HYACINTHOS 28749

[César Lozada  - Dao Thanh Oai]:

 

Dao Thanh Oai (Dec. 20, 2018):

Let ABC be a triangle and {B_a, C_a} two points on BC such that AA_bA_c is an equilateral triangle. Construct {C_b, A_b} and {A_c, B_c} cyclically.

Let P=X(n)=u:v:w (trilinears) a triangle center whose exact coordinates with respect to ABC are U:V:W.

 

Denote

1)      A’ = X(n)-of-AA_bA_c, B’=X(n)-of-BB_cB_a, C’=X(n)-of-CC_aC_b

2)      A”= X(n)-of-AB_cC_b, B”=X(n)-of-BC_aA_c, C”=X(n)-of-CA_bB_a

 

Then:

1)      A’B’C’ and A”B”C” are equilateral

2)      ABC and A’B’C’ are perspective

3)      ABC and A”B”C” are perspective

 

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César Lozada (Dec. 21, 2018):

 

·         Trilinear coordinates are:

B_a = 0 : sec(B+π/6) : sec(C-π/6), and cyclically C_b, A_c

C_a = 0 : sec(B-π/6) : sec(C+π/6), and cyclically A_b, B_c

 

A’ = sqrt(3/4)*u+cos(C-π/6)*v+cos(B-π/6)*w :  w*cos(A+π/6) : v*cos(A+π/6)

A” = sqrt(3/4)*u+cos(B+π/6)*v+cos(C+π/6)*w : v*cos(A-π/6) : w*cos(A-π/6)

 

·         Triangles A’B’C’ and A”B”C” are equilateral and have respective squared-sidelengths:

a’² = (4/3)*∑ [U^2+2*cos(A-π/3)*V*W]

a”² = (4/3)*∑ [U^2+2*cos(A+π/3)*V*W].

 

·         ABC, A’B’C’, A”B”C” are perspective by pairs and the perpector is always the isogonal conjugate of P.

 

·         A’B’C’ and A”B”C” share the same center O* = u+2*cos(C)*v+2*cos(B)*w : :

 

ETC pairs (P,O*(P)):

(1,5902), (2,5640), (3,381), (4,5890), (5,5946), (6,6), (10,15049), (15,13), (16,14), (20,15305), (23,9140), (25,26869), (30,5663), (32,5309), (39,7753), (50,1989), (58,3017), (67,9971), (69,11188), (74,4), (98,6785), (99,6787), (106,6788), (110,2), (112,6794), (113,9730), (115,15544), (125,51), (128,15537), (140,13364), (141,16776), (146,15072), (182,5476), (186,14644), (187,115), (193,15531), (265,568), (323,110), (351,8371), (352,111), (353,6032), (376,16261), (395,11626), (396,11624), (399,3), (468,12099), (511,542), (512,690), (513,8674), (514,2774), (515,2779), (516,2772), (517,2771), (518,2836), (519,2842), (520,9033), (521,2850), (522,2773), (523,526), (524,2854), (525,9517), (526,523), (541,14915), (542,511), (574,5475), (616,16259), (617,16260), (647,1637), (649,4120), (652,14400), (667,14431), (669,9148), (690,512), (691,9144), (729,14700), (840,10773), (842,11005), (858,12824), (895,1992), (902,3120), (974,16657), ……..

 

·         More: Six points {A_b, A_c, B_c, B_a, C_a, C_b} lie on a conic with center X(6) and perspector X(18434). No ETC center or known point lie on this conic.

 

Some others O*(P):

 

O*(X(7)) = X(7)X(2808) ∩ X(674)X(7671)

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

= X(7)-4*X(29957)

= on lines: {7, 2808}, {674, 7671}, {942, 15058}, {2772, 5902}, {2836, 11188}, {5889, 10399}, {8236, 9052}, {10122, 11444}

= [ 1.4492936891692840, 1.7009424869838610, 1.7941841343789590 ]

 

O*(X(8)) = X(8)X(29958) ∩ X(392)X(23155)

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

= X(8)-4*X(29958), 5*X(3616)-2*X(23154), 3*X(5640)-2*X(5902), 3*X(5640)-4*X(15049), 2*X(5693)+X(5889), 4*X(5694)-X(11412), 4*X(5883)-5*X(11451), 4*X(5884)-7*X(15043), 8*X(5885)-11*X(15024), 3*X(7998)-4*X(10176), 5*X(10574)-2*X(15071), 5*X(11444)-8*X(20117), 10*X(15016)-13*X(15028)

= on lines: {8, 29958}, {392, 23155}, {511, 7985}, {513, 17579}, {758, 3060}, {1464, 19245}, {2392, 2979}, {2771, 5890}, {2779, 15305}, {2810, 3241}, {2836, 11188}, {2842, 5640}, {3616, 23154}, {3877, 8679}, {4511, 26892}, {5693, 5889}, {5694, 11412}, {5752, 11684}, {5883, 11451}, {5884, 15043}, {5885, 15024}, {6126, 10546}, {7998, 10176}, {10574, 15071}, {11346, 24482}, {11444, 20117}, {15016, 15028}

= reflection of X(i) in X(j) for these (i,j): (2979, 5692), (5902, 15049), (23155, 392)

= {X(5902), X(15049)}-harmonic conjugate of X(5640)

= [ 1.0374458669256180, -0.6622607913206142, 3.6203315527022110 ]

 

O*(X(13)) = REFLECTION OF X(13) IN X(11624)

= (SB+SC)*(2*S^2+3*sqrt(3)*R^2*S+(9*R^2-2*SW)*SA) : : (barys)

= 3*X(5640)-X(16259)

= on lines: {3, 6}, {4, 11581}, {13, 5663}, {14, 5640}, {17, 11459}, {18, 13363}, {531, 25165}, {1154, 16962}, {3411, 12006}, {3412, 5889}, {5946, 11626}, {6104, 14170}, {6780, 16637}, {7998, 16241}, {8929, 15441}, {10654, 11002}, {11455, 12816}, {13754, 16267}, {15045, 16963}, {15072, 16965}, {16261, 16808}

= reflection of X(13) in X(11624)

= [ 1.8836111656604600, 2.1671587645755060, 1.2709647992042020 ]

 

O*(X(14)) = REFLECTION OF X(14) IN X(11626)

= (SB+SC)*(2*S^2-3*sqrt(3)*R^2*S+(9*R^2-2*SW)*SA) : : (barys)

= 3*X(5640)-X(16260)

= on lines: {3, 6}, {4, 11582}, {13, 5640}, {14, 5663}, {17, 13363}, {18, 11459}, {530, 25155}, {1154, 16963}, {3411, 5889}, {3412, 12006}, {5946, 11624}, {6105, 14169}, {6779, 16636}, {7998, 16242}, {8930, 15442}, {10653, 11002}, {11455, 12817}, {13754, 16268}, {15045, 16962}, {15072, 16964}, {16261, 16809}

= reflection of X(14) in X(11626)

= [ -2.2908008624718520, -1.0094342334803110, 5.3967962723808360 ]

 

César Lozada