HYACINTHOS 26766

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

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

 

Naa, Nab, Nac = the orthogonal projections of Na on AP, AB, AC, resp.

Nba, Nbb, Nbc = the orthogonal projections of Nb on BA, BP, BC, resp.

Nca, Ncb, Ncc = the orthogonal projections of Nc on CA, CB, CP, resp.

 

O1, O2, O3 = the circumcenters of NaaNabNac, NbaNbbNbc, NcaNcbNcc, resp.

N1, N2, N3 = the NPC centers of NaaNabNac, NbaNbbNbc, NcaNcbNcc, resp.

 

For P = I:

 

1. ABC, O1O2O3 are orthologic

 

2. ABC, N1N2N3 are orthologic.


[Peter Moses]:

Hi Antreas,

1).
(ABC,  O1O2O3): X(79).
O1O2O3, ABC): X(9955).

2).
(ABC,  N1N2N3): 

 

a^2 (a^4-2 a^2 b^2+b^4+a^3 c-a^2 b c-a b^2 c+b^3 c+2 a^2 c^2+a b c^2+2 b^2 c^2-a c^3-b c^3-3 c^4) (a^4+a^3 b+2 a^2 b^2-a b^3-3 b^4-a^2 b c+a b^2 c-b^3 c-2 a^2 c^2-a b c^2+2 b^2 c^2+b c^3+c^4):: 

on lines {{500,3746},{1830,1844}}.

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The isogonal conjugate of this point is "nicer".:

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

on lines {{1,550},{3,5443},{4,5445},{11,5131},{20,5903},{30,80},{35,79},{36,516},{40,4333},{46,2955},{65,4324},{109,477},{149,4973},{165,6907},{214,5180},{515,3245},{517,4316},{519,9963},{529,5541},{942,4330},{1155,3583},{1478,9778},{1479,5435},{1768,5842},{1836,5010},{2099,3534},{3017,9340},{3057,4325},{3336,6284},{3337,15171},{3474,3488},{3476,4299},{3529,10573},{3579,3585},{3582,5122},{3601,4338},{3648,3678},{3746,4292},{4084,11015},{4297,11009},{4304,5425},{4312,8255},{5127,5196},{5442,7741},{5493,5559},{5535,5840},{5536,12750},{7280,12699},{7354,11010},{7411,14799},{11495,15175},{11813,13587},{12047,12512}}.
reflection of X(i) in X(j) for these {i,j}: {{80, 484}, {149, 4973}, {3583, 1155}, {5180, 214}}
3 X[3582] - 4 X[5122], 2 X[11] - 3 X[5131], 4 X[4316] - X[7972], 7 X[80] - 8 X[11545], 7 X[484] - 4 X[11545], 2 X[11813] - 3 X[13587].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (35, 1770, 79), (40, 4333, 10483), (65, 4324, 5441), (3474, 4302, 5902), (4299, 6361, 5697).
----------------

(N1N2N3,  ABC): 

a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^5 b^2 c-a^4 b^3 c-2 a^3 b^4 c+2 a^2 b^5 c+a b^6 c-b^7 c+a^6 c^2+a^5 b c^2+4 a^4 b^2 c^2-a^3 b^3 c^2-7 a^2 b^4 c^2+2 b^6 c^2-a^4 b c^3-a^3 b^2 c^3+2 a^2 b^3 c^3-a b^4 c^3+b^5 c^3-3 a^4 c^4-2 a^3 b c^4-7 a^2 b^2 c^4-a b^3 c^4-2 b^4 c^4+2 a^2 b c^5+b^3 c^5+3 a^2 c^6+a b c^6+2 b^2 c^6-b c^7-c^8):: 

on lines {{2801,12006},{12005,13630}}.

Best regards,
Peter Moses.