4 X[12619],2 X[12531] + 7 X[20057],X[12532] + 8 X[18240],2 X[25416] + 25 X[31272].

= lies on these lines: {2, 2802}, {8, 1387}, {11, 2476}, {20, 16174}, {80, 3622}, {100, 5550}, {145, 6702}, {149, 1125}, {153, 8227}, {214, 31418}, {952, 5055}, {962, 6713}, {1145, 19877}, {1320, 6667}, {2771, 5886}, {2829, 9779}, {3035, 9802}, {3036, 20050}, {3090, 12737}, {3091, 11715}, {3523, 14217}, {3623, 15863}, {3624, 21630}, {3634, 12653}, {3636, 9897}, {3890, 6797}, {4323, 12832}, {4678, 26726}, {5047, 22560}, {5056, 12751}, {5180, 15325}, {5541, 19862}, {5901, 12247}, {9624, 10265}, {9778, 21154}, {9803, 11729}, {9809, 20418}, {9955, 12248}, {10588, 20586}, {10595, 12619}, {11373, 27529}, {11376, 25414}, {12531, 20057}, {12532, 18240}, {13205, 17531}, {15558, 18220}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 3616, 6224}, {1320, 6667, 9780}, {1387, 31272, 8}

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>2. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent (on X(1)X(11))  

X(7972).

Best regards,
Peter Moses.