VARIATIONS (of Hyacinthos 28502)
Let ABC be a triangle. P a point and A'B'C' :1. the pedal triangle of O,
or
2. the pedal triangle of P.
Denote:
(Na), (Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
D = the Poncelet point of ABCP
DA' intersects again (Nb), (Nc) at Ab, Ac, resp.
DB' intersects again (Nc), (Na) at Bc, Ba, resp.
DC' intersects again (Na), (Nb) at Ca, Cb resp.
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
1. Ma, Mb, Mc and D are concyclic.
Centers (for the two cases of A'B'C')?
2.The perpendicular bisectors of AbAc, BcBa, CaCb, are concurrent at the antipode of D on the circle (MaMbMcD)
Points (for the two cases of A'B'C')?
[César Lozada]:
1)
For A’B’C’ = medial triangle of ABC = pedal(O)
If P=x:y:z (barys) then the center of the circle is:
O*(P) = x^2*(SB*c^2*y^2+SC*b^2*z^2)-(5*S^2+SB*SC)*x^2*y*z-x*y*z*(S^2+SB*SC)*(y+z)-a^4*y^2*z^2 : :
and the given perpendicular bisectors concur at:
D*(P) = SB*c^6*x^4*y^4+(-4*S^2+(SB-SC)*SA)*c^4*x^4*y^3*z+(6*S^4-(10*SA-SB-SC)*SA*S^2+(SB+SC)*SA^3)*x^4*y^2*z^2+(-4*S^2-(SB-SC)*SA)*b^4*x^4*y*z^3+SC*b^6*x^4*z^4+(-S^2+(2*SB+SC)*SB)*c^4*x^3*y^4*z+(6*S^4+(-5*SA^2+SA*SB+2*SA*SC-8*SB^2)*S^2+(2*SA^2*SB+2*SA^2*SC+SA*SB^2+3*SB^3)*SA)*x^3*y^3*z^2+(6*S^4+(-5*SA^2+2*SA*SB+SA*SC-8*SC^2)*S^2+(2*SA^2*SB+2*SA^2*SC+SA*SC^2+3*SC^3)*SA)*x^3*y^2*z^3+(-S^2+(SB+2*SC)*SC)*b^4*x^3*y*z^4+SA*a^4*c^2*x^2*y^4*z^2-(4*S^2-(2*SA+3*SB+3*SC)*SA)*a^4*x^2*y^3*z^3+SA*a^4*b^2*x^2*y^2*z^4-(-S^2+(2*SB+SC)*SB)*a^4*x*y^4*z^3-(-S^2+(SB+2*SC)*SC)*a^4*x*y^3*z^4-a^8*y^4*z^4 : :
which is the antipode of Q in the cabove circle.
ETC pairs (P,O*(P)): (1,5901), (2,5), (4,5), (13,24015), (14,24016), (74,20417), (98,11623), (104,20418), (110,16534), (671,11623), (3414,542), (5627,22104)
For P on the circumcircle of ABC, PO* = PH/4
ETC pairs (P,D*(P)): (1,19907), (2,114), (671,98)
For P on the circumcircle of ABC, D*(P) = P
Some others:
O*(X(6)) = MIDPOINT OF X(5) AND X(575)
= 2*a^6-4*(b^2+c^2)*a^4+(b^4-8*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2) : : (barys)
= 3*X(2)+X(576), X(3)+3*X(5476), X(3)-3*X(10168), X(4)+3*X(182), X(4)+15*X(3618), X(4)-9*X(14561), 13*X(4)+3*X(14927), X(4)-3*X(19130), X(182)-5*X(3618), X(182)+3*X(14561), 13*X(182)-X(14927), 5*X(3618)+3*X(14561), 5*X(3618)+X(19130), 3*X(14561)-X(19130), X(14927)+13*X(19130)
= on lines: {2, 576}, {3, 5476}, {4, 83}, {5, 542}, {6, 17}, {30, 20190}, {51, 7495}, {61, 9117}, {62, 9115}, {69, 15520}, {110, 7605}, {114, 3329}, {115, 5038}, {125, 15018}, {140, 143}, {141, 5097}, {262, 7875}, {373, 5972}, {382, 10541}, {384, 10992}, {395, 14136}, {396, 14137}, {468, 5943}, {524, 3628}, {546, 11645}, {550, 5092}, {599, 5070}, {631, 20423}, {1350, 15720}, {1352, 5056}, {1428, 5270}, {1503, 3850}, {1657, 5085}, {1992, 5067}, {2030, 7745}, {2330, 4857}, {2781, 12006}, {3090, 7856}, {3091, 11179}, {3095, 7889}, {3098, 3523}, {3398, 10991}, {3522, 17508}, {3526, 11477}, {3549, 11511}, {3564, 6329}, {3642, 22683}, {3643, 22685}, {3763, 5093}, {3815, 6721}, {3818, 3851}, {4045, 10796}, {4663, 11230}, {5012, 7533}, {5055, 15069}, {5068, 6776}, {5073, 12017}, {5094, 10601}, {5182, 16044}, {5355, 7697}, {5422, 21243}, {5486, 9813}, {5642, 16042}, {5892, 15122}, {6034, 7765}, {6036, 7792}, {6639, 8538}, {6688, 23292}, {6689, 15026}, {6759, 23327}, {7499, 21849}, {7505, 8541}, {7527, 10990}, {7570, 11225}, {7607, 7806}, {7749, 11261}, {7797, 12177}, {7804, 23698}, {7831, 22521}, {7907, 22486}, {8252, 9974}, {8253, 9975}, {8537, 14940}, {8584, 15699}, {8681, 9820}, {9019, 10095}, {9730, 20417}, {9976, 14643}, {10110, 16618}, {10112, 14788}, {10249, 22802}, {11180, 15022}, {11255, 16511}, {11303, 16002}, {11304, 16001}, {11451, 15073}, {12007, 18358}, {13353, 13419}, {13364, 25338}, {13434, 15462}, {14791, 19136}, {14810, 15712}, {14864, 23300}, {15000, 18114}, {15037, 19140}, {15074, 16776}, {15534, 15703}, {18381, 19153}
= midpoint of X(i) and X(j) for these {i,j}: {5, 575}, {6, 24206}, {141, 5097}, {182, 19130}, {3589, 18583}, {4045, 10796}, {5092, 5480}, {5476, 10168}, {6593, 20301}, {8550, 18553}, {12007, 18358}, {14810, 21850}
= reflection of X(15516) in X(6329)
= complement of the complement of X(576)
= complement of the complementary conjugate of X(15850)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 597, 575), (5, 7817, 20398), (5, 8550, 18553), (17, 18, 1506), (182, 14561, 19130), (373, 14389, 5972), (575, 18553, 8550), (3526, 14848, 11477), (3618, 14561, 182), (5070, 11482, 599), (6694, 6695, 6680), (14632, 14633, 140)
= [ 1.4965719182416210, 1.1363265588773730, 2.1632513634962180 ]
D*(X(6)) = REFLECTION OF X(125) IN O*(X(6))
= a^2*(2*a^10-6*(b^2+c^2)*a^8+4*(b^4+b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4-(6*b^8+6*c^8-(7*b^4-8*b^2*c^2+7*c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4-b^2*c^2+2*c^4)) : : (barys)
= (SB+SC)*((27*R^2-5*SW)*S^2-3*(3*R^2-SA)*SA*SW) : : (barys)
= 3*X(6)+X(399), 3*X(6)-X(9976), X(74)-3*X(182), X(74)+3*X(9970), X(113)+3*X(15303), X(146)+3*X(11179), X(265)-3*X(5476), X(399)-3*X(19140), 3*X(597)-X(10264), X(895)-3*X(15520), X(1352)+3*X(25321), X(1511)-3*X(6593), 3*X(1992)+5*X(20125), 2*X(7687)-3*X(19130), X(9976)+3*X(19140)
= lies on the cubic K802 and these lines: {3, 19379}, {5, 25329}, {6, 13}, {74, 182}, {110, 576}, {125, 15018}, {146, 11179}, {323, 5642}, {511, 1511}, {524, 10272}, {575, 5663}, {597, 10264}, {729, 6236}, {895, 14491}, {1351, 12584}, {1352, 25321}, {1495, 11649}, {1539, 11645}, {1974, 3043}, {1992, 20125}, {2781, 5092}, {2854, 5097}, {2914, 5095}, {2930, 5093}, {3098, 10298}, {4663, 11699}, {5191, 18114}, {5505, 9813}, {5609, 22330}, {5972, 15066}, {6699, 10168}, {6723, 15106}, {6759, 13248}, {9969, 11702}, {10296, 13202}, {10541, 15041}, {11061, 14561}, {11470, 15463}, {11597, 18374}, {12041, 20190}, {12367, 18449}, {12383, 20423}, {12900, 24206}, {13289, 19153}, {14094, 22234}, {15032, 15063}, {15037, 16003}, {16163, 19924}, {18583, 20301}, {19150, 22336}
= midpoint of X(i) and X(j) for these {i,j}: {5, 25329}, {6, 19140}, {110, 576}, {182, 9970}, {399, 9976}, {1351, 12584}, {3098, 10752}, {4663, 11699}, {6759, 13248}
= reflection of X(i) in X(j) for these (i,j): (125, O*(X(6))), (12041, 20190), (20301, 18583),
= X(575)-of-anti-orthocentroidal triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 399, 9976), (9976, 19140, 399), (10752, 15462, 3098)
= [ -0.4971278651989889, -0.8070852709047360, 4.4288594533948840 ]
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O*(X(7)) = MIDPOINT OF X(7) AND X(1001)
= 2*(b+c)*a^2-(b^2-4*b*c+c^2)*a-(b^2-c^2)*(b-c) : : (barys)
= X(1)+3*X(6173), 3*X(7)+5*X(3616), 3*X(7)+X(5698), 3*X(9)-7*X(3624), X(10)-3*X(142), 2*X(10)-3*X(3826), X(10)+3*X(5542), 5*X(10)-3*X(24393), 5*X(142)-X(24393), 3*X(1001)-5*X(3616), 3*X(1001)-X(5698), 5*X(3616)-X(5698), X(3826)+2*X(5542), 5*X(3826)-2*X(24393), 5*X(5542)+X(24393), X(5880)-3*X(6173)
= on lines: {1, 528}, {2, 3715}, {5, 2801}, {7, 21}, {8, 17297}, {9, 583}, {10, 141}, {11, 10129}, {12, 7705}, {37, 24231}, {48, 5829}, {57, 6690}, {75, 4966}, {145, 2550}, {226, 3660}, {238, 17365}, {244, 5718}, {320, 16823}, {329, 8167}, {335, 16593}, {354, 2886}, {392, 11551}, {442, 18398}, {496, 2486}, {499, 5729}, {516, 550}, {524, 16825}, {527, 1125}, {551, 5126}, {553, 4640}, {673, 20132}, {726, 17243}, {740, 7263}, {750, 17724}, {940, 17061}, {946, 15726}, {954, 8071}, {971, 9955}, {982, 17056}, {1329, 5439}, {1376, 3475}, {1386, 3664}, {1401, 18165}, {1445, 15296}, {1621, 11246}, {1757, 17337}, {1836, 4666}, {3008, 4663}, {3035, 3306}, {3058, 20292}, {3120, 17450}, {3243, 3632}, {3246, 4896}, {3296, 19843}, {3337, 7483}, {3416, 17298}, {3452, 3848}, {3474, 4428}, {3555, 9710}, {3576, 5735}, {3629, 4974}, {3634, 4407}, {3662, 4026}, {3663, 15569}, {3685, 7321}, {3696, 4684}, {3703, 17140}, {3720, 3782}, {3751, 17278}, {3756, 17717}, {3790, 17241}, {3813, 5045}, {3820, 3833}, {3824, 10916}, {3829, 3838}, {3871, 9782}, {3873, 3925}, {3914, 4883}, {3923, 7228}, {3932, 17234}, {3962, 24564}, {3976, 4443}, {4023, 24589}, {4133, 4726}, {4310, 4648}, {4355, 5436}, {4421, 10578}, {4423, 5905}, {4444, 4806}, {4644, 16020}, {4649, 17366}, {4654, 10582}, {4655, 7238}, {4699, 4733}, {4719, 24171}, {4743, 5853}, {4888, 7290}, {4995, 9352}, {5219, 6667}, {5223, 20195}, {5248, 24470}, {5284, 17483}, {5450, 5901}, {5550, 6172}, {5572, 16193}, {5696, 24390}, {5708, 10198}, {5794, 11518}, {5805, 18481}, {5847, 17376}, {5850, 6666}, {5851, 5886}, {5855, 11529}, {5904, 17529}, {6006, 23814}, {6361, 11495}, {6594, 17564}, {6691, 8257}, {6701, 20116}, {7277, 16468}, {7288, 12848}, {7354, 18450}, {7680, 10202}, {7958, 12528}, {7965, 11220}, {7988, 13257}, {8256, 15888}, {8301, 14828}, {8545, 11375}, {8581, 15844}, {10177, 12047}, {10580, 11235}, {10785, 16112}, {10861, 11025}, {10944, 14151}, {11036, 12635}, {11037, 12513}, {11260, 12577}, {11269, 17070}, {15298, 17437}, {17000, 20072}, {17307, 19877}
= midpoint of X(i) and X(j) for these {i,j}: {1, 5880}, {7, 1001}, {142, 5542}
= reflection of X(i) in X(j) for these (i,j): (3826, 142), (15254, 1125), (15481, 6666)
= complement of X(5220)
= X(5480)-of-Fuhrmann triangle
= X(5880)-of-anti-Aquila triangle
= X(8550)-of-Wasat triangle
= X(9970)-of-K798i triangle
= X(15069)-of-3rd Euler triangle
= X(15581)-of-2nd Zaniah triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6173, 5880), (7, 3616, 5698), (226, 3742, 3816), (354, 5249, 2886), (3306, 17718, 3035), (3475, 9776, 1376), (3616, 5698, 1001), (3812, 21620, 12607), (3838, 11019, 3829), (4654, 10582, 24703), (4684, 24199, 3696), (5045, 12609, 3813), (5439, 13407, 1329), (17140, 18139, 3703), (17234, 24349, 3932)
= [ 1.6876507916748680, 1.3775837170889340, 1.9080369277651780 ]
D*(X(7)) = REFLECTION OF X(11) IN O*(X(7))
= 4*(b+c)*a^5-(7*b^2-2*b*c+7*c^2)*a^4-2*(b^2-c^2)*(b-c)*a^3+2*(4*b^4+4*c^4-(5*b^2-4*b*c+5*c^2)*b*c)*a^2-2*(b^3-c^3)*(b^2-c^2)*a-(b^2-c^2)^2*(b-c)^2 : : (barys)
= 3*X(7)+X(6224), X(80)-3*X(6173), 3*X(142)-2*X(6702), X(1145)-3*X(10427), 3*X(2550)-X(12531), 3*X(5542)-X(21630)
= on lines: {7, 528}, {11, 10129}, {80, 6173}, {104, 1001}, {116, 119}, {214, 527}, {518, 1145}, {908, 12831}, {952, 5880}, {971, 12611}, {1537, 15726}, {2550, 12531}, {3035, 5220}, {3218, 6174}, {4996, 5852}, {5542, 21630}, {5735, 12119}, {5770, 20400}, {5784, 17660}, {5850, 6594}, {17768, 18450}
= midpoint of X(i) and X(j) for these {i,j}: {5735, 12119}, {5784, 17660}
= reflection of X(5220) in X(3035)
= [ 0.7321664621045320, -0.6175831111610631, 3.7302990378937930 ]
-----------------------------------
D*(X(13)) = REFLECTION OF X(115) IN X(24015)
= 4*S^4+(SA-2*SW)*SW*S^2-sqrt(3)*(3*SW*SB*SC+(3*SA-2*SW)*S^2)*S+SB*SC*SW^2 : : (barys)
= 3*X(13)-X(6321), X(147)-3*X(5613), 3*X(5464)-X(13188), 3*X(5470)-X(13102), X(5611)-3*X(16529), X(10723)-3*X(25154), X(16002)-4*X(20415), 2*X(22505)-3*X(22797)
= on lines: {2, 98}, {3, 6778}, {5, 20378}, {13, 5611}, {61, 115}, {99, 628}, {303, 5983}, {511, 6783}, {575, 6114}, {619, 13349}, {1569, 3107}, {5464, 13188}, {5470, 13102}, {5474, 13103}, {5479, 20252}, {5965, 5982}, {6777, 20416}, {10723, 25154}, {14137, 15516}, {14185, 16770}, {16001, 22513}, {16530, 22848}, {22505, 22797}, {22686, 23024}
= midpoint of X(i) and X(j) for these {i,j}: {3, 6778}, {98, 22509}, {5474, 13103}, {5613, 6770}
= reflection of X(i) in X(j) for these (i,j): (115, 20415), (5479, 20252), (6774, 6771), (6777, 20416), (16002, 115)
= complement of X(22507)
= [ 1.2804107128960740, -0.3261052626423511, 3.2754708731693580 ]
D*(X(14)) = REFLECTION OF X(115) IN X(24016)
= 4*S^4+(SA-2*SW)*SW*S^2+sqrt(3)*(3*SW*SB*SC+(3*SA-2*SW)*S^2)*S+SB*SC*SW^2 : : (barys)
= 3*X(14)-X(6321), 3*X(5463)-X(13188), 3*X(5469)-X(13103), X(5615)-3*X(16530), X(10723)-3*X(25164), 2*X(22505)-3*X(22796)
= on lines: {2, 98}, {3, 6777}, {5, 20377}, {14, 5615}, {62, 115}, {99, 627}, {302, 5982}, {511, 6782}, {575, 6115}, {618, 13350}, {1569, 3106}, {5463, 13188}, {5469, 13103}, {5473, 13102}, {5478, 20253}, {5965, 5983}, {6778, 20415}, {10723, 25164}, {14136, 15516}, {14187, 16771}, {16002, 22512}, {16529, 22892}, {22505, 22796}, {22684, 23018}
= midpoint of X(i) and X(j) for these {i,j}: {3, 6777}, {98, 22507}, {5473, 13102}, {5617, 6773}
= reflection of X(i) in X(j) for these (i,j): (115, 20416), (5478, 20253), (6771, 6774), (6778, 20415), (16001, 115)
= complement of X(22509)
= [ 7.7266406432092800, 9.6088415029678040, -6.5777522400899220 ]
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César Lozada
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