HYACINTHOS 29121

[Antreas P. Hatzipolakis]:

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

Denote:

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

La, Lb, Lc = the reflections of NbNc, NcNa, NaNb in BC, CA, AB, resp.

L1, L2, L3 = the perpendiculars from Na, Nb, Nc to La, Lb, Lc, resp.

L11, L21, L31 = the reflections of L1, L2, L3 in BC, CA, AB, resp.  

L11', L21', L31' = the reflections of L1, L2, L3 in  IA', IB', IC', resp.   

1.  L11, L21, L31 concur at a point Q
2.  L11', L21', L31' concur at a point Q'
3. The parallels to L11, L21, L31 (or L11', L21', L31') through A, B, C, resp. concur at a point Q"

 

Points Q, Q', Q" ?

 

The centroid of QQ'Q" is the incenter I.

 

[César Lozada]: 

 

 

Q = COMPLEMENT OF X(9897)

= 4*a^4-3*(b+c)*a^3-3*(b-c)^2*a^2+(b+c)*(3*b^2-5*b*c+3*c^2)*a-(b^2-c^2)^2 :: (barys)

= 3*X(1)-X(149), X(8)-3*X(15015), 3*X(10)-4*X(3035), 5*X(10)-4*X(3036), 3*X(10)-2*X(15863), 2*X(11)-3*X(551), X(100)+3*X(10031), X(149)+3*X(6224), 2*X(149)-3*X(21630), 3*X(214)-2*X(3035), 5*X(214)-2*X(3036), 3*X(214)-X(15863), 3*X(1317)+X(6154), 5*X(1317)+X(12732), 3*X(1317)-X(25416), 5*X(3035)-3*X(3036), 6*X(3036)-5*X(15863), 3*X(3244)+2*X(6154), X(3244)+2*X(10609), 5*X(3244)+2*X(12732), 3*X(3244)-2*X(25416), X(6154)-3*X(10609), 5*X(6154)-3*X(12732), 2*X(6224)+X(21630), X(7972)-3*X(10031), 3*X(10165)-2*X(12619), 5*X(10609)-X(12732), 3*X(10609)+X(25416), 3*X(12732)+5*X(25416)

= lies on these lines: {1, 149}, {2, 9897}, {8, 15015}, {10, 140}, {11, 551}, {20, 13253}, {36, 100}, {65, 1317}, {80, 1125}, {104, 5267}, {145, 3336}, {226, 18976}, {515, 6265}, {516, 10698}, {528, 5542}, {758, 17660}, {944, 6326}, {946, 11567}, {950, 12740}, {993, 3655}, {997, 5531}, {1145, 3625}, {1320, 5557}, {1483, 5885}, {1484, 15178}, {1768, 5731}, {2550, 6264}, {2771, 3878}, {2800, 4297}, {2932, 8715}, {3241, 12653}, {3576, 12247}, {3616, 20085}, {3623, 9802}, {3626, 12531}, {3636, 16173}, {3754, 17636}, {3814, 11698}, {3884, 17638}, {4084, 4311}, {4301, 5840}, {4304, 12758}, {4314, 15558}, {4669, 6174}, {4973, 5855}, {5493, 24466}, {5727, 10199}, {5836, 32900}, {5854, 9945}, {5883, 6797}, {5886, 12747}, {6667, 19883}, {6684, 19914}, {6702, 19862}, {7982, 13199}, {9803, 18231}, {10057, 13411}, {10106, 12739}, {10197, 13384}, {10738, 13464}, {11571, 21578}, {12019, 15808}, {12053, 12743}, {12331, 25440}, {12573, 14151}, {12611, 31673}, {12737, 13607}, {12751, 28236}, {13205, 25439}, {13605, 31525}, {14792, 17100}, {17439, 21090}, {25558, 30424}

= midpoint of X(i) and X(j) for these {i,j}: {1, 6224}, {20, 13253}, {100, 7972}, {145, 5541}, {944, 6326}, {1317, 10609}, {6154, 25416}, {7982, 13199}, {10698, 12119}, {12653, 20095}

= reflection of X(i) in X(j) for these (i,j): (10, 214), (80, 1125), (946, 19907), (1484, 15178), (3244, 1317), (3625, 1145), (4084, 11570), (4301, 25485), (4669, 6174), (5493, 24466), (10265, 1385), (10738, 13464), (12531, 3626), (12737, 13607), (13605, 31525), (15863, 3035), (17636, 3754), (17638, 3884), (19914, 6684), (21630, 1), (21635, 6265), (30424, 25558), (31673, 12611)

= complement of X(9897)

= (anti-Aquila)-anticomplement of-X(11)

= (3rd Euler)-anticomplement of-X(6246)

= (Wasat)-anticomplement of-X(6246)

= (2nd circumperp)-complement of-X(12247)

= (hexyl)-complement of-X(12247)

= (5th mixtilinear)-complement of-X(12653)

= X(6224)-of-anti-Aquila triangle

= X(7722)-of-Wasat triangle

= X(11807)-of-excenters-reflections triangle

= X(12308)-of-2nd Zaniah triangle

= X(21630)-of-5th mixtilinear triangle

= X(21650)-of-2nd circumperp triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (214, 15863, 3035), (3035, 15863, 10)

= [ 1.1932457104282190, -1.5806695340035960, 4.1842453698661530 ]

 

Q’  = X(3244)

Q” = X(80)

 

César Lozada