HYACINTHOS 24303

[Antreas P. Hatzipolakis]:

 

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

Denote:

N1, N2, N3 = the NPC centers of IBC, ICA, IAB, resp.

Na, Nb, Nc = the reflections of N1, N2, N3 in IA, IB, IC, resp.

N'a, N'b, N'c = the reflections of N1, N2, N3 in IA', IB', IC', resp.

R1, R2, R3 = the perpendicular bisectors of NaN'a, NbN'b, NcN'c, resp.

 

1. R1 =  R2 = R3 = OI line

2. Na, N'a, Nb, N'b, Nc, N'c are concyclic.

The center of the circle lies on the OI line

[César Lozada]:

 

> 1.      R1 =  R2 = R3 = OI line

Confirmed

> 2.      Na, N'a, Nb, N'b, Nc, N'c are concyclic. The center of the circle lies on the OI line

Very nice!

Radius = |R/2-r|

Center:

Z = 2*a^3-3*(b+c)*a^2-2*(b^2-3*b* c+c^2)*a+3*(b^2-c^2)*(b-c) : : (trilinears)

= 3*X(1)-X(3)

= midpoint of X(i),X(j) for these {i,j}: {1,1482}, {3,7982}, {40,8148}, {145,355}, {946,3244}, {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), (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), (3625,10175), (3632,5079), (3633,5072), (3636,6684), (3680,6918), (3892,5884), (3913,6911), (3915,5398), (3940,4853), (3962,5288), (3991,4919), (4004,5253), (4511,6946), (4677,5055), (4691,10172), (4701,10171), (4870,6980), (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)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1,46,1388), (1,2098,9957), (1,2099,942), (1,3340,999), (1,5697,2646), (1,7962,3295), (1,7982,3), (3,1482,7982), (4,5734,3656), (8,5886,9956), (46,1388,5126), (145,5603,355), (355,5603,9955), (1466,2099,3340), (3241,5734,4), (3632,8227,5790), (3679,9624,1656)

= [ -0.855718682912623, -0.41201269918015, 4.320851127299541 ]

 

César Lozada