Let ABC be a triangle, A'B'C' the pedal triangle of O and P a point.
Denote:
O, H lie on the locus
Locus = L∞ ∪ {Gibert’s Q050, through ETC’s 2, 3, 4, 13, 14, 67, 1113, 1114, 11058}
Orthologic centers Z = A->A” and Z”=A”->A:
· For P = X(2): Z = X(4); Z”=X(3)
· For P=X(3):
Z = isogonal conjugate of X(1658)
= (3*SB^2-10*R^2*SB+3*S^2-4*SC* SA)*(3*SC^2-10*R^2*SC+3*S^2-4* SA*SB) : : (barys)
= on the Jerabek hyperbola and lines: {3, 12278}, {6, 7547}, {54, 7577}, {68, 3153}, {265, 5889}, {382, 11559}, {3431, 6143}, {3521, 6241}, {4846, 11457}, {6288, 11464}, {11270, 13619}, {11744, 12290}, {12161, 15002}, {13622, 15073}
= isogonal conjugate of X(1658)
= [ 1.1859314135369550, 1.8986314682717730, 1.7788743514714800 ]
Z” = X(11750)
· For P=H
Z = X(11270); Z”=X(382)
· For P=X(13)
Z = X(18)
Z” = X(3)X(13) ∩ X(4)X(542)
= (3*SA-2*SW)*S^2-SB*SC*(SW+4* sqrt(3)*S) : : (barys)
= X(3)-3*X(13), 5*X(3)-3*X(5473), 2*X(3)-3*X(6771), X(3)+3*X(13103), 5*X(13)-X(5473), 2*X(140)-3*X(5459), 2*X(546)-3*X(5478), 3*X(616)-7*X(3090), 3*X(618)-4*X(3628), 5*X(632)-6*X(6669), 5*X(1656)-3*X(5463), 5*X(3091)-3*X(5617), 2*X(5473)-5*X(6771), X(5473)+5*X(13103), X(6771)+2*X(13103)
= on lines: {3, 13}, {4, 542}, {5, 530}, {18, 9115}, {61, 5472}, {62, 115}, {140, 5459}, {182, 5335}, {397, 575}, {511, 5318}, {546, 5478}, {616, 3090}, {618, 3628}, {632, 6669}, {1656, 5463}, {3091, 5617}, {3146, 6770}, {3303, 10062}, {3304, 10078}, {3746, 13076}, {5097, 5321}, {5334, 15520}, {5339, 11482}, {6321, 6778}, {7982, 9901}, {10594, 12142}, {11306, 12155}, {11542, 13350}, {11705, 15178}
= midpoint of X(i) and X(j) for these {i,j}: {13, 13103}, {6321, 6778}
= reflection of X(i) in X(j) for these (i,j): (6771, 13), (13350, 11542)
= [ -3.2772637193912510, -2.8332708757796890, 7.1147429587047410 ]
· For P=X(14)
Z = X(17)
Z” = X(3)X(14) ∩ X(4)X(542)
= (3*SA-2*SW)*S^2-SB*SC*(SW-4* sqrt(3)*S) : : (barys)
= X(3)-3*X(14), 5*X(3)-3*X(5474), 2*X(3)-3*X(6774), X(3)+3*X(13102), 5*X(14)-X(5474), 2*X(140)-3*X(5460), 2*X(546)-3*X(5479), 3*X(617)-7*X(3090), 3*X(619)-4*X(3628), 5*X(632)-6*X(6670), 5*X(1656)-3*X(5464), 5*X(3091)-3*X(5613), 2*X(5474)-5*X(6774), X(5474)+5*X(13102), X(6774)+2*X(13102)
= on lines: {3, 14}, {4, 542}, {5, 531}, {17, 9117}, {61, 115}, {62, 5471}, {140, 5460}, {182, 5334}, {398, 575}, {511, 5321}, {546, 5479}, {617, 3090}, {619, 3628}, {632, 6670}, {1656, 5464}, {3091, 5613}, {3146, 6773}, {3303, 10061}, {3304, 10077}, {3746, 13075}, {5097, 5318}, {5335, 15520}, {5340, 11482}, {6321, 6777}, {7982, 9900}, {10594, 12141}, {11305, 12154}, {11543, 13349}, {11706, 15178}
= midpoint of X(i) and X(j) for these {i,j}: {14, 13102}, {6321, 6777}
= reflection of X(i) in X(j) for these (i,j): (6774, 14), (13349, 11543)
= [ 3.1689662109219680, 7.1016758898304890, -2.7384801545545570 ]
· For P=X(67)
Z = X(54)
Z” = complement of X(14094)
= 2*(b^2+c^2)*a^8-(7*b^4-6*b^2* c^2+7*c^4)*a^6+(b^2+c^2)*(9*b^ 4-16*b^2*c^2+9*c^4)*a^4-(b^2- c^2)^2*(5*b^4+8*b^2*c^2+5*c^4) *a^2+(b^4-c^4)*(b^2-c^2)^3 : : (barys)
= (-18*R^2+3*SA+2*SW)*S^2+(18*R^ 2-SW)*SB*SC : : (barys)
= X(4)-3*X(9140), 5*X(4)-7*X(15044), 4*X(5)-3*X(113), 2*X(5)-3*X(125), X(5)-3*X(10264), 11*X(5)-12*X(15088), 3*X(67)-X(15069), X(113)-4*X(10264), 3*X(113)-2*X(15063), 11*X(113)-16*X(15088), 3*X(125)-X(15063), 11*X(125)-8*X(15088), 15*X(9140)-7*X(15044), 3*X(9140)+X(15054), 6*X(10264)-X(15063), 11*X(10264)-4*X(15088), X(14981)-3*X(15357), 7*X(15044)+5*X(15054)
= on the Steiner circle and these lines: {2, 14094}, {3, 67}, {4, 541}, {5, 113}, {20, 68}, {30, 6070}, {52, 2781}, {64, 265}, {110, 631}, {140, 5609}, {146, 3832}, {155, 15106}, {184, 15132}, {343, 14855}, {376, 15021}, {381, 15027}, {399, 3526}, {427, 13148}, {539, 2071}, {548, 12041}, {549, 11693}, {569, 5622}, {575, 15303}, {858, 13754}, {1112, 1907}, {1205, 10625}, {1209, 12827}, {1503, 8262}, {1511, 3530}, {1539, 3861}, {1656, 5655}, {1906, 12133}, {1986, 11806}, {2771, 12665}, {2931, 9715}, {2948, 9588}, {3028, 15888}, {3031, 9568}, {3043, 9706}, {3047, 9705}, {3091, 10706}, {3292, 15122}, {3520, 10116}, {3523, 9143}, {3524, 15020}, {3528, 12383}, {3545, 15025}, {3564, 10564}, {3580, 14915}, {3843, 7687}, {3853, 10113}, {3855, 15081}, {4301, 13605}, {4309, 10065}, {4317, 10081}, {4330, 12896}, {5067, 12900}, {5070, 6053}, {5071, 15029}, {5169, 5890}, {5449, 6241}, {5462, 12824}, {5504, 9936}, {5576, 13382}, {5734, 7978}, {5881, 13211}, {6000, 11799}, {6102, 14448}, {7722, 15100}, {8550, 12506}, {9589, 9904}, {9624, 11723}, {9643, 12888}, {9644, 10118}, {9656, 12373}, {9657, 12903}, {9670, 12904}, {9671, 12374}, {9680, 10819}, {9693, 10817}, {9714, 10117}, {9970, 15118}, {10111, 13293}, {10574, 14789}, {10575, 12359}, {10733, 12244}, {11579, 13352}, {11623, 11656}, {11694, 12108}, {12105, 15361}, {12121, 15041}, {12163, 15133}, {12219, 12284}, {12236, 13417}, {12301, 12302}, {13336, 15462}, {14683, 15035}, {15023, 15698}, {15039, 15720}
= midpoint of X(i) and X(j) for these {i,j}: {4, 15054}, {74, 3448}, {110, 12317}, {265, 10620}, {7722, 15100}, {10733, 12244}, {12163, 15133}, {12219, 12284}
= reflection of X(i) in X(j) for these (i,j): (110, 6699), (113, 125), (125, 10264), (155, 15115), (399, 5972), (1539, 11801), (1986, 11806), (3292, 15122), (5609, 140), (6053, 6723), (7728, 7687), (9970, 15118), (11562, 974), (12162, 15738), (12295, 265), (12308, 6053), (13202, 10113), (13417, 12236), (14448, 6102), (15063, 5)
= complement of X(14094)
= antipode of X(15063) in the Steiner circle
= X(15054) of Euler triangle
= X(15063) of Johnson triangle
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (5, 15063, 113), (125, 15063, 5), (140, 5609, 5642), (399, 15061, 5972), (631, 15057, 6699), (3523, 9143, 15034), (6053, 6723, 14643), (7689, 11457, 11750), (9140, 15054, 4), (12308, 14643, 6053)
= [ 9.3650716217768900, 9.8754285077324120, -7.5185113873427780 ]
César Lozada
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου