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24380Re: I, NPC, Points

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• Sep 13, 2016

[Antreas Hatzipolakis]:

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

A1, B1, C1 = the orthogonal projections of Na, Nb, Nc on IA, IB, IC, resp.

A2, B2, C2 = the reflections of Na, Nb, Nc, in IA, IB, IC, resp.

1. The orthocenter of A1B1C1 lies on the OI line of ABC

*** X(942)
https://beta.groups.yahoo. com/neo/groups/Hyacinthos/ conversations/messages/24090

2. The Euler line of A2B2C2 is the OI line of ABC.
Which points are the O,I wrt the triangle A2B2C2?

*** The O wrt the triangle A2B2C2 is  W = 3 I - O

W = (a (2 a^3 - 3 a^2 (b + c) + 3 (b - c)^2 (b + c) -
2 a (b^2 - 3 b c + c^2)) : ... : ... ),

with (6-9-13)-search numbers (-0.855718682912623, -0.412012699180152, 4.32085112729954).

Midpoint of X(i) and X(j) for these {i,j}:  {1, 1482}, {3, 7982}, {40, 8148}, {145, 355}, {381, 2487}, {946, 3244}, {1320, 6265}, {1657, 9589}, {3241, 3656}, {3555, 5887}, {4301, 5882}.

Reflection of X(i) in X(j) for these {i,j}:  {8, 9956}, {10, 5901}, {65, 6583}, {355, 9955}, {1385, 1}, {1483, 3635}, {3579, 1385}, {4669, 547}, {5493, 548}, {5690, 1125}, {6684, 3636}.

On lines: {1, 3}, {4, 1392}, {5, 519}, {8, 3090}, {10, 3628}, {20, 3655}, {30,
4301}, {72, 1173}, {140, 551}, {145, 355}, {381, 5881}, {392, 5047}, {515, 1483}, {518, 576}, {546, 946}, {547, 4669}, {548, 5493}, {573, 3723}, {575, 1386}, {631, 3654}, {632, 1125}, {944, 3146}, {956, 3951}, {962, 3529}, {1000, 5703}, {1056, 4323}, {1058, 4345}, {1210, 1387}, {1320, 1389}, {1339, 6048}, {1457, 5399}, {1656, 3679}, {1657, 9589}, {1837, 7743}, {1870, 1872}, {2771, 7984}, {2800, 3881}, {3058, 7491}, {3419, 6984}, {3485, 6982}, {3488, 5812}, {3523, 3653}, {3525, 3616}, {3555, 5887}, {3584, 5559}, {3585, 7972}, {3621, 5818}, {3622, 5657}, {3632, 5079}, {3633, 5072}, {3636, 6684}, {3680, 6918}, {3872, 3984}, {3892, 5884}, {3913, 6911}, {3915, 5398}, {3940, 4853}, {3962, 5288}, {3991, 4919}, {4004, 5253}, {4511, 6946}, {4677, 5055}, {4870, 6980}, {4902, 5059}, {4930, 6913}, {5044, 5289}, {5054, 9588}, {5076, 5691}, {5258, 7489}, {5722, 5761}, {5727, 9669}, {6419, 7969}, {6420, 7968}, {6447, 9583}, {6519, 9616}, {6863, 10056}, {6914, 8666}, {6924, 8715}, {6958, 10072}, {6988, 7320}.

A.Montesdeoca

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