A*, B*, C* = the orthogonal projections of A", B", C" on IA', IB', IC', resp.
Na, Nb, Nc = the NPC centers of A*IA", B*IB", C*IC", resp.
ABC, NaNbNc are orthologic.
Orthologic centers?
Let P, Pa, Pb, Pc be same points on the Euler lines of ABC, A*IA", B*IB", C*IC", resp.
ABC, PaPbPc are orthologic.
The locus of the orthologic center (PaPbPc, ABC) as P moves on the Euler line is the OI line.
Which is the locus of other orthologic center (ABC, PaPbPc) ?
Hi Antreas,
Please check. With your configuration A*, I, A” are collinear.
Anyway, I have checked your conjecture for some similar constructions using other triangles derived from I. Assume P is on the Euler line such that OP=t*OH and Pa =P-of-A*IA” and similarly Pb and Pc, where A*= orthogonal-projection of A” in IA’ and cyclically B*, C*.
- A’B’C’ = anticevian-triangle-of-I and A”B”C” = pedal-triangle-of-I, or
A’B’C’ = cevian-triangle-of I and A”B”C” = pedal-triangle-of-I, or
A’B’C’ = circumcevian-triangle-of-I and A”B”C” = pedal-triangle-of-I, or
A’B’C’ = circumcevian-triangle-of-I and A”B”C” = reflection-of-I in sidelines of ABC
Orthologic centers:
Za = ABC->PaPbPc = 1/(a*(-(b^2+c^2-a^2)*t-b*c)) : : (trilinears)
= on Feuerbach hyperbola
Zp=PaPbPc->ABC = ((2*a^4-(b^2+c^2)*a^2-(b^2-c^ 2)^2)*t-4*a^2*b*c)/a : : (trilinears)
= on line {1, 79} = {1, 30}
ETC-pairs (P, Za): (2,5557), (3,1), (4,79), (5,7), (20,80), (26,1063), (140,3296), (376,5559), (382,5561), (548,1000), (549,5558), (550,8), (1657,5560), (3149,7284), (3522,13606), (3528,13602), (3560,5665), (3627,5556), (3628,5551), (6911,7091), (6924,1476), (6985,84), (7387,1041), (7580,90), (8703,7320), (12084,1061), (12085,1039)
ETC-pairs (P, Zp): (3,1), (381,3058), (382,6284), (1657,7354), (2070,10149), (3534,5434), (7387,8144), (13743,10543)
Some others:
Zp( X(2) ) = X(1)X(30) ∩ X(2)X(496)
= 2*a^4-(b^2+12*b*c+c^2)*a^2-(b^ 2-c^2)^2 : : (barys)
= 3*X(1)-X(5434) = 5*X(1)+X(6284) = 7*X(1)-X(7354) = 13*X(1)-X(10483) = 3*X(3058)+X(5434) = 5*X(3058)-X(6284) = 7*X(3058)+X(7354) = 13*X(3058)+X(10483) = 2*X(5045)+X(10624) = 5*X(5434)+3*X(6284) = 7*X(5434)-3*X(7354) = 13*X(5434)-3*X(10483) = 7*X(6284)+5*X(7354) = 13*X(6284)+5*X(10483) = 13*X(7354)-7*X(10483)
= On lines: {1, 30}, {2, 496}, {3, 10385}, {5, 3303}, {11, 547}, {12, 5066}, {35, 5298}, {55, 549}, {56, 8703}, {140, 3582}, {142, 214}, {149, 6175}, {329, 3241}, {376, 390}, {381, 495}, {388, 3830}, {428, 6198}, {499, 11539}, {516, 5049}, {519, 960}, {546, 4857}, {548, 5563}, {550, 3304}, {553, 5045}, {942, 12575}, {952, 5919}, {1056, 3543}, {1062, 10691}, {1478, 8162}, {1479, 3845}, {1500, 9300}, {1697, 3654}, {2241, 5306}, {3057, 12433}, {3085, 5055}, {3086, 5054}, {3524, 14986}, {3534, 4294}, {3545, 9669}, {3583, 14893}, {3585, 12101}, {3600, 11001}, {3601, 3653}, {3614, 14892}, {3627, 9670}, {3679, 4863}, {3748, 4870}, {3813, 6675}, {3829, 10197}, {3839, 9654}, {3853, 5270}, {3858, 9671}, {4030, 4975}, {4330, 12103}, {4366, 6661}, {5010, 14891}, {5071, 5274}, {5225, 14269}, {5433, 11812}, {5603, 8236}, {5886, 10389}, {7741, 10109}, {7743, 13405}, {7951, 11737}, {7956, 10596}, {10054, 13183}, {10086, 12351}
= midpoint of X(i) and X(j) for these {i,j}: {1, 3058}, {553, 10624}, {3241, 11113}
= reflection of X(553) in X(5045)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1, 12701, 6147), (11, 3584, 547), (35, 5298, 12100), (55, 10072, 549), (497, 6767, 495), (1058, 3295, 496), (1479, 11237, 3845), (3303, 11238, 10056), (3304, 4309, 550), (3582, 3746, 4995), (3582, 4995, 140), (10056, 11238, 5)
= [ 2.163584190563316, 2.16233567736650, 1.145085540393732 ]
Zp( X(4) ) = X(1)X(30) ∩ X(3)X(496)
= 2*a^4-(b^2+4*b*c+c^2)*a^2-(b^ 2-c^2)^2 : : (barys)
= X(1)-3*X(3058) = 5*X(1)-3*X(5434) = 3*X(1)-X(7354) = 5*X(1)-X(10483) = X(8)-3*X(11113) = 5*X(3058)-X(5434) = 3*X(3058)+X(6284) = 9*X(3058)-X(7354) = 15*X(3058)-X(10483) = 3*X(5434)+5*X(6284) = 9*X(5434)-5*X(7354) = 3*X(5434)-X(10483) = 3*X(6284)+X(7354) = 5*X(6284)+X(10483) = 5*X(7354)-3*X(10483)
= On lines: {1, 30}, {2, 9669}, {3, 496}, {4, 390}, {5, 55}, {8, 11113}, {10, 528}, {11, 35}, {12, 546}, {20, 999}, {21, 149}, {26, 10833}, {33, 6756}, {34, 13488}, {36, 548}, {40, 5722}, {56, 550}, {65, 12433}, {80, 7161}, {100, 4187}, {145, 11114}, {192, 7762}, {221, 5878}, {350, 7767}, {354, 1770}, {355, 1697}, {376, 14986}, {381, 3085}, {382, 388}, {397, 7005}, {398, 7006}, {405, 3434}, {428, 3920}, {442, 1621}, {452, 5082}, {499, 549}, {515, 9856}, {516, 942}, {517, 950}, {519, 4127}, {529, 3244}, {535, 3635}, {547, 4995}, {595, 1834}, {614, 10691}, {631, 5274}, {938, 6361}, {943, 8226}, {946, 4314}, {952, 1898}, {956, 6872}, {962, 3488}, {993, 3813}, {1001, 8728}, {1012, 12116}, {1056, 3146}, {1062, 4319}, {1089, 4030}, {1125, 7743}, {1210, 3579}, {1250, 11543}, {1329, 8715}, {1352, 10387}, {1385, 1387}, {1478, 3303}, {1482, 3486}, {1483, 2098}, {1484, 10058}, {1500, 7745}, {1532, 11491}, {1595, 11393}, {1596, 11398}, {1656, 5218}, {1657, 4293}, {1699, 6253}, {1837, 5119}, {1870, 1885}, {1914, 5305}, {2066, 7583}, {2067, 9660}, {2192, 9833}, {2241, 5254}, {2478, 3820}, {2550, 11108}, {2646, 5901}, {2829, 4342}, {2886, 5248}, {2901, 5846}, {3035, 3825}, {3052, 5292}, {3056, 3564}, {3090, 5281}, {3149, 7956}, {3189, 3940}, {3270, 6146}, {3297, 6560}, {3298, 6561}, {3304, 4299}, {3419, 5250}, {3474, 5708}, {3487, 9812}, {3526, 10589}, {3528, 5265}, {3529, 3600}, {3530, 5010}, {3575, 6198}, {3576, 11373}, {3582, 12100}, {3584, 3614}, {3585, 3853}, {3601, 5886}, {3612, 11376}, {3616, 11112}, {3628, 5432}, {3685, 3695}, {3703, 4894}, {3748, 13407}, {3815, 9665}, {3830, 5229}, {3832, 8164}, {3843, 10590}, {3845, 10056}, {3850, 7951}, {3851, 10588}, {3871, 5046}, {3883, 5295}, {3886, 5814}, {3925, 5259}, {3927, 5698}, {4114, 15007}, {4205, 5263}, {4292, 5045}, {4298, 5049}, {4305, 10246}, {4313, 5603}, {4324, 5563}, {4326, 5805}, {4354, 9630}, {4366, 6656}, {4428, 10198}, {4512, 5791}, {4514, 7283}, {4640, 10916}, {5084, 9709}, {5086, 12690}, {5122, 12512}, {5204, 8703}, {5268, 10128}, {5272, 7734}, {5310, 6676}, {5414, 7584}, {5533, 14792}, {5534, 10388}, {5559, 9897}, {5697, 5844}, {5714, 10578}, {5719, 12047}, {5762, 14100}, {5787, 12705}, {5853, 12572}, {5874, 10928}, {5875, 10927}, {5881, 9819}, {6244, 6865}, {6644, 10046}, {6796, 7681}, {6827, 10306}, {6842, 10738}, {6863, 11928}, {6882, 11849}, {6907, 10267}, {6914, 10943}, {6922, 11248}, {6928, 10679}, {6934, 10596}, {6938, 10806}, {6940, 13199}, {7159, 12896}, {7191, 7667}, {7483, 11680}, {7502, 9672}, {7526, 10831}, {7530, 10037}, {7982, 11827}, {8162, 9657}, {8727, 11496}, {9538, 12225}, {9589, 11529}, {9598, 15048}, {9612, 10389}, {9629, 11819}, {9848, 14110}, {9955, 13411}, {10053, 13183}, {10065, 10264}, {10081, 14677}, {10086, 12185}, {10087, 11698}, {10088, 12374}, {10609, 11015}, {10638, 11542}, {10942, 10953}, {10948, 14793}, {11277, 14799}, {11446, 14516}, {11517, 14022}, {12736, 13145}, {12955, 13311}, {13116, 13297}, {13901, 13925}, {13958, 13993}
= midpoint of X(i) and X(j) for these {i,j}: {1, 6284}, {950, 10624}, {1482, 7491}, {3057, 10572}, {5697, 10950}, {6238, 12428}, {7982, 11827}, {12743, 12758}
= reflection of X(i) in X(j) for these (i,j): (65, 12433), (4292, 5045), (9957, 12575)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1, 1836, 6147), (1, 9580, 12699), (1, 10483, 5434), (2, 9669, 10593), (3, 497, 496), (4, 390, 3295), (4, 3295, 495), (5, 10386, 55), (55, 4309, 10386), (390, 9668, 495), (497, 4294, 3), (498, 1479, 10896), (498, 10896, 5), (1836, 6147, 11544), (3058, 6284, 1), (3295, 9668, 4)
= [ 3.110135552775880, 3.10639001318544, 0.054639602267130 ]
Zp( X(5) ) = X(1)X(30) ∩ X(5)X(497)
= 2*a^4-(b^2+8*b*c+c^2)*a^2-(b^ 2-c^2)^2 : : (barys)
= X(1)+3*X(3058) = 7*X(1)-3*X(5434) = 3*X(1)+X(6284) = 5*X(1)-X(7354) = 9*X(1)-X(10483) = X(145)+3*X(11113) = 7*X(3058)+X(5434) = 9*X(3058)-X(6284) = 15*X(3058)+X(7354) = 9*X(5434)+7*X(6284) = 15*X(5434)-7*X(7354) = 27*X(5434)-7*X(10483) = 5*X(6284)+3*X(7354) = 3*X(6284)+X(10483) = 9*X(7354)-5*X(10483)
= On lines: {1, 30}, {3, 390}, {4, 6767}, {5, 497}, {11, 3628}, {12, 3850}, {20, 7373}, {35, 3530}, {55, 140}, {56, 548}, {145, 11113}, {149, 442}, {382, 1056}, {388, 3627}, {495, 546}, {498, 547}, {516, 5045}, {517, 6738}, {519, 3988}, {528, 1125}, {529, 3635}, {549, 3086}, {550, 999}, {612, 10128}, {614, 7734}, {632, 5218}, {938, 12702}, {942, 10624}, {946, 5719}, {950, 952}, {971, 15006}, {1000, 12645}, {1012, 10806}, {1385, 4314}, {1387, 2646}, {1478, 3853}, {1482, 3488}, {1483, 3486}, {1621, 6675}, {1656, 5274}, {1657, 3600}, {1697, 5690}, {1870, 13488}, {2241, 5305}, {2829, 13607}, {3149, 10596}, {3159, 9053}, {3304, 4302}, {3434, 8728}, {3487, 8236}, {3526, 5281}, {3579, 11019}, {3582, 11812}, {3583, 3861}, {3584, 7173}, {3585, 12102}, {3601, 11373}, {3614, 11737}, {3622, 11112}, {3623, 11114}, {3695, 4514}, {3748, 12047}, {3813, 5248}, {3816, 8715}, {3820, 3913}, {3843, 5261}, {3845, 5225}, {3851, 8164}, {3858, 10590}, {3859, 9671}, {3871, 4187}, {4292, 5049}, {4313, 10246}, {4366, 7819}, {4421, 10200}, {4995, 10124}, {5044, 5853}, {5066, 10056}, {5082, 11108}, {5217, 10072}, {5268, 13361}, {5298, 14891}, {5433, 12108}, {5708, 6361}, {5842, 13464}, {5843, 14100}, {5887, 9848}, {5901, 12053}, {5919, 10572}, {6198, 6756}, {6253, 11522}, {6283, 13081}, {6405, 13082}, {6922, 10679}, {6928, 12000}, {6987, 8158}, {7191, 10691}, {7330, 10384}, {7491, 10247}, {7525, 10832}, {7553, 9642}, {7743, 13411}, {7951, 12811}, {7956, 11500}, {8162, 12953}, {8727, 12116}, {9614, 10389}, {9955, 13405}, {10039, 12019}, {10046, 12106}, {10198, 11235}, {11237, 14893}, {11276, 14799}, {12915, 13369}
= midpoint of X(i) and X(j) for these {i,j}: {942, 10624}, {950, 9957}
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1, 12699, 6147), (3, 390, 10386), (55, 496, 140), (388, 9668, 3627), (390, 1058, 3), (495, 1479, 546), (497, 3085, 9669), (497, 3295, 5), (498, 10593, 547), (498, 11238, 10593), (999, 4294, 550), (1479, 3303, 495), (3085, 9669, 5), (3295, 9669, 3085), (3488, 9785, 1482)
= [ 2.400222031116457, 2.39834926132124, 0..872474055862082 ]
Za( X(21) ) = X(3)X(5424) ∩ X(4)X(5425)
= a*(a^3-(b+2*c)*a^2-(b^2+b*c+c^ 2)*a+(b^2-c^2)*(b-2*c))*(a^3-( 2*b+c)*a^2-(b^2+b*c+c^2)*a+(b^ 2-c^2)*(2*b-c)) : : (barys)
= On Feuerbach hyperbola and these lines: {3, 5424}, {4, 5425}, {7, 10483}, {8, 3822}, {21, 5902}, {79, 382}, {80, 10895}, {90, 11529}, {1000, 11009}, {1482, 13606}, {2098, 13602}, {2099, 5559}, {2320, 5563}, {2346, 5697}, {3296, 4317}, {3340, 7162}, {6596, 15015}, {7284, 11518}, {10308, 15071}
=[ 1.101002004563726, 1.15322344351537, 2.334124249675099 ]
Zp( X(21) ) = X(1)X(30) ∩ X(21)X(145)
= 6*a^4-4*(b+c)*a^3-(5*b^2+4*b* c+5*c^2)*a^2+4*(b^2-c^2)*(b-c) *a-(b^2-c^2)^2 : : (barys)
= 5*X(1)-X(79) = 3*X(1)-X(3649) = 3*X(1)+X(5441) = 4*X(1)-X(11544) = 2*X(10)-3*X(6675) = 3*X(21)+X(145) = 3*X(79)-5*X(3649) = 3*X(79)+5*X(5441) = X(79)+5*X(10543) = 4*X(79)-5*X(11544) = X(3649)+3*X(10543) = 4*X(3649)-3*X(11544) = X(5441)-3*X(10543) = 4*X(5441)+3*X(11544) = 4*X(10543)+X(11544)
= On lines: {1, 30}, {10, 6675}, {21, 145}, {35, 5427}, {55, 5428}, {442, 496}, {495, 3486}, {517, 10122}, {548, 5902}, {758, 3635}, {950, 9955}, {999, 3651}, {1058, 2475}, {1387, 3636}, {1483, 3303}, {2099, 10386}, {2646, 12433}, {2771, 12735}, {3241, 11684}, {3244, 3647}, {3530, 5442}, {3622, 6175}, {3623, 3650}, {3624, 5722}, {3632, 5426}, {3746, 5844}, {3812, 9945}, {4313, 6361}, {5126, 6744}, {5221, 8703}, {5558, 5731}, {5703, 7319}, {5719, 10572}, {6767, 13743}, {8148, 10385}, {10021, 10950}, {11246, 12103}, {12019, 13411}
= midpoint of X(i) and X(j) for these {i,j}: {1, 10543}, {3244, 3647}, {3649, 5441}
= reflection of X(6701) in X(3636)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1, 5441, 3649), (3636, 6701, 11281), (3649, 10543, 5441)
= [ 2.101157060326708, 2.10007323147572, 1.217002832273475 ]
- A’B’C’ = pedal-triangle-of-I and A”B”C” = anticevian-triangle-of-I
Orthologic centers:
Za = ABC->PaPbPc = 1/(((b+c)*a^2-b*c*a-(b^2-c^2)* (b-c))*t+b*c*a) : : (trilinears)
= on Feuerbach hyperbola
Zp=PaPbPc->ABC = ((b+c)*a^2-2*b*c*a-(b^2-c^2)*( b-c))*t+a*(-a^2+b^2+c^2) : : (trilinears)
= on line {1, 3} = IO
ETC-pairs (P, Za): (2,943), (3,1), (4,21), (20,104), (376,1476), (404,1000), (411,4), (474,7160), (631,2346), (1816,7049), (3149,9), (3651,7), (4192,987), (6905,8), (6906,2320), (6911,7162), (6924,5559), (6940,7320), (6942,1320), (6985,90), (7411,3296), (7549,2335), (7580,84), (11337,1061)
ETC-pairs (P, Zp): (2,3576), (3,3), (4,1), (5,1385), (20,40), (21,10902), (140,13624), (376,165), (381,10246), (382,1482), (411,11012), (443,8726), (550,3579), (631,7987), (1012,55), (1532,1319), (1657,12702), (3146,7982), (3149,56), (3529,7991), (3560,10267), (3830,10247), (5073,8148), (6831,2646), (6833,3612), (6844,13384), (6847,3601), (6848,1420), (6869,5709), (6881,13151), (6905,36), (6906,35), (6909,2077), (6911,10269), (6934,46), (6938,5119), (6942,7280), (6948,3359), (6950,5010), (6985,11249), (7411,7688), (7580,3428), (11414,8193), (13737,1622)
Some others:
Za( X(5) ) = X(4)X(35) ∩ X(7)X(36)
= a*(a^3-c*a^2-(b^2+b*c+c^2)*a-( b^2-c^2)*c)*(a^3-b*a^2-(b^2+b* c+c^2)*a+(b^2-c^2)*b) : : (barys)
= On Feuerbach hyperbola and lines: {1, 2361}, {2, 11604}, {3, 79}, {4, 35}, {7, 36}, {8, 3746}, {9, 584}, {21, 5692}, {46, 5665}, {55, 80}, {56, 5557}, {84, 3612}, {90, 3601}, {119, 4995}, {405, 6598}, {993, 2320}, {1001, 3254}, {1006, 5902}, {1156, 10058}, {1172, 2074}, {1320, 1621}, {1389, 5697}, {1478, 14799}, {1479, 6884}, {1896, 11107}, {2646, 5694}, {3065, 6326}, {3085, 14795}, {3295, 5559}, {3296, 5563}, {3303, 13606}, {3560, 5441}, {3576, 7284}, {3577, 5119}, {4189, 10266}, {4428, 12641}, {5010, 5219}, {5047, 15079}, {5426, 6596}, {5432, 8068}, {5444, 14793}, {5445, 11507}, {5553, 14803}, {5560, 5587}, {6767, 13602}, {6861, 7741}, {6875, 14804}, {6906, 10308}, {10526, 10902}, {11375, 14794}
= isogonal conjugate of X(5902)
= [ 2.540214860436932, 2.21444684780075, 0.935179036305279 ]
Za( X(22) ) = X(8)X(1062) ∩ X(56)X(1063)
= a*(a^6-(b+c)^2*a^4+2*b^2*c*a^ 3-(b^2+c^2)*(b-c)^2*a^2-2*(b^ 2-c^2)*b^2*c*a+(b^4-c^4)*(b^2- c^2))*(a^6-(b+c)^2*a^4+2*b*c^ 2*a^3-(b^2+c^2)*(b-c)^2*a^2+2* (b^2-c^2)*b*c^2*a+(b^4-c^4)*( b^2-c^2)) : : (barys)
= On Feuerbach hyperbola and lines: {8, 1062}, {56, 1063}, {1061, 9630}
= [ 6.732353133964532, 0.67804032356344, 0.064012042226061 ]
Zp( X(22) ) = X(1)X(3) ∩ X(10)X(24)
= a^2*(a^8-2*(b^2+c^2)*a^6+2*b^ 2*c^2*a^4-2*b^2*c^2*(b+c)*a^3+ 2*(b^2-b*c+c^2)^2*(b+c)^2*a^2+ 2*(b^2-c^2)*(b-c)*b^2*c^2*a-( b^4-c^4)^2) : : (barys)
= On lines: {1, 3}, {8, 7488}, {10, 24}, {22, 515}, {25, 5587}, {26, 355}, {186, 5657}, {378, 516}, {944, 7512}, {946, 7503}, {952, 7502}, {1125, 7509}, {1593, 9911}, {1631, 7420}, {1658, 5690}, {1698, 6642}, {1699, 9818}, {1995, 10175}, {2070, 5790}, {2071, 9778}, {2550, 7501}, {2915, 11500}, {2917, 12785}, {2931, 13211}, {2948, 12412}, {3167, 9621}, {3518, 5818}, {3520, 6361}, {3556, 5693}, {3586, 10833}, {3624, 7393}, {3679, 9590}, {4297, 10323}, {5687, 10913}, {5691, 7387}, {5731, 6636}, {5794, 9712}, {5881, 9626}, {5886, 7514}, {6326, 9912}, {6796, 11337}, {7395, 8227}, {7412, 11392}, {7485, 10165}, {7506, 9956}, {7526, 12699}, {7527, 9812}, {7529, 7989}, {8276, 13893}, {8277, 13947}, {9578, 10037}, {9581, 10046}, {9896, 9937}, {9907, 12973}, {10117, 12368}, {12310, 12407}
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3, 8193, 40), (26, 355, 8185), (5587, 9625, 25), (5691, 9591, 7387), (5881, 9626, 9798), (7395, 11365, 8227), (9715, 9798, 9626)= [ -5.947101469869723, -4.61610056060946, 9.581242471500030 ]
César Lozada
Continued from Hyacinthos 26753
[César Lozada]:
Sorry. I forgot to complete the lists in last message.
Group 1 applies to:
A'B'C' = anticevian-of-I and A"B"C" = pedal-of-I
A'B'C' = anticevian-of-I and A"B"C" = reflection-of-I
A'B'C' = antipedal-of-I and A"B"C" = pedal-of-I
A'B'C' = antipedal-of-I and A"B"C" = reflection-of-I
A'B'C' = cevian-of-I and A"B"C" = pedal-of-I
A'B'C' = cevian-of-I and A"B"C" = reflection-of-I
A'B'C' = circumcevian-of-I and A"B"C" = pedal-of-I
Group 2 applies to:
A'B'C' = pedal-of-I and A"B"C" = anticevian-of-I
A'B'C' = pedal-of-I and A"B"C" = antipedal-of-I
A'B'C' = reflection-of-I and A"B"C" = anticevian-of-I
A'B'C' = reflection-of-I and A"B"C" = antipedal-of-I
Continued …
3) A'B'C' = pedal-of-I and A"B"C" = circumcevian-of-I, or
A'B'C' = reflection-of-I and A"B"C" = circumcevian-of-I
Orthologic centers:
Za = ABC->PaPbPc = 1/(((b+c)*a^2-b*c*a-(b^2-c^2)* (b-c))*t+b*c*a) : : (trilinears) (Same Za as in group 2)
Zp = PaPbPc->ABC = ((b+c)*a^2-2*b*c*a-(b^2-c^2)*( b-c))*t-2*a^3+(b+c)*a^2+2*(b^ 2-b*c+c^2)*a-(b^2-c^2)*(b-c) : : (trilinears)
= on line {1, 3} = IO
ETC-pairs (P, Zp): (2,10246), (3,1385), (4,1), (20,3), (376,3576), (550,13624), (1657,3579), (3146,1482), (3529,40), (3543,10247), (5059,12702), (5073,11278), (6834,1388), (6868,10267), (6869,11249), (6905,1319), (6906,2646), (6934,56), (6935,13384), (6938,55), (6948,10269), (7411,13151), (11001,165), (11541,11531)
Example:
Zp( X(5) ) = X(1)X(3) ∩ X(5)X(551)
= a*(4*a^3-3*(b+c)*a^2-2*(2*b^2- 3*b*c+2*c^2)*a+3*(b^2-c^2)*(b- c)) : : (barys)
= 3*X(1)+X(3) = 7*X(1)+X(40) = 13*X(1)+3*X(165) = 5*X(1)-X(1482) = 5*X(1)+3*X(3576) = 5*X(1)+X(3579) = 9*X(1)-X(7982) = 11*X(1)+5*X(7987) = 15*X(1)+X(7991) = 13*X(1)-X(8148) = X(1)+3*X(10246) = 7*X(1)-3*X(10247) = 19*X(1)-3*X(11224) = 7*X(1)-X(11278) = 17*X(1)-X(11531) = 11*X(1)+X(12702) = 2*X(1)+X(13624) = 7*X(3)-3*X(40) = 13*X(3)-9*X(165) = X(3)-3*X(1385) = 5*X(3)+3*X(1482) = 5*X(3)-9*X(3576) = 5*X(3)-3*X(3579) = 3*X(3)+X(7982) = 11*X(3)-15*X(7987) = 5*X(3)-X(7991) = 13*X(3)+3*X(8148) = X(3)-9*X(10246) = 7*X(3)+9*X(10247) = 7*X(3)+3*X(11278) = 17*X(3)+3*X(11531) = 11*X(3)-3*X(12702) = 2*X(3)-3*X(13624) = X(40)-7*X(1385) = 5*X(40)+7*X(1482) = 5*X(40)-7*X(3579) = 9*X(40)+7*X(7982) = 15*X(40)-7*X(7991) = 13*X(40)+7*X(8148) = X(40)+3*X(10247) = 11*X(40)-7*X(12702) = 2*X(40)-7*X(13624) = 3*X(165)-13*X(1385) = 5*X(165)-13*X(3576) = 15*X(165)-13*X(3579) = 3*X(165)+X(8148) = X(165)-13*X(10246) = 7*X(165)+13*X(10247)
= On lines: {1, 3}, {4, 3655}, {5, 551}, {8, 3525}, {10, 632}, {20, 3656}, {30, 13464}, {72, 13472}, {140, 519}, {145, 10303}, {214, 5836}, {355, 3090}, {376, 5734}, {381, 9624}, {382, 11522}, {392, 3897}, {515, 546}, {516, 12103}, {518, 575}, {549, 11362}, {550, 4301}, {572, 3723}, {576, 1386}, {631, 3241}, {758, 12104}, {944, 3091}, {946, 3627}, {950, 1387}, {952, 1125}, {956, 3984}, {960, 1493}, {1056, 6049}, {1058, 10525}, {1201, 5396}, {1317, 10039}, {1656, 5881}, {1698, 12645}, {1699, 5076}, {1702, 6447}, {1703, 6448}, {1829, 3518}, {1902, 14865}, {2320, 3890}, {2771, 5609}, {2948, 15039}, {3146, 5603}, {3244, 5690}, {3476, 11374}, {3486, 6982}, {3523, 3654}, {3526, 3679}, {3529, 5731}, {3534, 9589}, {3623, 5657}, {3624, 5790}, {3635, 5844}, {3816, 10942}, {3817, 3857}, {3884, 12005}, {3898, 5884}, {3951, 5730}, {4315, 6147}, {4342, 10386}, {4669, 11539}, {4745, 10124}, {4861, 5440}, {4870, 5270}, {5072, 8227}, {5079, 5587}, {5265, 11041}, {5434, 7491}, {5453, 10105}, {5493, 8703}, {5777, 6265}, {6419, 7968}, {6420, 7969}, {6519, 9615}, {6713, 12735}, {6863, 10072}, {6883, 12513}, {6889, 11240}, {6958, 10056}, {6967, 11239}, {7504, 10031}, {7743, 10572}, {7978, 15021}, {7984, 15034}, {9709, 11530}, {9812, 11541}, {10586, 10786}, {10587, 10785}, {10594, 11363}, {11699, 14094}, {12778, 15020}, {12898, 15027}
= midpoint of X(i) and X(j) for these {i,j}: {1, 1385}, {5, 5882}, {10, 1483}, {40, 11278}, {550, 4301}, {1125, 13607}, {1317, 12619}, {1482, 3579}, {3244, 5690}, {3635, 6684}, {3884, 12005}, {6713, 12735}, {7967, 11230}
= reflection of X(i) in X(j) for these (i,j): (4745, 10124), (5885, 13373), (5901, 3636), (6583, 5045), (9955, 5901), (9956, 1125), (13145, 9940), (13624, 1385)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1, 35, 5048), (1, 36, 11011), (1, 40, 10247), (1, 1319, 942), (1, 2646, 9957), (1, 3576, 1482), (1, 3612, 2098), (1, 10246, 1385), (1, 13384, 3295), (3, 1482, 7991), (3, 7991, 3579), (40, 10247, 11278), (1385, 3579, 3576), (1482, 3576, 3579), (3576, 7991, 3), (8162, 10310, 12000)
= [ 2.963322105641861, 2.74146911377563, 0.375037200535779 ]
4) A'B'C' = pedal-of-I and A"B"C" = circumanticevian-of-I, or
A'B'C' = reflection-of-I and A"B"C" = circumanticevian-of-I
Orthologic centers:
Za = ABC->PaPbPc = 1/(((b+c)*a^2-3*b*c*a-(b^2-c^ 2)*(b-c))*t-b*c*a) : : (trilinears)
= On Feuerbach hyperbola
Zp = PaPbPc->ABC = ((b+c)*a^2-2*b*c*a-(b^2-c^2)*( b-c))*t-2*a^3+(b+c)*a^2+2*(b^ 2-b*c+c^2)*a-(b^2-c^2)*(b-c) : : (trilinears) (Same as Zp in group 3)
ETC pairs (P,Za): (3,1), (4,1476), (20,104), (21,3296), (376,21), (1006,5558), (1012,7091), (3522,943), (3528,2346), (3651,2320), (6906,7), (6909,4), (6914,5557)
Some others:
Za( X(2) ) = isogonal conjugate of X(9957)
= a*(a^3-b*a^2-(b^2-6*b*c+c^2)* a+(b^2-c^2)*b)*(a^3-c*a^2-(b^ 2-6*b*c+c^2)*a-(b^2-c^2)*c) : : (barys)
= on Feuerbach hyperbola and lines: {3, 7320}, {4, 3304}, {7, 7373}, {8, 474}, {9, 8666}, {35, 13602}, {36, 13606}, {56, 1000}, {79, 12053}, {80, 10106}, {354, 1389}, {942, 1320}, {943, 1319}, {1056, 1329}, {1385, 2346}, {1420, 7160}, {3065, 10074}, {3255, 5542}, {3333, 3680}, {3600, 5555}, {4308, 6985} , {5558, 5734}, {5559, 5563}, {5560, 9613}, {5561, 9614}, {5603, 10309} , {6919, 10586}, {10305, 10595}, {10307, 12114} , {11813, 12577}, {13143, 13375}
= isogonal conjugate of X(9957)
= [ 1.114104070208635, 1.16603788924167, 2.319205603105393 ]
Za( X(5) ) = X(35)X(7320) ∩ X(36)X(1000)
= a*(a^3-b*a^2-(b^2-5*b*c+c^2)* a+(b^2-c^2)*b)*(a^3-c*a^2-(b^ 2-5*b*c+c^2)*a-(b^2-c^2)*c) : : (barys)
= on Feuerbach hyperbola and lines: {3, 13606}, {8, 5563}, {35, 7320}, {36, 1000}, {55, 13602}, {56, 5559}, {79, 3304}, {80, 999}, {1156, 10074}, {1320, 5902}, {1420, 7162}, {3338, 3680}, {5424, 10246}, {5425, 14497}, {5557, 7373}
= [ 0.919431605110653, 0.97265852091306, 2.542932457378112 ]
5) A'B'C' = circumcevian-of-I and A"B"C" = reflection-of-I
Orthologic centers:
Za = ABC->PaPbPc = 1/(a*((-a^2+b^2+c^2)*t+b*c)) : : (trilinears) (Same as Zp in group 1)
Zp = PaPbPc->ABC = ((2*a^4-(b^2+c^2)*a^2-(b^2-c^ 2)^2)*t-2*a^2*b*c)/a : : (trilinears)
= on line {1, 79} = {1, 30}
ETC pairs (P, Zp): (2,3058), (3,1), (4,6284), (20,7354), (21,10543), (23,5160), (26,8144), (186,10149), (199,7073), (376,5434), (1650,11906), (1657,10483), (3149,12701), (3651,3649), (4220,4854), (6097,5453), (6985,12699), (7464,7286), (7580,1836), (12113,11905), (13743,5441)
Example:
Zp(X(5)) = 2*a^4-(b^2+4*b*c+c^2)*a^2-(b^ 2-c^2)^2 : : (barys) = Zp(X(4))-of-group 1
César Lozada
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