HYACINTHOS 26117

[Antreas P. Hatzipolakis]

 

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

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI. resp.

La = the Euler line of IAbAc. Similarly Lb, Lc

Na = the NPC center of IAbAc. Similarly Nb, Nc.

1. The NPCs (Na), (Nb), (Nc) are concurrent at the midpoint of IFeuerbach point

 

2. ABC, NaNbNc are bilogic (perspective and orthologic)

3. La, Lb, Lc are concurrent.

4. The parallels to La, Lb, Lc through A, B, C, resp. are concurrent.

 

[Peter Moses]:

 

 

Hi Antreas,

 

1). X(1387).

 

2). perspector: X(5557).

 

(ABC, NaNBNc) orthology: X(104).

 

(NaNbNc,  ABC) orthology:

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

on lines {{1,4},{2,5734},{3,551}, {5,519},{8,5056},{10,1482},{ 11,11011},{12,5048},{40,3306}, {84,5558},{104,5557},{140,517} ,{145,5068},{165,10299},{214, 10993},{354,5884},{355,3244},{ 376,9589},{382,3655},{495, 7681},{496,6738},{499,4848},{ 516,550},{527,3560},{547,4745} ,{553,5563},{631,7991},...}.

Complement X[11362].

Midpoint of X(i) and X(j) for these {i,j}: {{1, 946}, {3, 4301}, {4, 5882}, {5, 10222}, {10, 1482}, {355, 3244}, {551, 3656}, {3817, 10247}, {5884, 12672}, {7982, 11362}}.

Reflection of X(i) in X(j) for these {i,j}: {{1125, 5901}, {1385, 3636}, {4745, 547}, {6684, 1125}, {10172, 5886}}.

3 X[1] + X[4], X[3] - 3 X[551], 5 X[1] - X[944], 5 X[4] + 3 X[944], X[4] - 3 X[946], X[944] + 5 X[946], 2 X[140] - 3 X[1125], X[550] - 3 X[1385], 3 X[10] - 5 X[1656], 3 X[1482] + 5 X[1656], 5 X[4] - 9 X[1699], 5 X[946] - 3 X[1699], 5 X[1] + 3 X[1699], X[944] + 3 X[1699], 3 X[40] - 7 X[3523], 7 X[3523] - 15 X[3616], X[40] - 5 X[3616], X[550] - 6 X[3636], 5 X[3] - 9 X[3653], 5 X[551] - 3 X[3653], X[382] + 3 X[3655], X[3] + 3 X[3656], 3 X[3653] + 5 X[3656], X[355] - 3 X[3817], X[3244] + 3 X[3817], 3 X[355] - 7 X[3851], 9 X[3817] - 7 X[3851], 3 X[3244] + 7 X[3851], 3 X[3656] - X[4301], 3 X[551] + X[4301], 9 X[3653] + 5 X[4301], 3 X[8] - 11 X[5056], 3 X[145] + 13 X[5068], 3 X[3] - X[5493], 9 X[551] - X[5493], 9 X[3656] + X[5493], 3 X[4301] + X[5493], 13 X[5068] - 9 X[5587], X[145] + 3 X[5587], X[4] - 9 X[5603], X[1699] - 5 X[5603], X[946] - 3 X[5603], X[1] + 3 X[5603], X[944] + 15 X[5603], 7 X[4] - 3 X[5691], 7 X[946] - X[5691], 7 X[1] + X[5691], 7 X[944] + 5 X[5691], 3 X[2] + 5 X[5734], 3 X[944] - 5 X[5882], 3 X[1] - X[5882], 3 X[946] + X[5882], 9 X[5603] + X[5882], 9 X[1699] + 5 X[5882], 3 X[5691] + 7 X[5882], 3 X[354] - X[5884], 5 X[1656] - 9 X[5886], X[10] - 3 X[5886], X[1482] + 3 X[5886], X[140] - 3 X[5901], 4 X[140] - 3 X[6684], 4 X[5901] - X[6684], 7 X[944] - 15 X[7967], 7 X[5882] - 9 X[7967], 7 X[1] - 3 X[7967], 7 X[5603] + X[7967], 7 X[946] + 3 X[7967], X[5691] + 3 X[7967], 7 X[1699] + 5 X[7967], 7 X[4] + 9 X[7967], 5 X[5734] - X[7982], 3 X[2] + X[7982], 5 X[631] - X[7991], 11 X[5056] - 15 X[8227], X[8] - 5 X[8227], 3 X[376] + X[9589], 3 X[2] - 7 X[9624], 5 X[5734] + 7 X[9624], X[7982] + 7 X[9624], 7 X[3523] - 9 X[10165], X[40] - 3 X[10165], 5 X[3616] - 3 X[10165], 10 X[1656] - 9 X[10172], 2 X[10] - 3 X[10172], 2 X[1482] + 3 X[10172], 11 X[5056] - 9 X[10175], X[8] - 3 X[10175], 5 X[8227] - 3 X[10175], X[3244] - 3 X[10247], X[355] + 3 X[10247], 7 X[3851] + 9 X[10247], X[550] - 9 X[10283], X[1385] - 3 X[10283], 2 X[3636] - 3 X[10283], 9 X[165] - 13 X[10299], X[5882] - 15 X[10595], X[1] - 5 X[10595], X[946] + 5 X[10595], 3 X[5603] + 5 X[10595], X[4] + 15 X[10595], 3 X[214] - X[10993], 7 X[9624] - X[11362], 5 X[5734] + X[11362], X[4] - 5 X[11522], 3 X[946] - 5 X[11522], 9 X[5603] - 5 X[11522], 3 X[10595] + X[11522], 3 X[1] + 5 X[11522], X[5882] + 5 X[11522], 3 X[354] + X[12672], ... .

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,5882),(1,1699,944),(1, 5603,946),(1,5691,7967),(1, 9612,3476),(1,9614,3486),(1, 11522,4),(1,12047,10106),(2, 5734,7982),(2,7982,11362),(3, 3656,4301),(4,5603,11522),(4, 11522,946),(8,8227,10175),(40, 3616,10165),(354,12672,5884),( 355,10247,3244),(551,4301,3),( 946,5882,4),(1125,3754,6692),( 1385,10283,3636),(1482,5886, 10),(3244,3817,355),(5603, 10595,1),(5734,9624,11362),( 7982,9624,2),(10531,10597, 1478),(10532,10596,1479).

crosssum of X(55) and X(2317).

Searches: {-0. 151423470638859220137139950320 ,-0. 365059685028567577229849027515 ,3. 96328586568363014280231594477} .

 

3). X(3649).

 

4). X(79).

 

Best regards,

Peter Moses