Let ABC be a triangle, P a point and A1B1C1, A2B2C2 the pedal, antipedal triangles of P, resp.
Denote:
D = the Poncelet point of ABCP
For P = I (D = X(11)):
1. The parallels to DA1, DB1, DC1 through A2, B2, C2, resp. are concurrent.
2. The parallels to DA2, DB2, DC2 through A1, B1, C1, resp. are concurrent.
Points?
Loci (if not complicated :) ?
********
Less complicated loci:
Let ABC be a triangle, Q a fixed point, A1B1C1, A2B2C2 the pedal, antipedal triangles of Q, resp. and P a point.
Which is the locus of P such that:
1. The parallels to PA1, PB1, PC1 through A2, B2, C2, resp. are concurrent ?
2. The parallels to PA2, PB2, PC2 through A1, B1, C1, resp. are concurrent ?
For Q = I, the locus is the entire plane.
[César Lozada]
> Less complicated loci:
> For Q = I, the locus is the entire plane.
Yes.
Let Z’ and Z” be the points of concurrence for 1 and 2 , resp., and X=X(57). Then
XZ’(P)/XP = 2*R/r and XZ”(P)/XP = r/(2*R)
For P=u:v:w (trilinears)
Z’(P) = a*(a+b+c)*(-a+b+c)*u-b*(a+b-c)*(a-b+c)*v-(a-b+c)*(a+b-c)*c*w : :
= (2*R-r)*X(57)-2*R*P
= on line PX(57)
and
Z”(P) = ((b+c)*u+v*b+w*c)*(a+b-c)*(a-b+c) : :
= (2*R-r)*X(57)+r*P
= on line PX(57)
Note: XZ’(P)/XZ”(P) = (r/(2R))^2
ETC pairs (P,Z’(P)): (1,40), (2,3928), (3,5709), (4,84), (6,7289), (7,9), (8,6762), (11,1768), (12,6763), (36,5535), (46,12704), (56,46), (57,57), (65,1), (79,7701), (85,21384), (145,2136), (174,20183), (177,164), (222,1763), (226,63), (234,362), (241,20367), (269,2270), (354,165), (481,6212), (482,6213), (497,10860), (553,2)
ETC pairs (P,Z”(P)): (1,65), (2,553), (3,942), (4,4292), (8,10106), (9,7), (10,4298), (19,3668), (20,950), (36,18838), (40,1), (43,1401), (46,56), (55,5173), (57,57), (63,226), (75,4032), (84,4), (90,7702), (100,5083), (102,12016), (103,11028), (104,12736), (164,177), (165,354), (167,17641), (169,10481), (173,2091), (191,3649), (200,17625), (269,14524), (355,18990), (362,234), (484,1319), (550,12433), (573,3664), (610,1439), (649,3676), (901,24201), (978,17114), (1019,7178), (1021,17094), (1054,1357), (1158,946), (1276,3639), (1277,3638), (1282,1362), (1697,3340), (1699,11246), (1706,3600), (1709,1836), (1721,12723), (1742,21746), (1743,1122), (1757,1463), (1759,3665), (1760,16888), (1763,222), (1764,3666), (1766,3663), (1768,11), (2077,5570), (2136,145), (2270,269), (2448,2447), (2449,2446), (2550,12573), (2900,16465), (2938,4890), (2941,4854), (2948,3028), (2950,1537), (2951,14100), (3057,13601), (3158,3873), (3169,3879), (3174,15185), (3218,3911), (3219,3982), (3220,1876), (3243,7672), (3294,4955), (3305,4114), (3306,4031), (3333,3339), (3338,5221), (3358,5805), (3359,999), (3464,1354), (3496,3674), (3509,9436), (3576,5902), (3579,5045), (3587,15934), (3632,10944), (3651,10122), (3652,11544), (3659,12814), (3679,5434), (3680,14923), (3751,1469), (3811,3874), (3868,15556), (3874,12432), (3928,2), (3929,4654), (4063,3669), (4091,14837), (4253,10521), (4297,6738), (4882,9850), (5011,1323), (5119,2099), (5128,1420), (5184,5194), (5223,8581), (5437,21454), (5493,12575), (5531,17660), (5535,36), (5536,1155), (5537,18839), (5539,1356), (5540,1358), (5541,1317), (5691,7354), (5709,3), (5732,5728), (5752,11573), (5787,20420), (6211,24231), (6212,481), (6213,482), (6244,12915), (6261,5884), (6326,11570), (6361,10624), (6762,8), (6763,12), (6765,3555), (6766,7991), (6796,12005), (6985,13369), (7171,5722), (7289,6), (7580,10391), (7701,79), (7713,1448), (7966,11041), (7982,5903), (7991,3057), (7992,12688), (7993,17636), (7994,17642), (7996,17635), (8001,17639), (8666,3754), (8715,3881), (9355,4014), (9841,938), (9851,17632), (9860,3023), (9875,18969), (9896,18970), (9897,18976), (9898,18979), (9899,6285), (9900,18975), (9901,18974), (9902,18982), (9903,18983), (9904,3024), (9905,18984), (9906,18989), (9907,18988), (10085,1837), (10476,986), (10860,497), (10864,5691), (11010,11011), (11012,13750), (11260,10107), (11372,4312), (11495,5572), (11500,12675), (11519,3893), (11523,3868), (12114,7686), (12387,12403), (12396,12402), (12404,17633), (12407,18968), (12408,6020), (12409,18985), (12511,12564), (12512,6744), (12513,5836), (12514,3671), (12515,1387), (12516,12855), (12517,12914), (12518,5571), (12519,12917), (12526,12709), (12565,12711), (12629,10914), (12658,12854), (12659,12912), (12660,12913), (12702,9957), (12704,46), (12705,4295), (12717,24248), (12767,17638), (12773,6797), (13069,17640), (13101,17643), (13174,3027), (13221,3320), (13679,18986), (13799,18987), (15071,1858), (15239,2096), (15624,13476), (16009,16006), (16118,18977), (16139,16137), (16143,17637), (16548,22464), (16549,7198), (16552,4059), (16560,1086), (16572,7195), (17594,10473), (18197,7180), (18206,16609), (18446,18389), (18540,18541), (18725,2262), (18788,20358), (20114,12809), (20183,174), (20367,241), (20368,982), (20606,24215), (21366,7217), (21375,3782), (21377,7251), (21381,1365), (21384,85), (21387,6063), (22650,18971), (22651,18972), (22652,18973), (22653,18978), (23361,20617)
Some non-ETC’s:
Z’(X(5)) = X(3)X(63) ∩ X(5)X(57)
= a*(-a^2+b^2+c^2)*(a^4-2*(b^2-b*c+c^2)*a^2+(b^2-c^2)^2) : : (barys)
= X(4)-3*X(5770), 2*R*X(5)-(2*R-r)*X(57), X(84)+3*X(3928), 3*X(165)-X(5534), 3*X(3928)-X(5709)
= on lines: {1, 1399}, {3, 63}, {4, 3218}, {5, 57}, {7, 6824}, {8, 6948}, {9, 140}, {11, 90}, {12, 17700}, {26, 3220}, {30, 84}, {36, 5693}, {38, 601}, {40, 550}, {46, 355}, {56, 920}, {65, 22758}, {104, 3869}, {144, 6926}, {155, 222}, {165, 5534}, {191, 3576}, {226, 6862}, {255, 1062}, {329, 6891}, {392, 16203}, {405, 10202}, {411, 13243}, {484, 5881}, {499, 7082}, {513, 8279}, {517, 1158}, {518, 11248}, {527, 6705}, {546, 18540}, {548, 3587}, {549, 3929}, {602, 896}, {603, 1060}, {631, 3219}, {632, 7308}, {758, 5450}, {908, 6958}, {938, 6930}, {942, 3560}, {960, 10269}, {971, 6985}, {982, 3073}, {993, 5884}, {997, 5694}, {999, 12709}, {1001, 13373}, {1147, 7193}, {1155, 11499}, {1181, 22129}, {1210, 6929}, {1216, 3784}, {1364, 6238}, {1385, 5289}, {1407, 17814}, {1445, 5779}, {1454, 1478}, {1465, 8757}, {1482, 4018}, {1483, 1697}, {1484, 9614}, {1565, 7183}, {1656, 3306}, {1699, 7701}, {1708, 5777}, {1709, 10943}, {1737, 18961}, {1776, 3086}, {1858, 8071}, {2003, 12161}, {2077, 5904}, {2096, 6850}, {2323, 16266}, {2771, 6261}, {2800, 8666}, {2801, 6796}, {3072, 4650}, {3091, 23958}, {3157, 17102}, {3305, 3526}, {3333, 5901}, {3336, 5587}, {3337, 8227}, {3338, 3649}, {3358, 5762}, {3359, 5690}, {3487, 6892}, {3555, 10679}, {3564, 7289}, {3628, 5437}, {3651, 11220}, {3781, 5447}, {3811, 12341}, {3868, 6906}, {3870, 11849}, {3876, 6940}, {3911, 6959}, {3937, 5562}, {4292, 6917}, {4293, 7098}, {4640, 10267}, {4995, 7162}, {5223, 10270}, {5248, 12005}, {5249, 6861}, {5250, 10246}, {5251, 15016}, {5273, 6989}, {5435, 5811}, {5442, 5660}, {5535, 5691}, {5704, 6973}, {5708, 6913}, {5714, 6859}, {5720, 6924}, {5744, 6825}, {5761, 6935}, {5768, 6868}, {5780, 16417}, {5789, 18541}, {5844, 6762}, {5851, 18243}, {5905, 6833}, {6001, 11249}, {6326, 7280}, {6734, 6923}, {6846, 21454}, {6847, 9965}, {6869, 9799}, {6875, 18444}, {6883, 9940}, {6887, 9776}, {6888, 17483}, {6890, 20078}, {6905, 12528}, {6922, 13226}, {6938, 12649}, {6972, 17484}, {7686, 18761}, {10200, 15297}, {10461, 15952}, {10525, 10916}, {10526, 12616}, {10680, 12672}, {10785, 11415}, {10942, 21031}, {11012, 15071}, {12532, 18861}, {12705, 22791}, {13465, 19861}, {15888, 17699}, {16574, 19543}, {18732, 22659}, {19549, 21371}
= midpoint of X(i) and X(j) for these {i,j}: {84, 5709}, {6869, 9799}
= reflection of X(i) in X(j) for these (i,j): (10525, 10916), (10526, 12616), (12699, 10943)
= X(156)-of-excentral triangle
= X(13561)-of-6th mixtilinear triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 7171, 550), (40, 10085, 18481), (57, 7330, 5), (84, 3928, 5709), (90, 17437, 11), (255, 7004, 1062), (1155, 14872, 11499), (1709, 12704, 12699), (1768, 6763, 40), (3587, 9841, 548), (4640, 12675, 10267), (4652, 18446, 3), (5435, 5811, 6944), (5720, 15803, 6924), (10884, 21165, 3)
= [ 1.8934096412964810, -7.5242498634252230, 7.9758791682957730 ]
Z’(X(35)) = X(9)X(114) ∩ X(57)X(98)
= a*(a^6-(b+c)*a^3*b*c-3*(b^2+b*c+c^2)*a^4+(b^2-c^2)*(b-c)*a*b*c+(b^2+b*c+c^2)*(3*b^2-2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b-c)^2) : : (barys)
= 3*X(40)-2*X(11010), 3*X(165)-2*X(11849)
= on lines: {1, 3}, {9, 6990}, {10, 6900}, {30, 6763}, {48, 16553}, {63, 2894}, {191, 12699}, {920, 9580}, {946, 6884}, {1732, 1766}, {3218, 20066}, {3219, 18483}, {3651, 3874}, {3868, 16132}, {4654, 14526}, {5127, 5358}, {5250, 15674}, {5587, 6894}, {5762, 15908}, {5904, 6985}, {6284, 16113}, {7098, 10624}, {7289, 9047}, {7411, 12005}, {10021, 16139}, {11499, 15104}, {12248, 12625}, {16127, 20078}
= reflection of X(7982) in X(11014)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 5709, 5535), (40, 6766, 12703), (40, 12704, 3576), (484, 12702, 40), (2095, 5584, 15016), (11009, 11010, 1697)
= [ 25.8651107352448500, 21.6520746627009100, -23.2869767777677300 ]
Z’(X(98)) = X(9)X(114) ∩ X(57)X(98)
= a*( a^10-2*(2*b^2+b*c+2*c^2)*a^8+(7*b^4+7*c^4-(2*b^2-7*b*c+2*c^2)*b*c)*a^6-(7*b^6+7*c^6-2*(2*b^4+2*c^4-(2*b^2-3*b*c+2*c^2)*b*c)*b*c)*a^4+(4*b^6+4*c^6+(6*b^4+6*c^4+(7*b^2+6*b*c+7*c^2)*b*c)*b*c)*(b-c)^2*a^2-(b^2-c^2)^2*(b^6+c^6-2*(b^4+b^2*c^2+c^4)*b*c)) :: (barys)
= 3*X(165)-2*X(12178)
= on lines: {1, 22504}, {9, 114}, {40, 99}, {46, 9860}, {57, 98}, {63, 147}, {84, 2794}, {165, 12178}, {542, 3928}, {1697, 7970}, {1709, 12182}, {2782, 5709}, {3218, 5984}, {3220, 9861}, {3333, 11710}, {3929, 6054}, {5437, 6036}, {6033, 7330}, {6769, 13173}, {7061, 7350}, {12514, 21636}, {18540, 22505}
= reflection of X(i) in X(j) for these (i,j): (1, 22504), (6769, 13173)
= X(22504)-of-Aquila triangle
= [ 85.8975779458338600, 112.0337348830553000, -113.5661879505926000 ]
Z”(X(5)) = X(3)X(7) ∩ X(5)X(57)
= 2*a^4+2*(b+c)*a^3-(b^2-4*b*c+c^2)*a^2-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : : (barys)
= X(1)+3*X(11246), 3*X(354)+X(1770), 3*X(354)-X(15171), 3*X(553)-X(942), 9*X(553)-X(950), 3*X(553)+X(4292), 6*X(553)-X(12433), 3*X(942)-X(950), X(950)+3*X(4292), 2*X(950)-3*X(12433), X(3868)+3*X(11112), 2*X(4292)+X(12433), 3*X(5049)-X(10624), 3*X(5434)+X(5903), 3*X(5902)+X(7354)
= on lines: {1, 550}, {3, 7}, {4, 5708}, {5, 57}, {11, 79}, {12, 3336}, {20, 15934}, {30, 553}, {36, 3649}, {40, 4355}, {46, 495}, {56, 5901}, {58, 1086}, {63, 8728}, {65, 952}, {78, 17563}, {84, 5805}, {140, 226}, {144, 17582}, {273, 7546}, {329, 16408}, {354, 1770}, {355, 3339}, {376, 11036}, {382, 938}, {386, 17365}, {388, 5690}, {390, 3296}, {404, 17483}, {442, 3218}, {443, 3927}, {474, 5905}, {484, 15888}, {496, 1836}, {516, 5045}, {517, 4298}, {527, 5044}, {528, 3881}, {529, 3754}, {540, 24176}, {545, 3159}, {546, 1210}, {548, 4114}, {549, 4654}, {596, 5846}, {632, 5219}, {944, 1159}, {946, 20418}, {962, 7373}, {975, 17276}, {990, 8144}, {999, 4295}, {1056, 12702}, {1125, 17235}, {1155, 13407}, {1385, 3671}, {1387, 5563}, {1399, 15253}, {1407, 5707}, {1420, 10283}, {1434, 1565}, {1466, 6924}, {1478, 5221}, {1479, 4860}, {1482, 3600}, {1483, 3340}, {1595, 1892}, {1656, 5435}, {1657, 3488}, {1788, 9654}, {1876, 6756}, {2094, 17528}, {2095, 6850}, {2099, 4317}, {2476, 23958}, {2646, 11551}, {3073, 15251}, {3075, 15252}, {3219, 17529}, {3295, 3474}, {3306, 17527}, {3333, 4312}, {3361, 5886}, {3526, 5226}, {3530, 3982}, {3534, 4313}, {3579, 21620}, {3585, 12019}, {3601, 8703}, {3616, 17571}, {3627, 5722}, {3628, 3911}, {3648, 5284}, {3662, 17698}, {3678, 5852}, {3824, 5745}, {3845, 9581}, {3851, 5704}, {3868, 11112}, {3916, 5249}, {3925, 6763}, {3928, 5791}, {3937, 18180}, {3940, 6904}, {3947, 11231}, {4252, 24159}, {4303, 5453}, {4304, 12103}, {4308, 10247}, {4315, 23339}, {4316, 10543}, {4325, 5425}, {4338, 12701}, {4757, 5855}, {4880, 21677}, {4973, 4999}, {5049, 10624}, {5253, 14450}, {5260, 9782}, {5266, 24231}, {5267, 11281}, {5298, 5443}, {5326, 5442}, {5434, 5903}, {5586, 11529}, {5735, 9841}, {5777, 5843}, {5779, 6864}, {5789, 6843}, {5840, 6583}, {5842, 12005}, {5844, 10106}, {5857, 12609}, {5902, 7354}, {6284, 18398}, {6361, 6767}, {6831, 13226}, {6887, 8732}, {6894, 13243}, {6915, 13257}, {7091, 12700}, {7682, 22792}, {9655, 18391}, {9657, 10573}, {9776, 11108}, {10593, 17728}, {11009, 12735}, {11019, 22793}, {11041, 18526}, {11496, 20330}, {11518, 15704}, {11544, 12047}, {11827, 15016}, {13374, 18260}, {14054, 17616}, {15170, 17609}, {15681, 15933}, {16056, 22458}, {16415, 20805}, {16863, 18228}, {17484, 17531}, {17580, 20059}
= midpoint of X(i) and X(j) for these {i,j}: {65, 18990}, {942, 4292}, {1770, 15171}
= reflection of X(i) in X(j) for these (i,j): (5044, 12436), (12433, 942), (15172, 5045)
= X(143)-of-intouch triangle
= X(5901)-of-2nd anti-circumperp-tangential triangle
= X(6101)-of-inverse-in-incircle triangle
= X(10263)-of-incircle-circles triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7, 6147), (3, 6147, 5719), (4, 21454, 5708), (46, 10404, 495), (79, 3337, 11), (354, 1770, 15171), (443, 9965, 3927), (553, 4292, 942), (999, 4295, 22791), (1836, 3338, 496), (4654, 15803, 11374), (5435, 5714, 1656), (5557, 15228, 1), (5708, 18541, 4), (11374, 15803, 549)
= [ 0.4001038898216197, 0.5031062285587133, 3.1076968360645230 ]
Z”(X(6)) = X(6)X(57) ∩ X(7)X(8)
= a*((b+c)*a+b^2+c^2)*(a+b-c)*(a-b+c) : : (barys)
= 3*X(354)-X(3056)
= on lines: {1, 1350}, {6, 57}, {7, 8}, {12, 3844}, {37, 1423}, {46, 611}, {56, 77}, {60, 757}, {72, 17272}, {86, 1431}, {141, 226}, {181, 9436}, {193, 17490}, {210, 5232}, {241, 1400}, {273, 1875}, {279, 959}, {307, 3665}, {354, 3056}, {511, 942}, {513, 21202}, {517, 3663}, {524, 553}, {599, 4654}, {613, 3338}, {674, 5173}, {742, 4032}, {760, 18252}, {946, 24213}, {960, 3674}, {1100, 1429}, {1108, 18161}, {1155, 2330}, {1210, 5480}, {1284, 15569}, {1319, 1442}, {1351, 5708}, {1357, 1366}, {1358, 2836}, {1401, 9025}, {1432, 3863}, {1466, 7013}, {1470, 1804}, {1503, 4292}, {1834, 5929}, {1836, 12589}, {1837, 21279}, {1843, 1876}, {1887, 7282}, {2099, 7190}, {2257, 18725}, {2260, 3942}, {2262, 4000}, {2264, 7291}, {2269, 3666}, {2285, 6180}, {3057, 3672}, {3218, 15988}, {3242, 3340}, {3339, 3751}, {3361, 16475}, {3435, 7053}, {3487, 10519}, {3589, 3911}, {3618, 5435}, {3619, 5226}, {3629, 4031}, {3630, 4114}, {3631, 3982}, {3668, 3827}, {3673, 10446}, {3676, 9002}, {3739, 15985}, {3740, 5224}, {3763, 5219}, {3812, 10436}, {3875, 3880}, {4021, 9957}, {4260, 10481}, {4267, 4719}, {4269, 16696}, {4298, 5847}, {4341, 22769}, {4359, 15983}, {4419, 21871}, {4452, 14923}, {4662, 17270}, {4663, 5221}, {4731, 5936}, {4862, 5903}, {4888, 5902}, {4909, 5049}, {5083, 9024}, {5085, 15803}, {5092, 5122}, {5572, 21746}, {5846, 10106}, {6385, 18033}, {7023, 7177}, {7197, 14256}, {7269, 11011}, {7274, 18421}, {7686, 17861}, {8679, 20617}, {9004, 22277}, {9021, 15556}, {9037, 18838}, {9612, 10516}, {10360, 18935}, {10391, 18650}, {10456, 10477}, {10520, 12915}, {10914, 17151}, {12722, 15310}, {12723, 15726}, {13374, 24179}, {16603, 17239}, {17084, 17322}, {17189, 18180}, {17276, 21853}, {17705, 22464}, {18440, 18541}, {21239, 24005}
= midpoint of X(i) and X(j) for these {i,j}: {65, 1469}, {17276, 21853}
= X(53)-of-intouch triangle
= X(1386)-of-2nd anti-circumperp-tangential triangle
= X(20477)-of-inverse-in-incircle triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 2097, 7289), (7, 3212, 75), (7, 5933, 6604), (65, 1122, 7), (1423, 7146, 37)
= [ 0.4502505283892282, 0.7090856488978003, 2.9419510964908730 ]
Z”(X(11)) = X(7)X(100) ∩ X(11)X(57)
= (a+b-c)*(a-b+c)*(2*a^4-2*(b+c)*a^3-(b^2-4*b*c+c^2)*a^2+(b^2-c^2)^2) : : (barys)
= X(11)+3*X(11246), 3*X(65)+X(18976), 3*X(553)-X(5083)
= on lines: {5, 7702}, {7, 100}, {11, 57}, {30, 18838}, {56, 1387}, {65, 952}, {80, 3339}, {109, 1086}, {149, 21454}, {214, 3671}, {226, 3035}, {388, 1145}, {516, 3660}, {528, 553}, {938, 10724}, {942, 5840}, {1155, 5762}, {1317, 3340}, {1320, 3600}, {1466, 10090}, {1470, 11729}, {1537, 4295}, {1617, 3474}, {2099, 11046}, {2802, 4298}, {2829, 4292}, {2834, 3937}, {3036, 4848}, {3218, 5857}, {3333, 14217}, {3336, 8068}, {3337, 5533}, {3361, 16173}, {3911, 5087}, {4355, 5541}, {4440, 14594}, {4654, 6174}, {4860, 13274}, {5221, 12019}, {5708, 10738}, {5854, 10106}, {6147, 10044}, {7972, 18421}, {9945, 12739}, {10404, 10956}, {10742, 18541}, {11529, 12119}, {14151, 20095}, {15803, 21154}, {17579, 18419}, {17724, 23703}
= midpoint of X(4292) and X(12736)
= X(1112)-of-intouch triangle
= X(1387)-of-2nd anti-circumperp-tangential triangle
= X(13416)-of-Ursa-minor triangle
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5221, 13273, 12832), (12832, 13273, 12019)
= [ 0.6383105427686622, 0.9237167375291875, 2.7065634131094040 ]
Z”(X(98)) = X(1)X(99) ∩ X(57)X(98)
= (b+c)*a^7-(b-c)^2*a^6-2*(b^3+c^3)*a^5+(b^2+c^2)*(b-c)^2*a^4+(b+c)*(2*b^4+2*c^4-(2*b^2-b*c+2*c^2)*b*c)*a^3+(2*b^2+3*b*c+2*c^2)*(b-c)^2*b*c*a^2-(b^2-c^2)*(b-c)*(b^4-b^2*c^2+c^4)*a+(b^2-c^2)^2*b^2*c^2 : : (barys)
= 3*X(354)-X(3027)
= on lines: {1, 99}, {7, 147}, {46, 10053}, {56, 11710}, {57, 98}, {65, 1355}, {114, 226}, {115, 1210}, {148, 938}, {291, 23996}, {354, 3027}, {388, 9864}, {542, 553}, {620, 13411}, {942, 2782}, {950, 23698}, {1281, 11031}, {1836, 12185}, {1837, 13182}, {1876, 12131}, {2783, 5083}, {2784, 4298}, {2785, 12016}, {2786, 11028}, {2787, 12736}, {2794, 4292}, {2795, 10122}, {3218, 5985}, {3338, 10069}, {3339, 9860}, {3340, 7970}, {3488, 13172}, {3586, 10723}, {3601, 21166}, {3671, 21636}, {3911, 6036}, {4654, 6054}, {4697, 16598}, {5708, 12188}, {5722, 6321}, {5984, 21454}, {5988, 9436}, {8591, 15933}, {9579, 10722}, {9581, 14639}, {10072, 12258}, {10404, 12184}, {11019, 11599}, {11374, 15561}, {11518, 23235}, {13178, 18391}, {13188, 15934}
= midpoint of X(65) and X(3023)
= incircle-inverse-of X(741)
= center of the circle {X(65), X(1356), X(3023)}
= X(129)-of-intouch triangle
= X(1298)-of-inverse-in-incircle triangle
= X(11710)-of-2nd anti-circumperp-tangential triangle
= {X(1), X(10089)}-harmonic conjugate of X(11711)
= [ 1.4910582594067680, 2.0557947261713040, 1.5292412440626610 ]
César Lozada
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