HYACINTHOS 29549

[Antreas P. Hatzipolakis]:

Intouch triangle version:

 

Let ABC be a triangle and A'B'C' the pedal triangle of I.

 

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
N1, N2, N3 = the reflections of Na, Nb, Nc in AI, BI, CI, resp.

MaMbMc = the medial triangle of NaNbNc
M1, M2, M3 = the reflections of Ma, Mb, Mc in AI, BI, CI, resp.

L = the Euler line of A'B'C' = OI line of ABC
A"B"C" = the reflection of A'B'C' in L

A'B'C', N1N2N3, M1M2M3 share the same Euler line L.

 

1. A"B"C", N1N2N3, M1M2M3 are homothetic.
Which are the homothetic centers of 

1.1. A"B"C", N1N2N3,
1.2. N1N2N3, M1M2M3, 

1.3. M1M2M3, A"B"C" ?

2. 
2.1. Which points wrt triangle ABC are O, H, etc......  of N1N2N3 ?

2.2. Which points wrt triangle ABC are O, H, etc..... of M1M2M3 ?

 

3. Which point is I of ABC wrt triangle
3.1. N1N2N3
3.2. M1M2M3 ?

[Peter Moses]:


Hi Antreas,

1.1 X(65).
1.2 X(3).
1.3 X(2646).

2.1
O -> X(10222).
H -> X(65).

2.2
O -> X(15178).

H ->  COMPLEMENT OF X(5887) =

 

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

= X[1] - 3 X[10202], X[1] - 5 X[15016], 3 X[1] - X[23340], 3 X[3] - X[14110], 3 X[4] + X[9961], X[40] + 3 X[5902], 3 X[65] + X[14110], X[72] - 3 X[26446], 3 X[354] - X[1482], X[355] - 3 X[3753], 3 X[381] - X[12688], 2 X[548] - 3 X[10178], 5 X[631] - X[3869], 3 X[942] - 2 X[6583], X[942] + 2 X[13145], 3 X[942] + X[31798], X[946] - 3 X[5883], X[1071] + 3 X[3753], X[1657] - 3 X[5918], 5 X[1698] - X[5693], X[3057] - 3 X[10246], 7 X[3526] - 5 X[25917], 3 X[3576] + X[5903], X[3579] + 2 X[31794], 5 X[3698] - 3 X[5790], 5 X[3698] - X[14872], 3 X[3740] - 2 X[31835], 3 X[3742] - 2 X[5901], 2 X[3754] + X[13369], 4 X[3812] - X[31937], X[3868] + 3 X[5657], 3 X[3873] + X[12245], X[3878] - 3 X[10165], X[3901] + 7 X[9588], 3 X[3919] + X[4297], 7 X[3922] + X[12680], 7 X[3922] - X[18525], 7 X[4002] - 3 X[18908], 5 X[4004] + 3 X[10167], 5 X[4004] + X[18481], X[4084] + 3 X[10164], 2 X[5044] - 3 X[11231], 3 X[5049] - X[13600], 3 X[5049] - 2 X[33179], 3 X[5054] - X[31165], 5 X[5439] - 3 X[5886], 5 X[5439] - X[12672], 3 X[5587] + X[15071], 3 X[5692] - 7 X[31423], X[5694] - 3 X[11231], 3 X[5790] - X[14872], 5 X[5818] - X[12528], 3 X[5885] - X[6583], 2 X[5885] + X[31788], 6 X[5885] + X[31798], 3 X[5886] - X[12672], 3 X[5902] - X[24474], X[5903] - 3 X[10273], X[6583] + 3 X[13145], 2 X[6583] + 3 X[31788], 2 X[6583] + X[31798], X[7672] + 3 X[21151], 3 X[7967] + X[14923], X[7982] - 5 X[18398], 3 X[10164] - X[31806], 3 X[10165] - 2 X[31838], 3 X[10167] - X[18481], 3 X[10175] - X[31803], 3 X[10202] - 2 X[13373], 3 X[10202] - 5 X[15016], 9 X[10202] - X[23340], 3 X[10246] + X[25413], 3 X[10247] - 5 X[17609], X[10693] - 3 X[15061], 3 X[11227] - 2 X[13624], 3 X[11227] - X[31786], 3 X[11246] + X[11827], 6 X[13145] - X[31798], 2 X[13373] - 5 X[15016], 6 X[13373] - X[23340], 15 X[15016] - X[23340], 3 X[15064] - 5 X[31399], X[16004] + 2 X[17706], 5 X[31666] - 6 X[33574], 3 X[31788] - X[31798]

= lies on these lines: {1, 3}, {2, 5887}, {4, 9961}, {5, 3812}, {8, 6897}, {10, 912}, {30, 7686}, {72, 5552}, {84, 18761}, {119, 125}, {140, 960}, {182, 3827}, {224, 5687}, {355, 377}, {381, 12688}, {392, 6910}, {404, 21740}, {515, 3754}, {516, 31870}, {518, 5690}, {519, 12005}, {548, 10178}, {601, 3924}, {631, 3869}, {758, 6684}, {944, 4190}, {946, 5883}, {950, 5840}, {952, 5836}, {958, 24467}, {962, 6899}, {971, 5880}, {1064, 24443}, {1125, 2800}, {1158, 3560}, {1483, 3880}, {1490, 18491}, {1519, 6831}, {1657, 5918}, {1698, 5693}, {1706, 5534}, {1737, 1858}, {1770, 7491}, {1788, 6825}, {1836, 6928}, {1837, 6923}, {1871, 14018}, {1888, 7510}, {1898, 10826}, {1902, 7414}, {2182, 16547}, {2390, 5892}, {2392, 31760}, {2649, 9356}, {2778, 12041}, {2801, 3918}, {2802, 13607}, {2818, 9729}, {2975, 26877}, {3474, 6868}, {3485, 6891}, {3486, 6948}, {3526, 25917}, {3555, 12648}, {3556, 6642}, {3577, 9841}, {3616, 6977}, {3627, 15726}, {3634, 20117}, {3654, 11239}, {3698, 5790}, {3740, 31835}, {3742, 5901}, {3868, 5657}, {3873, 12245}, {3874, 11362}, {3878, 10165}, {3881, 28234}, {3901, 9588}, {3919, 4297}, {3922, 12680}, {4002, 18908}, {4004, 6934}, {4084, 10164}, {4292, 5841}, {4295, 6827}, {4511, 6940}, {4848, 18389}, {5044, 5694}, {5054, 31165}, {5057, 6902}, {5086, 6951}, {5437, 7971}, {5439, 5886}, {5450, 30147}, {5553, 30513}, {5587, 15071}, {5603, 6890}, {5692, 31423}, {5721, 23604}, {5722, 10525}, {5770, 19843}, {5784, 5833}, {5787, 18517}, {5806, 22793}, {5818, 12528}, {5927, 6984}, {6259, 17649}, {6261, 6911}, {6265, 17614}, {6824, 14647}, {6836, 10531}, {6850, 18391}, {6862, 10200}, {6863, 24914}, {6882, 12047}, {6883, 12514}, {6913, 12686}, {6942, 9352}, {6947, 11415}, {6955, 10914}, {6958, 11375}, {6971, 17605}, {6980, 17606}, {6985, 12520}, {7672, 21151}, {7967, 14923}, {7992, 18540}, {8094, 8130}, {8129, 12445}, {9947, 17528}, {10039, 10956}, {10044, 10404}, {10085, 18519}, {10175, 31803}, {10265, 25639}, {10391, 31775}, {10543, 24466}, {10609, 32900}, {10693, 15061}, {10935, 11047}, {11112, 26201}, {11246, 11827}, {11374, 12709}, {11491, 18444}, {11499, 18446}, {11684, 26878}, {12433, 12710}, {12515, 12775}, {12564, 16004}, {12594, 24476}, {12664, 31828}, {12749, 17660}, {12778, 13217}, {12905, 13211}, {13374, 22791}, {14054, 20612}, {15064, 31399}, {15952, 18165}, {16049, 18180}, {16154, 17637}, {19860, 22758}, {31752, 31836}, {33597, 33858}

= complement of X(5887)
= midpoint of X(i) and X(j) for these {i,j}: {3, 65}, {10, 5884}, {40, 24474}, {355, 1071}, {942, 31788}, {1770, 7491}, {3057, 25413}, {3576, 10273}, {3654, 24473}, {3874, 11362}, {4084, 31806}, {5690, 24475}, {5836, 12675}, {5885, 13145}, {6259, 17649}, {6265, 17654}, {7686, 9943}, {12680, 18525}, {17660, 19914}, {31787, 31794}

= reflection of X(i) in X(j) for these {i,j}: {1, 13373}, {5, 3812}, {942, 5885}, {960, 140}, {1385, 9940}, {3579, 31787}, {3627, 16616}, {3878, 31838}, {5694, 5044}, {5777, 9956}, {9856, 9955}, {9957, 15178}, {10222, 5045}, {10284, 31792}, {13600, 33179}, {18857, 18856}, {20117, 3634}, {22791, 13374}, {22793, 5806}, {31786, 13624}, {31788, 13145}, {31793, 31663}, {31836, 31752}, {31837, 6684}, {31870, 33815}, {31937, 5}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40, 10679}, {1, 46, 11509}, {1, 2077, 33596}, {1, 3359, 11248}, {1, 5119, 10965}, {1, 10202, 13373}, {1, 10269, 1385}, {1, 12703, 12000}, {1, 14803, 2646}, {1, 15016, 10202}, {1, 18838, 942}, {1, 24927, 15178}, {3, 10246, 3612}, {3, 11507, 26285}, {3, 16203, 22768}, {40, 3576, 16208}, {40, 5902, 24474}, {40, 18443, 10267}, {65, 13750, 942}, {119, 24982, 9956}, {1071, 3753, 355}, {1385, 3579, 32613}, {1385, 10225, 33862}, {1385, 23961, 13624}, {1385, 32612, 18857}, {1385, 33281, 15178}, {3339, 30503, 5709}, {3359, 11248, 3579}, {3698, 14872, 5790}, {3878, 10165, 31838}, {4084, 10164, 31806}, {5049, 13600, 33179}, {5439, 12672, 5886}, {5554, 10940, 377}, {5554, 12115, 355}, {5690, 32213, 10915}, {5694, 11231, 5044}, {6862, 28628, 11230}, {10225, 33862, 31663}, {10246, 25413, 3057}, {10267, 18443, 1385}, {11227, 31786, 13624}, {12000, 12702, 12703}, {12609, 12616, 5}, {14647, 28629, 6824}

3.1 X(186).
3.2 X(2071).

Best regards,
Peter Moses.