HYACINTHOS 26229

[Antreas P. Hatzipolakis]:

 

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

Denote:

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

N1, N2, N3 = the reflections of Na, Nb, Nc in I, resp.

The NPCs of A'NaN1, B'NbN2, C'NcN3 are coaxial.
The 2nd intersection (other than I) lies on the OI line.

[César Lozada]:

 

2nd intersection (other than I) ?

 

Q2 = X(1)X(3) ∩ X(30)X(1125)

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

= X(1)+3*X(3) = 5*X(1)+3*X(40) = 7*X(1)+9*X(165) = X(1)-3*X(1385) = 7*X(1)-3*X(1482) = X(1)-9*X(3576) = 11*X(1)-3*X(7982) = 3*X(4)-11*X(5550) = X(4)-3*X(11230) = X(5)-3*X(10165) = X(4297)+3*X(10165)

= On lines:

{1,3}, {4,5550}, {5,4297}, {8,3524}, {10,549}, {20,5886}, {21,4881}, {30,1125}, {44,572}, {72,3431}, {74,11699}, {79,4870}, {104,6986}, {140,515}, {145,3654}, {182,4663}, {186,1829}, {214,960}, {229,4221}, {355,631}, {376,3616}, {378,11363}, {381,3624}, {382,8227}, {392,4189}, {495,4311}, {496,4304}, {500,1193}, {518,5092}, {582,1468}, {912,12038}, {944,3617}, {952,3626}, {956,4855}, {993,5044}, {1000,6049}, {1480,1616}, {1483,11362}, {2975,4420}, {3488,5265}, {3585,5444}, {3621,5657}, {3622,3656}, {3625,5690}, {3689,5288}, {3811,11194}, {3916,4511}, {4004,9352}, {4292,11544}, {4293,11374}, {4298,5719}, {4299,11375}, {4301,10283}, {4302,11376}, {4316,5443}, {4652,5730}, {4816,9588}, {5219,9655}, {5258,12773}, {5298,10543}, {5433,10572}, {5438,9708}, {5844,13607}, {5887,10167}, {6051,8143}, {6284,7743}, {6713,12019}, {6734,10609}, {6759,12262}, {8546,9004}, {8715,11260}, {9778,10595}, {10610,12675}, {11711,12042}

= midpoint of X(i) and X(j) for these {i,j}: {1,3579}, {3,1385}, {5,4297}, {74,11699}, {548,5901}, {550,946}, {551,8703}, {960,13369}, {1386,3098}, {1483,11362}, {1511,11709}, {5690,5882}, {5731,11231}, {6759,12262}, {8715,11260}, {11278,12702}, {11711,12042}, {11720,12041}, {12512,13464}

= reflection of X(i) in X(j) for these (i,j): (3828,11812), (3853,12571), (5885,9940), (6583,13373), (6684,3530), (9955,1125), (9956,140)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1,3,3579), (1,40,8148), (1,12702,11278), (3,1482,165), (3,3576,1385), (3,10246,40), (3,10680,5584), (3,13151,942), (35,1319,9957), (36,2646,942), (36,3576,13151), (65,7280,5122), (214,5267,960), (355,631,11231), (631,5731,355), (1385,3579,1), (2646,13151,1385), (3576,7987,3), (3579,11278,12702), (4297,10165,5), (8148,10246,1)

= [ 5.509349298011517, 4.84379032241281, -2.255505417306729 ]

 

César Lozada