X(11) = FEUERBACH POINT

Trilinears

\(1 - cos(B - C) : 1 - cos(C - A) : 1 - cos(A - B)\)

\(sin2(B/2 - C/2) : :\)

\(bc(b + c - a)(b - c)2 : :\)

\(1 - cos A - 2 cos B cos C : :\)

\(1 + cos A - 2 sin B sin C : :\)

Barycentrics

\(a(1 - cos(B - C)) : b(1 - cos(C - A)) : c(1 - cos(A -B))\)

\((b + c - a)(b - c)2 : :\)

Notes

X(11) is the point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle. Feuerbach’s famous theorem states that the nine-point circle is tangent to the incircle and the three excircles.

Let LA be the line through A parallel to X(1)X(3), and define LB and LC cyclically. Let MA be the reflection of BC in LA, and define MB and MC cyclically. Let A’ = MB∩MC, and define cyclically B’ and C’ cyclically. The triangle A’B’C’ is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A’B’C’ in line X(1)X(3). The triangle A″B″C″ is homothetic to ABC, with center of homothety X(11); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

The circumcircle of the incentral triangle intersects the incircle at 2 points, X(11) and X(3024), and the nine-point circle at 2 points, X(11) and X(115). (Randy Hutson, April 9, 2016)

X(11) lies on the bicevian conic of X(1) and X(2), which is also QA-Co1 (Nine-point Conic) of quadrangle ABC:ref:X(1) <X(1)> (see http://www.chrisvantienhoven.nl/other-quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/76-qa-co1.html) (Randy Hutson, April 9, 2016)

Let Na = X(5) of BC:ref:X(1) <X(1)>, Nb = X(5) of CA:ref:X(1) <X(1)>, Nc = X(5) of AB:ref:X(1) <X(1)>. Then

X(11) = X(186) of NaNbNc. (Randy Hutson, April 9, 2016)

Let JaJbJc be the excentral triangle and FaFbFc be the Feuerbach triangle. Let Fa’ = {X(5),Ja}-harmonic conjugate of Fa, and define Fb’, Fc’ cyclically. The lines AFa’, BFb’, CFc’ concur in X(11).

Let A’B’C’ be the orthic triangle. Let La be the antiorthic axis of AB’C’, and define Lb and Lc cyclically. Let A&Prime; = Lb&cap;Lc, and define B’’ and C’’ cyclically. The triangle A&Prime;B&Prime;C&Prime; is inversely similar to ABC, with similitude center X(9), and

X(11) = X(55)-of-A&Prime;B&Prime;C&Prime;. (Randy Hutson, December 10, 2016)

Let A’B’C’ be the orthic triangle. Let Na be the Nagel line of AB’C’, and define Nb and Nc cyclically. Let A&Prime; = Nb&cap;Nc, and define B’’ and C’’ cyclically. The triangle A&Prime;B&Prime;C&Prime; is inversely similar to ABC, and

X(11) = X(36)-of-A&Prime;B&Prime;C&Prime;. (Randy Hutson, December 10, 2016)

Let A’B’C’ be the orthic triangle. The lines IO of AB’C’, BC’A’, CA’B’ concur in X(11). (Randy Hutson, December 10, 2016)

Let A’B’C’ be the orthic triangle. The lines IO of AB’C’, BC’A’, CA’B’ concur in X(11). (Randy Hutson, June 27, 2018)

If you have The Geometer’s Sketchpad, you can view Feuerbach point. If you have GeoGebra, you can view Feuerbach point.

X(11) lies on the following curves:

incircle, nine point circle, 2nd Lester circle (Yiu), Mandart circle cevian circle of every point on the Feuerbach hyperbola Mandart inconic circumellipse of the medial and incentral triangles (see X(34585)) K672, K877, K925, K1051, Q015

X(11) lies on the following lines: {1, 5}, {2, 55}, {3, 499}, {4, 56}, {6, 1985}, {7, 658}, {8, 1320}, {9, 3254}, {10, 121}, {13, 202}, {14, 203}, {15, 11755}, {16, 11773}, {17, 7005}, {18, 7006}, {19, 37432}, {20, 5204}, {21, 4996}, {22, 9673}, {24, 9672}, {25, 10829}, {28, 1852}, {29, 40980}, {30, 36}, {31, 5348}, {32, 9665}, {33, 427}, {34, 235}, {35, 140}, {37, 5513}, {38, 4415}, {39, 13077}, {40, 6922}, {42, 37662}, {43, 33141}, {46, 12515}, {54, 2477}, {57, 1360}, {58, 37357}, {59, 33562}, {60, 3615}, {63, 7082}, {64, 12920}, {65, 117}, {67, 32287}, {68, 1069}, {69, 10755}, {72, 10395}, {74, 10767}, {75, 16067}, {76, 12836}, {78, 25681}, {79, 3065}, {81, 14008}, {83, 10799}, {84, 12676}, {86, 14009}, {88, 19636}, {90, 17437}, {98, 10768}, {99, 10769}, {101, 10770}, {102, 10771}, {103, 10772}, {106, 10774}, {107, 10775}, {109, 10777}, {110, 215}, {111, 10779}, {112, 10780}, {113, 942}, {114, 2783}, {115, 1015}, {116, 3022}, {118, 226}, {122, 2803}, {123, 2804}, {124, 1364}, {125, 3024}, {126, 2805}, {127, 2806}, {128, 7159}, {131, 37361}, {132, 1848}, {133, 1838}, {137, 3327}, {141, 3056}, {142, 5580}, {145, 5154}, {150, 24203}, {153, 388}, {155, 10071}, {165, 9580}, {171, 33106}, {172, 7745}, {174, 8086}, {177, 12614}, {181, 2051}, {182, 38119}, {184, 9667}, {185, 26955}, {190, 17777}, {192, 7777}, {197, 37366}, {198, 37367}, {200, 4863}, {210, 3452}, {212, 748}, {214, 442}, {222, 34029}, {225, 37368}, {229, 37369}, {230, 1914}, {233, 21860}, {238, 1936}, {239, 26019}, {241, 33331}, {243, 17923}, {244, 676}, {255, 7299}, {262, 12837}, {265, 10091}, {273, 6046}, {278, 1857}, {279, 15511}, {312, 3703}, {314, 11609}, {321, 21333}, {325, 350}, {329, 5825}, {333, 37373}, {334, 18149}, {344, 30741}, {345, 4387}, {371, 9661}, {372, 13977}, {377, 22768}, {381, 999}, {382, 4299}, {384, 26686}, {386, 9555}, {395, 7127}, {398, 2307}, {402, 11903}, {403, 1870}, {404, 6691}, {405, 19755}, {428, 5322}, {429, 1104}, {430, 40956}, {431, 40985}, {440, 23207}, {443, 31418}, {452, 30478}, {474, 2932}, {479, 36620}, {480, 6601}, {481, 31555}, {482, 31556}, {484, 28174}, {485, 1124}, {486, 1335}, {493, 10945}, {494, 10946}, {498, 1656}, {500, 10035}, {511, 15974}, {512, 46671}, {513, 3025}, {514, 3328}, {515, 1319}, {516, 1155}, {517, 1737}, {518, 908}, {519, 3814}, {521, 34940}, {522, 3326}, {523, 1090}, {524, 4396}, {527, 41555}, {529, 5080}, {537, 21093}, {546, 3585}, {547, 3584}, {548, 4324}, {549, 5010}, {550, 7280}, {551, 3822}, {553, 13159}, {559, 51749}, {573, 9554}, {574, 9664}, {578, 9653}, {590, 2066}, {594, 17452}, {595, 45939}, {596, 44040}, {597, 38090}, {609, 18907}, {611, 14561}, {612, 37439}, {613, 1352}, {615, 5414}, {618, 13076}, {619, 13075}, {620, 15452}, {625, 5148}, {626, 6029}, {629, 22910}, {630, 22865}, {631, 4294}, {632, 10386}, {641, 13082}, {642, 13081}, {644, 26074}, {650, 1566}, {656, 38981}, {659, 33311}, {661, 20974}, {662, 19642}, {671, 12348}, {672, 17747}, {693, 23989}, {740, 51411}, {750, 33104}, {756, 29690}, {758, 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{1327, 13693}, {1328, 13813}, {1346, 2463}, {1347, 2464}, {1348, 1674}, {1349, 1675}, {1354, 4292}, {1356, 44950}, {1357, 2827}, {1365, 4934}, {1366, 3664}, {1385, 6842}, {1386, 38050}, {1397, 13478}, {1398, 37197}, {1399, 3073}, {1402, 45189}, {1420, 5691}, {1425, 43820}, {1427, 1856}, {1428, 1503}, {1435, 7008}, {1440, 7023}, {1447, 4872}, {1455, 1877}, {1456, 34049}, {1457, 51421}, {1458, 2635}, {1460, 2050}, {1465, 8758}, {1466, 6847}, {1469, 5480}, {1482, 6971}, {1491, 42771}, {1500, 1506}, {1511, 12896}, {1513, 38646}, {1519, 6001}, {1560, 40941}, {1575, 21956}, {1587, 18995}, {1588, 18996}, {1591, 3083}, {1592, 3084}, {1594, 6198}, {1598, 9913}, {1617, 19541}, {1650, 11909}, {1657, 38754}, {1672, 1676}, {1673, 1677}, {1697, 1698}, {1728, 5812}, {1731, 7359}, {1738, 5121}, {1739, 45269}, {1746, 44085}, {1751, 6056}, {1758, 43056}, {1770, 22793}, {1776, 3218}, {1785, 39535}, {1795, 10747}, {1804, 7318}, {1827, 16580}, {1844, 18402}, {1853, 2192}, {1854, 17054}, {1855, 46830}, 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10703}, {6833, 10531}, {6834, 11500}, {6862, 11507}, {6863, 10267}, {6871, 10586}, {6879, 10596}, {6880, 37000}, {6881, 11230}, {6883, 40292}, {6891, 10310}, {6893, 10629}, {6906, 18861}, {6908, 8273}, {6911, 8069}, {6914, 14793}, {6917, 22766}, {6923, 10269}, {6924, 38722}, {6928, 11249}, {6929, 22758}, {6944, 10321}, {6949, 11491}, {6952, 14882}, {6958, 11248}, {6959, 10320}, {6968, 12115}, {6980, 10246}, {6985, 7742}, {6996, 17798}, {7040, 31387}, {7063, 40608}, {7073, 7281}, {7074, 37679}, {7137, 50574}, {7160, 12857}, {7202, 48055}, {7333, 44949}, {7334, 44947}, {7352, 22660}, {7353, 45860}, {7356, 20424}, {7358, 46663}, {7362, 45861}, {7377, 21010}, {7384, 41245}, {7395, 10831}, {7397, 31230}, {7491, 26286}, {7503, 9659}, {7507, 11392}, {7529, 10037}, {7537, 41227}, {7580, 37578}, {7603, 31476}, {7649, 13999}, {7671, 8255}, {7677, 36002}, {7679, 8236}, {7687, 46683}, {7697, 10063}, {7727, 10264}, {7728, 10081}, {7770, 38655}, {7819, 30103}, {7968, 49240}, {7969, 49241}, {7982, 41687}, {7987, 37424}, {8053, 30944}, {8087, 8097}, {8088, 8098}, {8113, 8377}, {8114, 8378}, {8143, 10036}, {8144, 13371}, {8163, 8165}, {8196, 12462}, {8200, 11879}, {8203, 12463}, {8207, 11880}, {8212, 12765}, {8213, 12766}, {8220, 11953}, {8221, 11954}, {8222, 11947}, {8223, 11948}, {8228, 8243}, {8230, 8239}, {8254, 47378}, {8256, 13463}, {8257, 25606}, {8261, 20288}, {8361, 30104}, {8380, 8390}, {8381, 8392}, {8382, 11924}, {8422, 12622}, {8580, 10388}, {8724, 10070}, {8808, 10374}, {8976, 13905}, {8988, 13883}, {9053, 32927}, {9317, 17044}, {9345, 17392}, {9539, 31074}, {9540, 9648}, {9597, 44518}, {9613, 18492}, {9628, 15559}, {9638, 34224}, {9645, 14790}, {9646, 10576}, {9647, 35821}, {9652, 10539}, {9658, 10594}, {9710, 9780}, {9819, 19875}, {9842, 9850}, {9843, 9848}, {9844, 51706}, {9880, 18969}, {9912, 11365}, {9927, 18970}, {9931, 23307}, {9947, 16215}, {9956, 9957}, {9970, 32308}, {9993, 12499}, {9996, 10047}, {10024, 18447}, {10031, 38314}, {10044, 18224}, {10052, 18223}, {10053, 38224}, {10055, 14852}, {10065, 15061}, {10075, 12856}, {10086, 15561}, {10088, 14643}, {10104, 10801}, {10106, 19925}, {10113, 18968}, {10118, 23315}, {10122, 33667}, {10129, 10394}, {10157, 12915}, {10199, 11112}, {10281, 32161}, {10309, 34256}, {10356, 10873}, {10358, 10797}, {10382, 10582}, {10383, 41867}, {10384, 38052}, {10398, 38036}, {10407, 10408}, {10428, 17101}, {10459, 50034}, {10473, 10478}, {10479, 10480}, {10500, 10506}, {10501, 17630}, {10502, 17631}, {10503, 17629}, {10514, 10923}, {10515, 10924}, {10516, 12588}, {10526, 10680}, {10532, 10894}, {10577, 35809}, {10585, 10587}, {10597, 10599}, {10638, 23302}, {10651, 37772}, {10652, 37773}, {10700, 24864}, {10749, 13312}, {10759, 14853}, {10786, 10806}, {10795, 10804}, {10796, 10802}, {10830, 10835}, {10872, 10879}, {10921, 10931}, {10922, 10932}, {10951, 11957}, {10952, 11958}, {11011, 13464}, {11012, 31789}, {11018, 15008}, {11108, 19854}, {11189, 23332}, {11193, 35967}, {11224, 30286}, {11236, 11240}, {11256, 32049}, {11263, 12267}, {11274, 51103}, {11281, 31936}, {11363, 12137}, {11364, 12198}, {11366, 12460}, {11367, 12461}, {11368, 12498}, {11369, 12551}, {11377, 12741}, {11378, 12742}, {11391, 11401}, {11429, 23292}, {11436, 13567}, {11446, 23293}, {11461, 23294}, {11529, 38021}, {11544, 34502}, {11700, 29008}, {11717, 14028}, {11735, 31525}, {11831, 12729}, {11867, 11883}, {11868, 11884}, {11897, 12752}, {11904, 11915}, {11927, 17115}, {11929, 12001}, {11992, 11993}, {12000, 15867}, {12005, 13751}, {12183, 12190}, {12233, 19366}, {12241, 19365}, {12349, 12357}, {12350, 23234}, {12372, 12382}, {12393, 12402}, {12394, 12400}, {12423, 12431}, {12587, 12595}, {12599, 18979}, {12610, 12723}, {12612, 12854}, {12613, 12912}, {12615, 12913}, {12617, 12709}, {12618, 12721}, {12621, 12876}, {12632, 27525}, {12635, 12649}, {12677, 12687}, {12691, 13374}, {12757, 16193}, {12809, 17607}, {12848, 36971}, {12858, 12875}, {12863, 12864}, {12868, 15998}, {12888, 23306}, {12890, 12906}, {12903, 14644}, {12910, 23309}, {12911, 23310}, {12918, 13117}, {12919, 13129}, {12930, 13095}, {12931, 13106}, {12932, 13107}, {12933, 13110}, {12934, 13113}, {12935, 13119}, {12936, 13122}, {12937, 13131}, {12938, 13133}, {12939, 13135}, {12942, 36765}, {13006, 34460}, {13043, 23313}, {13044, 23314}, {13080, 13089}, {13181, 13190}, {13182, 14639}, {13214, 13218}, {13266, 46403}, {13295, 13314}, {13384, 25055}, {13411, 37080}, {13561, 32168}, {13624, 37401}, {13687, 18986}, {13692, 13715}, {13694, 13717}, {13699, 13701}, {13724, 23361}, {13741, 25992}, {13743, 35451}, {13747, 25440}, {13807, 18987}, {13812, 13838}, {13814, 13840}, {13819, 13821}, {13852, 13995}, {13860, 38657}, {13881, 16781}, {13893, 31432}, {13896, 13907}, {13936, 13976}, {13951, 13963}, {13953, 13965}, {14112, 16185}, {14189, 37757}, {14193, 31227}, {14330, 44012}, {14527, 33961}, {14563, 39782}, {14839, 24250}, {15298, 38108}, {15313, 15608}, {15494, 15509}, {15519, 38255}, {15607, 38388}, {15614, 48183}, {15616, 39007}, {15726, 30379}, {15761, 32047}, {15803, 41869}, {15819, 22711}, {15911, 15932}, {15914, 40166}, {15971, 28385}, {16137, 46028}, {16139, 16155}, {16202, 26487}, {16252, 26888}, {16434, 37577}, {16484, 29640}, {16569, 32865}, {16586, 34976}, {16589, 31488}, {16602, 21949}, {16626, 22930}, {16627, 22885}, {16686, 40560}, {16726, 38960}, {16727, 17198}, {16752, 46515}, {16784, 43291}, {16819, 33034}, {17009, 44238}, {17024, 37353}, {17063, 17889}, {17067, 45275}, {17122, 33109}, {17123, 33138}, {17167, 18165}, {17232, 30948}, {17234, 30947}, {17243, 29643}, {17246, 46901}, {17279, 29857}, {17334, 36263}, {17340, 33161}, {17357, 30768}, {17365, 24725}, {17427, 35593}, {17435, 36197}, {17449, 32856}, {17451, 21049}, {17469, 29683}, {17535, 26060}, {17559, 19855}, {17591, 33154}, {17595, 24248}, {17596, 33095}, {17597, 33144}, {17598, 33152}, {17608, 17623}, {17609, 17624}, {17633, 17645}, {17634, 17650}, {17635, 17651}, {17639, 17655}, {17640, 17656}, {17641, 17657}, {17643, 17659}, {17665, 25414}, {17669, 26558}, {17734, 40091}, {17737, 33854}, {17800, 38637}, {17861, 41007}, {17863, 41003}, {17888, 48182}, {17917, 44695}, {18101, 27010}, {18201, 32857}, {18239, 34293}, {18389, 38039}, {18407, 28452}, {18412, 20330}, {18440, 39901}, {18480, 24928}, {18481, 37406}, {18491, 34746}, {18493, 48667}, {18517, 18544}, {18518, 18543}, {18583, 38168}, {18635, 34830}, {18641, 40946}, {18743, 29641}, {18922, 23291}, {18971, 22682}, {18972, 22831}, {18973, 22832}, {18978, 22833}, {18991, 19077}, {18992, 19078}, {19025, 19049}, {19026, 19050}, {19182, 23295}, {19434, 23298}, {19435, 23299}, {19512, 40910}, {19515, 23832}, {19522, 20878}, {19540, 23853}, {19635, 37128}, {19637, 37137}, {19794, 37090}, {20255, 23497}, {20262, 39050}, {20292, 27003}, {20335, 51384}, {20358, 26012}, {20420, 37583}, {20530, 20541}, {20582, 51158}, {20594, 21025}, {20872, 37449}, {20939, 35960}, {20989, 33849}, {20992, 30943}, {21029, 39244}, {21051, 45213}, {21073, 25066}, {21090, 24036}, {21139, 23766}, {21155, 32613}, {21172, 44014}, {21221, 26141}, {21621, 40961}, {21666, 44426}, {21746, 24220}, {21842, 34773}, {21856, 40606}, {21912, 40945}, {21944, 23938}, {22051, 35197}, {22071, 46838}, {22072, 27627}, {22148, 36280}, {22466, 22956}, {22704, 22732}, {22858, 22887}, {22903, 22932}, {22954, 23308}, {22955, 22981}, {22957, 22983}, {22965, 22966}, {23301, 38978}, {23383, 27622}, {23690, 45916}, {23773, 23776}, {23843, 28077}, {23847, 36571}, {23869, 25436}, {24025, 43048}, {24165, 48643}, {24179, 41004}, {24185, 31890}, {24232, 47016}, {24302, 43731}, {24317, 24424}, {24319, 25371}, {24325, 25385}, {24336, 25369}, {24388, 40965}, {24433, 26611}, {24443, 28096}, {24627, 24723}, {24685, 25342}, {24774, 34847}, {24816, 24828}, {24833, 24846}, {24835, 24848}, {24851, 24852}, {24880, 25648}, {24926, 51700}, {24997, 25135}, {25328, 32286}, {25351, 25377}, {25357, 25383}, {25405, 28204}, {25492, 33833}, {25542, 50205}, {25591, 36568}, {25641, 33964}, {25642, 33966}, {25741, 25754}, {25957, 30957}, {25958, 33172}, {26001, 26002}, {26005, 26010}, {26102, 33111}, {26128, 29668}, {26326, 26380}, {26327, 26404}, {26328, 26433}, {26329, 26434}, {26330, 26435}, {26331, 26436}, {26332, 26437}, {26351, 26359}, {26352, 26360}, {26355, 26361}, {26356, 26362}, {26365, 48501}, {26366, 48502}, {26367, 49410}, {26368, 49409}, {26369, 49068}, {26370, 49069}, {26386, 45373}, {26389, 26401}, {26410, 45374}, {26413, 26425}, {26466, 45614}, {26467, 45613}, {26468, 49032}, {26469, 49033}, {26483, 26501}, {26484, 26510}, {26485, 26519}, {26486, 26524}, {26580, 46909}, {26725, 33857}, {26904, 26906}, {26942, 44707}, {26949, 26951}, {27020, 32992}, {27064, 33121}, {27739, 50316}, {27747, 50305}, {28186, 36975}, {28224, 37006}, {28234, 36920}, {28530, 32845}, {28606, 29680}, {28795, 30826}, {28797, 30847}, {29349, 34583}, {29677, 31237}, {29685, 31264}, {29820, 33130}, {29827, 32784}, {29830, 30834}, {29835, 46897}, {29840, 32926}, {29844, 32920}, {29846, 32943}, {29849, 32915}, {29861, 33159}, {29872, 33157}, {30117, 45272}, {30304, 41706}, {30312, 30332}, {30313, 30333}, {30314, 30334}, {30315, 30337}, {30316, 30338}, {30317, 30339}, {30375, 30380}, {30376, 30381}, {30377, 30382}, {30378, 30383}, {30447, 50757}, {30823, 49768}, {30824, 36479}, {30831, 33173}, {30963, 37664}, {30971, 36635}, {31003, 39691}, {31137, 33087}, {31145, 50894}, {31221, 37586}, {31263, 48696}, {31393, 31434}, {31401, 31448}, {31402, 31404}, {31408, 31412}, {31409, 31415}, {31426, 31428}, {31433, 31441}, {31451, 31455}, {31459, 31463}, {31461, 31467}, {31471, 31481}, {31473, 31484}, {31477, 31489}, {31500, 35255}, {31557, 31567}, {31558, 31568}, {31586, 31588}, {31587, 31589}, {31654, 44048}, {31657, 38124}, {31658, 38131}, {31659, 37621}, {31775, 37561}, {31946, 38979}, {31996, 33033}, {32243, 32274}, {32288, 32310}, {32336, 32369}, {32351, 32378}, {32379, 32404}, {32381, 32406}, {32390, 32391}, {32631, 43757}, {32911, 33142}, {32913, 33096}, {32918, 32947}, {32930, 33119}, {32931, 33120}, {33112, 37633}, {33139, 37680}, {33170, 41242}, {33178, 38336}, {33504, 44049}, {33650, 34234}, {33898, 41426}, {33925, 42884}, {34030, 34040}, {34467, 34948}, {34595, 41859}, {34699, 45701}, {34747, 50893}, {34855, 51364}, {34903, 34904}, {35065, 42337}, {35094, 45320}, {35104, 51407}, {35250, 35252}, {35508, 38973}, {35580, 44047}, {35581, 44050}, {35582, 44051}, {35587, 44052}, {35591, 44053}, {35663, 35671}, {35664, 35669}, {35678, 35679}, {35762, 35852}, {35763, 35853}, {35798, 35818}, {35799, 35819}, {36436, 36451}, {36439, 36443}, {36457, 36461}, {36473, 36493}, {36487, 36526}, {36495, 36513}, {36508, 36557}, {36530, 36542}, {36545, 36547}, {36561, 36574}, {36577, 36579}, {36999, 50701}, {37520, 50307}, {37525, 38028}, {37592, 44954}, {37596, 44951}, {37607, 49745}, {37660, 50295}, {37681, 38293}, {37686, 41324}, {37756, 50533}, {37764, 49704}, {37787, 38454}, {38018, 38238}, {38074, 50907}, {38076, 50906}, {38163, 45977}, {38188, 51150}, {38472, 51377}, {38941, 43038}, {38985, 47601}, {38992, 41224}, {39004, 41220}, {39816, 39822}, {39845, 39851}, {39890, 39903}, {40964, 46878}, {41697, 49177}, {41877, 45223}, {42312, 44316}, {42770, 47123}, {43728, 46041}, {43817, 43819}, {43821, 43858}, {43860, 43862}, {43986, 44001}, {44229, 45630}, {44620, 44645}, {44621, 44646}, {44661, 51410}, {44663, 51423}, {45305, 49537}, {45398, 49337}, {45399, 49338}, {45404, 45440}, {45405, 45441}, {45456, 45496}, {45457, 45497}, {45470, 45472}, {45471, 45473}, {45506, 45544}, {45507, 45545}, {45554, 45582}, {45555, 45583}, {45558, 45586}, {45559, 45587}, {46436, 47019}, {46659, 47020}, {48641, 50117}, {49658, 49659}, {50194, 51709}, {50605, 50621}

X(11) = midpoint of X(i) and X(j) for these {i,j}: {1, 80}, {2, 10707}, {3, 10738}, {4, 104}, {5, 1484}, {7, 1156}, {8, 1320}, {9, 3254}, {10, 21630}, {20, 10724}, {21, 11604}, {36, 3583}, {40, 14217}, {56, 12764}, {65, 17638}, {69, 10755}, {74, 10767}, {76, 32454}, {79, 3065}, {84, 46435}, {98, 10768}, {99, 10769}, {100, 149}, {101, 10770}, {102, 10771}, {103, 10772}, {105, 10773}, {106, 10774}, {107, 10775}, {108, 10776}, {109, 10777}, {110, 10778}, {111, 10779}, {112, 10780}, {119, 37726}, {145, 12531}, {153, 38669}, {314, 11609}, {355, 12737}, {382, 38753}, {390, 20119}, {551, 50889}, {901, 31512}, {908, 26015}, {943, 24298}, {946, 10265}, {1000, 24297}, {1061, 24300}, {1109, 2611}, {1113, 10781}, {1114, 10782}, {1172, 43735}, {1290, 36175}, {1387, 12019}, {1479, 10090}, {1482, 19914}, {1650, 13268}, {1699, 11219}, {1737, 30384}, {1768, 34789}, {1837, 12740}, {2481, 14947}, {3057, 17636}, {3218, 5057}, {3241, 50890}, {3259, 6075}, {3632, 26726}, {3679, 50891}, {3680, 12641}, {3828, 50892}, {3868, 12532}, {3874, 47320}, {3937, 38389}, {4010, 13277}, {4106, 42322}, {4440, 36237}, {5176, 38460}, {5533, 39692}, {5559, 13143}, {5620, 13604}, {6246, 11715}, {6264, 12751}, {6326, 49176}, {6595, 10266}, {6596, 6598}, {6597, 6599}, {6601, 34894}, {7972, 9897}, {9318, 24712}, {9809, 13243}, {10309, 34256}, {10609, 12690}, {10698, 12247}, {10728, 12248}, {10742, 12773}, {10914, 17652}, {11256, 32049}, {12114, 12761}, {12515, 12699}, {12672, 17654}, {12868, 15998}, {13205, 13271}, {13266, 46403}, {13272, 22560}, {13463, 32198}, {15635, 44013}, {16173, 37718}, {17668, 36868}, {17763, 32844}, {22799, 51529}, {22938, 38602}, {24302, 43731}, {24851, 24852}, {31145, 50894}, {32843, 32919}, {34501, 34503}, {34747, 50893}, {39144, 39145}, {40565, 40566}, {43728, 46041}, {43740, 45393}, {45981, 45982}, {51402, 51442} reflection of X(i) in X(j) for these {i,j}: {1, 1387}, {2, 45310}, {3, 6713}, {8, 3036}, {10, 6702}, {12, 8068}, {20, 38759}, {36, 15325}, {59, 33562}, {65, 12736}, {72, 18254}, {80, 12019}, {100, 3035}, {104, 20418}, {105, 33970}, {119, 5}, {153, 38757}, {214, 1125}, {500, 10035}, {650, 10006}, {908, 5087}, {946, 16174}, {1071, 15528}, {1125, 33709}, {1145, 10}, {1155, 3911}, {1317, 1}, {1319, 44675}, {1519, 22835}, {1537, 946}, {1768, 13226}, {2720, 28347}, {3025, 14115}, {3035, 6667}, {3057, 15558}, {3649, 33593}, {3689, 6745}, {4996, 4999}, {5083, 18240}, {5298, 3582}, {5520, 47399}, {5948, 10277}, {6068, 9}, {6154, 100}, {6174, 2}, {6265, 11729}, {6594, 6666}, {6735, 5123}, {7972, 12735}, {10036, 8143}, {10427, 142}, {10609, 214}, {10993, 33814}, {11274, 51103}, {11570, 942}, {11700, 29008}, {12611, 9955}, {12665, 5777}, {12743, 950}, {12831, 226}, {12832, 1210}, {13257, 21635}, {13996, 1145}, {14115, 33646}, {14740, 46694}, {15326, 36}, {17660, 5083}, {17757, 3814}, {18239, 34293}, {18801, 8255}, {18802, 8256}, {19907, 5901}, {21578, 5126}, {22799, 546}, {24466, 3}, {24685, 25342}, {25485, 13464}, {25558, 25557}, {25606, 8257}, {27778, 17660}, {30384, 7743}, {31235, 31272}, {31525, 11735}, {33667, 10122}, {33812, 3636}, {33814, 140}, {34123, 32557}, {35204, 6675}, {35604, 34949}, {37725, 119}, {37726, 1484}, {38389, 38390}, {38752, 38319}, {38760, 34126}, {38761, 38602}, {39144, 45981}, {39145, 45982}, {39776, 5836}, {39778, 11281}, {40663, 1737}, {41541, 13411}, {41553, 13405}, {41556, 11019}, {41558, 6738}, {41684, 11545}, {44238, 17009}, {46409, 6714}, {50841, 3828}, {50842, 3679}, {50843, 551}, {50846, 3241}, {51007, 141}, {51008, 597}, {51062, 37}, {51157, 3589}, {51158, 20582}, {51198, 6}, {51377, 38472}, {51390, 24250}, {51409, 11813}, {51463, 26015}, {51569, 6701}

X(11) = isogonal conjugate of X(59)

X(11) = isotomic conjugate of X(4998)

X(11) = complement of X(100)

X(11) = anticomplement of X(3035)

X(11) = complementary conjugate of X(513)

X(11) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,523), (4,513), (7,514), (8,522), (262,1491)

X(11) = crosspoint of X(i) and X(j) for these (i,j): (7,514), (8,522)

X(11) = crosssum of X(i) and X(j) for these (i,j): (6,692), (55,101), (56,109), (1381,1382), (1397,1415)

X(11) = crossdifference of every pair of points on line X(101)X(109)

X(11) = circumcircle-inverse of X(14667)

X(11) = Fuhrmann-circle-inverse of X(1837)

X(11) = Stevanovic-circle-inverse of X(1566)

X(11) = Conway-circle-inverse of X(13244)

X(11) = Spieker-radical-circle-inverse of X(3030)

X(11) = polar-circle-inverse of X(108)

X(11) = orthoptic-circle-of-Steiner-inellipse-inverse of X(105)

X(11) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(20097)

X(11) = inverse-in-{circumcircle, nine-point circle}-inverter of X(105)

X(11) = inverse-in-excircles-radical-circle of X(3030)

X(11) = polar conjugate of X(46102)

X(11) = anticomplement of the isogonal conjugate of X(18771)

X(11) = complement of the isogonal conjugate of X(513)

X(11) = complement of the isotomic conjugate of X(693)

X(11) = isogonal conjugate of the anticomplement of X(46100)

X(11) = isotomic conjugate of the anticomplement of X(46101)

X(11) = isotomic conjugate of the complement of X(17036)

X(11) = isotomic conjugate of the isogonal conjugate of X(3271)

X(11) = isogonal conjugate of the isotomic conjugate of X(34387)

X(11) = isotomic conjugate of the polar conjugate of X(8735)

X(11) = polar conjugate of the isotomic conjugate of X(26932)

X(11) = polar conjugate of the isogonal conjugate of X(7117)

X(11) = medial-isogonal conjugate of X(513)

X(11) = orthic-isogonal conjugate of X(513)

X(11) = psi-transform of X(18343)

X(11) = X(i)-beth conjugate of X(j) for these (i,j): (11,244), (522,11), (693,11)

X(11) = orthopole of line X(1)X(3)

X(11) = anticenter of cyclic quadrilateral ABC:ref:X(104) <X(104)>

X(11) = perspector of ABC and extraversion triangle of X(12)

X(11) = homothetic center of intouch and 3rd Euler triangles

X(11) = trilinear square root of X(6728)

X(11) = perspector of Feuerbach triangle and Schroeter triangle

X(11) = X(110)-of-intouch-triangle

X(11) = X(403) of Fuhrmann triangle

X(11) = perspector of circumconic centered at X(650)

X(11) = center of circumconic that is locus of trilinear poles of lines passing through X(650)

X(11) = X(2)-Ceva conjugate of X(650)

X(11) = trilinear pole wrt intouch triangle of Soddy line

X(11) = trilinear pole wrt extouch triangle of line X(8)X(9)

X(11) = midpoint of PU(i) for these i: 121, 123

X(11) = bicentric sum of PU(i) for these i: 121, 123

X(11) = homothetic center of medial triangle and Mandart-incircle triangle

X(11) = X(100) of Mandart-incircle triangle

X(11) = X(3659) of orthic triangle if ABC is acute

X(11) = homothetic center of intangents triangle and reflection of extangents triangle in X(100)

X(11) = homothetic center of 3rd Euler triangle and intouch triangle

X(11) = QA-P2 (Euler-Poncelet Point) of quadrangle ABC:ref:X(1) <X(1)> (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/12-qa-p2.html)

X(11) = intersection of tangents to Steiner inellipse at X(1086) and X(1146)

X(11) = crosspoint wrt medial triangle of X(1086) and X(1146)

X(11) = perspector of orthic triangle and tangential triangle of hyperbola {{A,B,C,:ref:X(1) <X(1)>,:ref:X(2) <X(2)>}}

X(11) = homothetic center of cyclic quadrilateral ABC:ref:X(104) <X(104)> and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time

X(11) = perspector of ABC and cross-triangle of ABC and Feuerbach triangle

X(11) = homothetic center of medial triangle and cross-triangle of ABC and inner Johnson triangle

X(11) = homothetic center of Euler triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

X(11) = homothetic center of medial triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

X(11) = orthic-isogonal conjugate of X(513)

X(11) = trilinear pole of line X(4530)X(14393)

X(11) = point of concurrence of cevian circles of vertices of anticevian triangle of X(7)

X(11) = homothetic center of cyclic quadrilateral ABC:ref:X(104) <X(104)> and congruent quadrilateral formed by orthocenters of vertices taken 3 at a time

X(11) = homothetic center of Ursa-minor and Ursa-major triangles

X(11) = homothetic center of ABC and inner Johnson triangle

X(11) = trilinear product X(57)

X(11) = barycentric product X(7)

X(11) = homothetic center of Garcia reflection triangle (aka Gemini triangle 8) and 2nd Schiffler triangle

X(11) = excentral-to-ABC functional image of X(3659)

X(11) = excentral-to-ABC barycentric image of X(100)

X(11) = Pelletier-isogonal conjugate of X(35604)

X(11) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {18771, 8}, {31628, 20295}, {38809, 21272}

X(11) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 513}, {2, 3835}, {3, 20315}, {4, 20316}, {6, 514}, {7, 17072}, {9, 20317}, {10, 31946}, {11, 124}, {25, 3239}, {27, 30476}, {28, 8062}, {31, 650}, {32, 6586}, {34, 521}, {37, 4129}, {42, 661}, {55, 4521}, {56, 522}, {57, 4885}, {58, 523}, {75, 21260}, {76, 21262}, {81, 4369}, {86, 512}, {87, 4083}, {88, 4928}, {89, 47779}, {100, 24003}, {101, 4422}, {105, 3716}, {106, 900}, {109, 3035}, {111, 45661}, {190, 27076}, {223, 20314}, {239, 27854}, {244, 11}, {256, 21051}, {269, 3900}, {274, 42327}, {278, 46396}, {279, 46399}, {282, 20318}, {286, 21259}, {291, 3837}, {292, 812}, {310, 23301}, {330, 21191}, {335, 21261}, {512, 1213}, {513, 10}, {514, 141}, {521, 34823}, {522, 1329}, {523, 3454}, {593, 21196}, {596, 44316}, {604, 905}, {608, 14837}, {614, 17115}, {647, 440}, {649, 2}, {650, 3452}, {651, 21232}, {656, 21530}, {657, 6554}, {659, 17793}, {661, 1211}, {663, 9}, {665, 16593}, {667, 37}, {669, 21838}, {676, 118}, {692, 24036}, {693, 2887}, {739, 4763}, {741, 9508}, {749, 23815}, {798, 16589}, {810, 18591}, {812, 20333}, {849, 31947}, {870, 788}, {875, 1575}, {876, 3836}, {884, 40869}, {893, 25666}, {899, 14434}, {900, 121}, {902, 6544}, {904, 3709}, {905, 18589}, {918, 20540}, {977, 832}, {985, 4874}, {987, 6133}, {996, 9002}, {998, 9001}, {1002, 24720}, {1015, 1086}, {1019, 3739}, {1022, 3834}, {1024, 34852}, {1027, 518}, {1042, 656}, {1073, 20319}, {1086, 116}, {1096, 14298}, {1106, 6129}, {1111, 21252}, {1120, 6085}, {1126, 4977}, {1169, 8045}, {1193, 50330}, {1201, 6615}, {1220, 6371}, {1222, 6363}, {1245, 47842}, {1252, 10196}, {1255, 48049}, {1333, 14838}, {1357, 3756}, {1358, 17059}, {1390, 48050}, {1395, 6588}, {1397, 6589}, {1398, 21172}, {1400, 1577}, {1407, 7658}, {1411, 3738}, {1412, 17069}, {1413, 8058}, {1415, 16578}, {1416, 676}, {1431, 3907}, {1434, 17066}, {1438, 918}, {1458, 3126}, {1459, 3}, {1461, 17044}, {1474, 525}, {1477, 4925}, {1577, 21245}, {1635, 16594}, {1638, 31844}, {1647, 3259}, {1769, 119}, {1911, 665}, {1914, 27929}, {1919, 39}, {1960, 4370}, {1973, 2509}, {1977, 6377}, {1980, 16584}, {2006, 46397}, {2107, 46390}, {2162, 31286}, {2163, 4777}, {2170, 26932}, {2191, 3309}, {2203, 16612}, {2206, 647}, {2215, 23882}, {2254, 120}, {2279, 4762}, {2291, 45326}, {2297, 8712}, {2308, 4988}, {2309, 40627}, {2310, 5514}, {2334, 4778}, {2350, 693}, {2382, 45666}, {2384, 45684}, {2423, 3911}, {2424, 516}, {2530, 21249}, {2605, 3647}, {2983, 29162}, {3063, 1212}, {3064, 41883}, {3120, 125}, {3121, 16592}, {3122, 115}, {3123, 5518}, {3124, 6627}, {3125, 8287}, {3223, 25142}, {3226, 6373}, {3248, 1015}, {3261, 626}, {3270, 40616}, {3271, 1146}, {3415, 4522}, {3445, 3667}, {3572, 3912}, {3667, 2885}, {3669, 142}, {3676, 2886}, {3699, 3038}, {3709, 38930}, {3720, 50497}, {3733, 1125}, {3737, 960}, {3756, 5510}, {3766, 20542}, {3768, 13466}, {3835, 21250}, {3837, 20551}, {3937, 2968}, {3939, 3039}, {4010, 45162}, {4017, 442}, {4025, 1368}, {4040, 40607}, {4057, 4075}, {4079, 6537}, {4083, 34832}, {4091, 6389}, {4105, 5574}, {4107, 39080}, {4164, 19563}, {4367, 51575}, {4391, 21244}, {4444, 20541}, {4455, 35068}, {4466, 127}, {4556, 620}, {4560, 21246}, {4581, 3831}, {4584, 40548}, {4598, 40562}, {4603, 40546}, {4724, 3789}, {4750, 126}, {4775, 16590}, {4777, 21251}, {4817, 21264}, {5029, 6651}, {5331, 8672}, {6129, 6260}, {6164, 11814}, {6186, 3700}, {6187, 1639}, {6373, 20532}, {6385, 21263}, {6548, 21241}, {6591, 226}, {6729, 2090}, {7004, 123}, {7050, 43061}, {7117, 16596}, {7121, 21348}, {7123, 11068}, {7151, 14331}, {7178, 17052}, {7180, 17056}, {7192, 3741}, {7199, 21240}, {7203, 3742}, {7216, 18635}, {7250, 1834}, {7252, 5745}, {7649, 5}, {7658, 2884}, {8050, 36951}, {8578, 39026}, {8632, 17755}, {8643, 3161}, {8656, 36911}, {8700, 45679}, {8747, 520}, {9262, 190}, {9268, 6550}, {9309, 4147}, {9315, 31287}, {9456, 3960}, {10013, 6005}, {10566, 3934}, {10579, 8713}, {11125, 113}, {13476, 50337}, {14413, 10427}, {14419, 16597}, {14936, 13609}, {16079, 4943}, {16726, 17761}, {16732, 21253}, {16757, 21247}, {16892, 21248}, {17096, 17050}, {17187, 3005}, {17924, 20305}, {17925, 34830}, {17954, 2787}, {17962, 2786}, {18001, 10026}, {18014, 20546}, {18098, 29512}, {18108, 1215}, {18191, 34589}, {18210, 34846}, {18344, 20262}, {20974, 40618}, {20979, 6376}, {21102, 1209}, {21109, 15116}, {21122, 40938}, {21123, 6292}, {21132, 46100}, {21143, 6547}, {21172, 2883}, {21758, 16586}, {21832, 46842}, {22350, 42769}, {22383, 1214}, {23189, 34851}, {23345, 519}, {23351, 5199}, {23355, 726}, {23472, 41771}, {23493, 798}, {23572, 6374}, {23838, 5123}, {23892, 536}, {24002, 17046}, {24012, 17426}, {25417, 4932}, {25426, 28840}, {25430, 4940}, {25576, 25107}, {26721, 23305}, {27789, 48041}, {27846, 38989}, {28615, 48003}, {30571, 4806}, {30650, 47778}, {30651, 48008}, {32674, 36949}, {32735, 24980}, {32739, 23988}, {34080, 25097}, {34444, 8714}, {34893, 2832}, {34916, 4160}, {35058, 23803}, {35348, 5087}, {35355, 3823}, {35365, 50752}, {36123, 8677}, {36127, 3042}, {36598, 29226}, {37129, 891}, {37627, 3880}, {38266, 3669}, {38346, 40619}, {38357, 46663}, {38986, 40610}, {39949, 4132}, {39961, 47996}, {39965, 48399}, {39972, 29198}, {40148, 649}, {40433, 6372}, {40495, 21235}, {40746, 824}, {40763, 3805}, {40958, 40628}, {41434, 28209}, {41436, 6006}, {43051, 20528}, {43531, 834}, {43922, 1647}, {43923, 1210}, {43924, 1}, {43925, 40940}, {43926, 50755}, {43927, 50605}, {43928, 4871}, {43929, 3008}, {43931, 3840}, {43932, 11019}, {46018, 4791}, {46107, 21243}, {47947, 17239}, {48032, 40609}, {48131, 51571}, {48144, 10472}, {48306, 51573}, {48340, 51572}, {50344, 3634}, {50521, 16587}, {50525, 28651}, {51223, 47843}, {51443, 4913}, {51476, 2490}, {51640, 18642}, {51642, 2092}, {51648, 35204}, {51650, 41540}, {51654, 6600}, {51658, 12640}, {51686, 23874}

X(11) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 523}, {2, 650}, {4, 513}, {5, 8819}, {7, 514}, {8, 522}, {9, 6362}, {10, 17420}, {75, 21120}, {79, 4977}, {80, 900}, {83, 3287}, {98, 659}, {100, 15914}, {261, 4560}, {262, 1491}, {273, 7178}, {278, 3064}, {312, 3700}, {314, 3910}, {333, 4976}, {497, 11934}, {514, 42462}, {522, 21132}, {596, 21119}, {650, 42454}, {693, 40166}, {941, 784}, {1000, 4777}, {1086, 4534}, {1088, 21104}, {1111, 1086}, {1156, 2826}, {1222, 40500}, {1320, 2804}, {1440, 3669}, {1751, 652}, {2051, 661}, {2298, 29142}, {2321, 48264}, {2481, 918}, {2997, 525}, {3254, 6366}, {3296, 4802}, {3434, 11927}, {3596, 4391}, {3615, 3737}, {3680, 42337}, {3701, 4985}, {4518, 50333}, {4858, 1146}, {4997, 1639}, {5397, 50349}, {5551, 28199}, {5556, 4778}, {5557, 28175}, {5558, 28147}, {5559, 28183}, {5560, 28217}, {5561, 28209}, {6384, 20508}, {6557, 3239}, {6601, 3900}, {7155, 3810}, {7249, 3004}, {7261, 812}, {7317, 28205}, {7318, 905}, {7319, 3667}, {7320, 28161}, {7607, 48226}, {7608, 47827}, {9221, 18116}, {11604, 3738}, {13478, 649}, {13606, 28187}, {14458, 50358}, {14484, 2526}, {14492, 50328}, {14554, 46393}, {15314, 29162}, {17197, 2170}, {17501, 39386}, {17758, 21127}, {18101, 3271}, {18490, 28151}, {18815, 10015}, {23836, 6550}, {24026, 38357}, {24298, 8674}, {24624, 654}, {26721, 21133}, {26856, 11998}, {30101, 23755}, {30479, 23880}, {30710, 47135}, {32016, 24113}, {32023, 693}, {34387, 26932}, {35097, 8677}, {36590, 23838}, {36620, 3676}, {38254, 7658}, {38255, 4521}, {39768, 6370}, {40419, 17494}, {40450, 1}, {40451, 244}, {40836, 7649}, {41527, 824}, {42318, 14330}, {43531, 17418}, {43672, 2254}, {43731, 28221}, {43732, 28213}, {43733, 28195}, {43734, 4926}, {43740, 521}, {43741, 35057}, {43749, 3907}, {43759, 4773}, {43972, 21106}, {44426, 42455}, {44733, 4841}, {45100, 48026}, {46103, 7252}, {51685, 4990}

X(11) = X(i)-cross conjugate of X(j) for these (i,j): {2170, 1086}, {2310, 1146}, {3259, 35015}, {3271, 8735}, {4516, 2170}, {4542, 4530}, {4953, 4534}, {5532, 42462}, {7117, 26932}, {7336, 21132}, {18210, 7004}, {21044, 4858}, {21132, 522}, {23615, 3064}, {42462, 514}, {42547, 43974}, {46101, 2}

X(11) = cevapoint of X(i) and X(j) for these (i,j): {1, 2957}, {2, 17036}, {650, 11193}, {1146, 4953}, {1647, 3259}, {2170, 2310}, {3120, 18210}, {3271, 7117}, {4516, 21044}, {4530, 4542}, {5532, 42462}, {7336, 21132}

X(11) = crosspoint of X(i) and X(j) for these (i,j): {1, 3737}, {2, 693}, {4, 44426}, {7, 514}, {8, 522}, {261, 4560}, {278, 3676}, {312, 18155}, {513, 34434}, {1111, 4858}, {3307, 3308}, {3596, 4391}, {4397, 6556}, {40437, 43728}

X(11) = crosssum of X(i) and X(j) for these (i,j): {1, 4551}, {3, 36059}, {6, 692}, {55, 101}, {56, 109}, {100, 2975}, {108, 41227}, {181, 4559}, {215, 1983}, {219, 3939}, {906, 6056}, {1110, 2149}, {1362, 2283}, {1381, 1382}, {1397, 1415}, {7066, 23067}, {17455, 23344}, {23981, 34586}

X(11) = trilinear pole of line {4530, 14393}

X(11) = crossdifference of every pair of points on line {101, 109}

X(11) = X(i)-lineconjugate of X(j) for these (i,j): {10770, 101}, {10777, 109}

X(11) = X(i)-isoconjugate of X(j) for these (i,j): {1, 59}, {2, 2149}, {3, 7012}, {6, 4564}, {7, 1110}, {8, 24027}, {9, 1262}, {12, 1101}, {19, 44717}, {31, 4998}, {41, 1275}, {48, 46102}, {55, 7045}, {56, 765}, {57, 1252}, {63, 7115}, {65, 4570}, {73, 5379}, {85, 23990}, {100, 109}, {101, 651}, {108, 1331}, {110, 4551}, {162, 23067}, {163, 4552}, {181, 24041}, {190, 1415}, {200, 7339}, {201, 250}, {213, 4620}, {219, 7128}, {249, 2171}, {269, 6065}, {312, 23979}, {480, 24013}, {603, 15742}, {604, 1016}, {644, 1461}, {649, 31615}, {650, 4619}, {653, 906}, {655, 1983}, {662, 4559}, {664, 692}, {728, 23971}, {872, 7340}, {883, 32666}, {901, 23703}, {919, 1025}, {934, 3939}, {1018, 4565}, {1020, 5546}, {1026, 32735}, {1088, 6066}, {1106, 4076}, {1259, 24033}, {1319, 9268}, {1332, 32674}, {1397, 7035}, {1400, 4567}, {1402, 4600}, {1404, 5376}, {1405, 5385}, {1414, 4557}, {1428, 5378}, {1458, 5377}, {1469, 5384}, {1783, 1813}, {1897, 36059}, {2190, 44710}, {2223, 39293}, {2283, 36086}, {2284, 36146}, {2289, 23984}, {2397, 32669}, {2427, 37136}, {3719, 23985}, {3869, 15386}, {3882, 8687}, {3888, 8685}, {4554, 32739}, {4556, 21859}, {4578, 6614}, {4579, 29055}, {4585, 32675}, {4587, 32714}, {5383, 41526}, {6056, 24032}, {6335, 32660}, {6358, 23357}, {6516, 8750}, {6602, 23586}, {7066, 24000}, {16945, 44724}, {17439, 38809}, {18026, 32656}, {18315, 35307}, {19614, 44699}, {23592, 34544}, {23981, 36037}, {23995, 34388}, {24029, 32641}, {32643, 42718}, {36074, 37211}, {36075, 37212}

X(11) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 765}, {1, 6615}, {2, 650}, {2, 4998}, {3, 59}, {4, 44699}, {5, 44710}, {6, 44717}, {7, 514}, {8, 522}, {8, 44724}, {9, 4564}, {11, 100}, {12, 523}, {55, 17115}, {56, 513}, {57, 661}, {63, 40628}, {65, 50330}, {69, 905}, {75, 1577}, {78, 656}, {99, 40625}, {101, 38991}, {108, 5521}, {109, 8054}, {115, 4552}, {125, 23067}, {181, 3005}, {190, 1146}, {200, 6608}, {223, 7045}, {226, 4988}, {244, 4551}, {345, 3239}, {478, 1262}, {480, 3900}, {518, 3126}, {521, 1259}, {644, 35508}, {647, 26942}, {651, 1015}, {653, 5190}, {658, 40615}, {664, 1086}, {665, 34253}, {668, 40624}, {676, 50441}, {692, 39025}, {798, 41526}, {883, 35094}, {900, 1317}, {918, 35509}, {934, 40617}, {1016, 3161}, {1025, 38980}, {1084, 4559}, {1249, 46102}, {1252, 5452}, {1275, 3160}, {1331, 38983}, {1332, 35072}, {1376, 4885}, {1400, 40627}, {1402, 50497}, {1403, 3835}, {1639, 51583}, {1788, 20315}, {1813, 39006}, {1897, 20620}, {2149, 32664}, {2170, 21362}, {2283, 38989}, {2284, 39014}, {2295, 3709}, {2968, 3699}, {3119, 35341}, {3162, 7115}, {3259, 23981}, {3667, 6049}, {3676, 17093}, {3700, 3969}, {3716, 8299}, {3738, 4996}, {3756, 43290}, {3837, 8850}, {3882, 17419}, {3911, 6544}, {3939, 14714}, {3952, 6741}, {3960, 41801}, {4076, 6552}, {4242, 13999}, {4417, 6589}, {4516, 22280}, {4521, 5435}, {4534, 25737}, {4554, 40619}, {4557, 40608}, {4561, 40626}, {4566, 40622}, {4567, 40582}, {4570, 40602}, {4571, 7358}, {4573, 40620}, {4585, 35128}, {4600, 40605}, {4620, 6626}, {5125, 7649}, {5375, 31615}, {5511, 40576}, {5515, 14594}, {6065, 6600}, {6067, 6362}, {6068, 6366}, {6129, 7080}, {6332, 51612}, {6516, 26932}, {6586, 33298}, {6609, 7339}, {6649, 16592}, {6735, 23757}, {7012, 36103}, {7952, 15742}, {14838, 40999}, {15632, 35014}, {17072, 34247}, {17780, 51402}, {18314, 34388}, {18838, 42769}, {21044, 22003}, {21232, 21320}, {23703, 38979}, {34467, 36059}

X(11) = barycentric product X(i)*X(j) for these {i,j}: {1, 4858}, {4, 26932}, {6, 34387}, {7, 1146}, {8, 1086}, {9, 1111}, {10, 17197}, {12, 26856}, {19, 17880}, {21, 16732}, {29, 4466}, {55, 23989}, {56, 23978}, {57, 24026}, {60, 338}, {69, 8735}, {75, 2170}, {76, 3271}, {85, 2310}, {86, 21044}, {92, 7004}, {100, 40166}, {115, 261}, {124, 13478}, {125, 46103}, {141, 18101}, {189, 38357}, {190, 21132}, {210, 16727}, {219, 2973}, {222, 21666}, {244, 312}, {257, 4459}, {264, 7117}, {270, 20902}, {273, 34591}, {274, 4516}, {278, 2968}, {279, 4081}, {281, 1565}, {284, 21207}, {314, 3125}, {318, 3942}, {321, 18191}, {331, 3270}, {333, 3120}, {335, 4124}, {339, 2189}, {345, 2969}, {346, 1358}, {348, 42069}, {479, 23970}, {513, 4391}, {514, 522}, {521, 17924}, {523, 4560}, {646, 764}, {649, 35519}, {650, 693}, {651, 42455}, {652, 46107}, {658, 23615}, {661, 18155}, {663, 3261}, {664, 42462}, {850, 7252}, {885, 918}, {903, 4530}, {905, 44426}, {1015, 3596}, {1016, 7336}, {1019, 4086}, {1021, 4077}, {1022, 4768}, {1088, 3119}, {1090, 4564}, {1109, 2185}, {1118, 23983}, {1275, 5532}, {1364, 2052}, {1365, 7058}, {1367, 36421}, {1440, 5514}, {1459, 46110}, {1461, 23104}, {1509, 4092}, {1577, 3737}, {1639, 6548}, {1647, 4997}, {1826, 17219}, {1977, 40363}, {2051, 34589}, {2150, 23994}, {2218, 17878}, {2320, 4957}, {2321, 17205}, {2325, 6549}, {2401, 2804}, {2481, 17435}, {2618, 39177}, {2995, 38345}, {3023, 40099}, {3035, 31611}, {3063, 40495}, {3064, 4025}, {3121, 40072}, {3122, 28660}, {3123, 27424}, {3124, 18021}, {3239, 3676}, {3248, 28659}, {3452, 40451}, {3613, 27010}, {3615, 8287}, {3669, 4397}, {3675, 36796}, {3699, 6545}, {3700, 7192}, {3701, 16726}, {3716, 4444}, {3756, 6557}, {3762, 23838}, {3790, 43266}, {3900, 24002}, {3910, 4581}, {3937, 7017}, {3939, 23100}, {4041, 7199}, {4089, 36910}, {4373, 4534}, {4451, 7200}, {4518, 27918}, {4522, 4817}, {4551, 40213}, {4582, 6550}, {4608, 4976}, {4811, 47915}, {4904, 6601}, {4939, 8056}, {4953, 27818}, {4985, 47947}, {6063, 14936}, {6332, 7649}, {6506, 7318}, {6556, 40617}, {6591, 35518}, {7068, 36419}, {7155, 21138}, {7178, 7253}, {8748, 17216}, {10015, 43728}, {10566, 48278}, {13426, 22106}, {13454, 22107}, {13609, 36620}, {14618, 23189}, {15413, 18344}, {15416, 43923}, {15633, 34050}, {15634, 40869}, {16082, 35014}, {16596, 40836}, {17094, 17926}, {17877, 39943}, {18210, 31623}, {21140, 36799}, {23062, 24010}, {26563, 40528}, {28132, 43042}, {31628, 42547}, {34234, 35015}, {34434, 40624}, {34896, 37771}, {35174, 46384}, {36795, 42753}, {37222, 51442}, {38347, 40216}, {38362, 44189}, {40086, 47793}, {40437, 46398}, {42754, 51565}, {43735, 46533}

X(11) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4564}, {2, 4998}, {3, 44717}, {4, 46102}, {6, 59}, {7, 1275}, {8, 1016}, {9, 765}, {19, 7012}, {21, 4567}, {25, 7115}, {31, 2149}, {34, 7128}, {41, 1110}, {55, 1252}, {56, 1262}, {57, 7045}, {60, 249}, {86, 4620}, {100, 31615}, {109, 4619}, {115, 12}, {116, 33298}, {124, 4417}, {125, 26942}, {149, 31633}, {216, 44710}, {220, 6065}, {244, 57}, {261, 4590}, {281, 15742}, {284, 4570}, {294, 5377}, {312, 7035}, {314, 4601}, {333, 4600}, {338, 34388}, {346, 4076}, {479, 23586}, {512, 4559}, {513, 651}, {514, 664}, {521, 1332}, {522, 190}, {523, 4552}, {604, 24027}, {647, 23067}, {649, 109}, {650, 100}, {652, 1331}, {657, 3939}, {661, 4551}, {663, 101}, {665, 2283}, {667, 1415}, {673, 39293}, {693, 4554}, {738, 24013}, {764, 3669}, {884, 919}, {885, 666}, {905, 6516}, {918, 883}, {926, 2284}, {1015, 56}, {1019, 1414}, {1021, 643}, {1024, 36086}, {1027, 36146}, {1086, 7}, {1090, 4858}, {1109, 6358}, {1111, 85}, {1118, 23984}, {1146, 8}, {1172, 5379}, {1249, 44699}, {1320, 5376}, {1357, 1407}, {1358, 279}, {1364, 394}, {1365, 6354}, {1397, 23979}, {1407, 7339}, {1459, 1813}, {1509, 7340}, {1565, 348}, {1566, 50441}, {1635, 23703}, {1639, 17780}, {1647, 3911}, {1769, 24029}, {1946, 906}, {1977, 1397}, {2087, 1319}, {2150, 1101}, {2170, 1}, {2175, 23990}, {2185, 24041}, {2189, 250}, {2254, 1025}, {2310, 9}, {2316, 9268}, {2320, 5385}, {2344, 5384}, {2423, 2720}, {2488, 35326}, {2611, 16577}, {2638, 2289}, {2643, 2171}, {2804, 2397}, {2968, 345}, {2969, 278}, {2973, 331}, {3022, 220}, {3023, 6645}, {3063, 692}, {3064, 1897}, {3119, 200}, {3120, 226}, {3121, 1402}, {3122, 1400}, {3123, 1423}, {3124, 181}, {3125, 65}, {3161, 44724}, {3239, 3699}, {3248, 604}, {3261, 4572}, {3269, 7066}, {3270, 219}, {3271, 6}, {3287, 4579}, {3310, 23981}, {3326, 26611}, {3328, 35110}, {3596, 31625}, {3669, 934}, {3675, 241}, {3676, 658}, {3680, 5382}, {3699, 6632}, {3700, 3952}, {3708, 201}, {3709, 4557}, {3716, 3570}, {3733, 4565}, {3737, 662}, {3738, 4585}, {3756, 5435}, {3810, 33946}, {3900, 644}, {3907, 18047}, {3937, 222}, {3942, 77}, {4014, 6180}, {4017, 1020}, {4041, 1018}, {4081, 346}, {4086, 4033}, {4089, 17078}, {4091, 6517}, {4092, 594}, {4124, 239}, {4130, 4578}, {4147, 4595}, {4163, 6558}, {4171, 4069}, {4369, 6649}, {4391, 668}, {4397, 646}, {4403, 7223}, {4435, 3573}, {4459, 894}, {4466, 307}, {4474, 4482}, {4475, 7146}, {4516, 37}, {4521, 43290}, {4522, 3807}, {4526, 23343}, {4528, 30731}, {4530, 519}, {4534, 145}, {4542, 4370}, {4546, 30720}, {4560, 99}, {4581, 6648}, {4582, 6635}, {4705, 21859}, {4768, 24004}, {4814, 4752}, {4820, 4756}, {4834, 36074}, {4858, 75}, {4876, 5378}, {4895, 1023}, {4904, 6604}, {4939, 18743}, {4944, 4767}, {4953, 3161}, {4965, 3759}, {4976, 4427}, {4990, 30729}, {5190, 5125}, {5432, 43986}, {5514, 7080}, {5532, 1146}, {5548, 6551}, {6075, 43043}, {6332, 4561}, {6377, 1403}, {6506, 5552}, {6545, 3676}, {6550, 30725}, {6590, 14594}, {6591, 108}, {6608, 35341}, {6615, 21362}, {6729, 6733}, {6741, 3969}, {7004, 63}, {7023, 23971}, {7058, 6064}, {7063, 7109}, {7117, 3}, {7155, 5383}, {7178, 4566}, {7192, 4573}, {7199, 4625}, {7200, 7176}, {7202, 1442}, {7203, 4637}, {7252, 110}, {7253, 645}, {7336, 1086}, {7337, 23985}, {7649, 653}, {8034, 7180}, {8042, 7203}, {8287, 40999}, {8648, 1983}, {8735, 4}, {8754, 8736}, {11124, 14589}, {11193, 5375}, {11927, 30626}, {11998, 2975}, {14304, 42718}, {14413, 23890}, {14430, 23891}, {14442, 39771}, {14827, 6066}, {14935, 7123}, {14936, 55}, {15280, 28743}, {15615, 39686}, {15635, 34051}, {15914, 43991}, {16726, 1014}, {16732, 1441}, {17059, 17234}, {17096, 4616}, {17197, 86}, {17205, 1434}, {17219, 17206}, {17420, 3882}, {17435, 518}, {17880, 304}, {17924, 18026}, {17926, 36797}, {18021, 34537}, {18101, 83}, {18155, 799}, {18191, 81}, {18210, 1214}, {18344, 1783}, {18771, 38809}, {20975, 2197}, {20982, 2594}, {21044, 10}, {21104, 35312}, {21120, 21272}, {21123, 46153}, {21127, 35338}, {21132, 514}, {21138, 3212}, {21139, 9312}, {21140, 43040}, {21143, 43924}, {21207, 349}, {21666, 7017}, {21789, 5546}, {21950, 4848}, {21963, 28387}, {22106, 13436}, {22107, 13453}, {22383, 36059}, {23062, 24011}, {23189, 4558}, {23615, 3239}, {23760, 31605}, {23761, 47676}, {23764, 30719}, {23838, 3257}, {23970, 5423}, {23978, 3596}, {23983, 1264}, {23989, 6063}, {24002, 4569}, {24010, 728}, {24012, 6602}, {24026, 312}, {24031, 3719}, {24840, 17350}, {26856, 261}, {26932, 69}, {26956, 26872}, {27010, 1078}, {27846, 1429}, {27918, 1447}, {28132, 36802}, {33573, 6745}, {34387, 76}, {34589, 14829}, {34591, 78}, {35015, 908}, {35072, 1259}, {35091, 6068}, {35092, 1317}, {35128, 4996}, {35348, 37139}, {35505, 1362}, {35506, 1682}, {35508, 480}, {35519, 1978}, {36123, 39294}, {36197, 210}, {36798, 5381}, {38345, 3869}, {38347, 1621}, {38357, 329}, {38358, 3681}, {38362, 196}, {38365, 4251}, {38374, 14256}, {38375, 3870}, {38389, 34048}, {38986, 41526}, {38989, 34253}, {39687, 6056}, {39786, 1284}, {40166, 693}, {40213, 18155}, {40451, 40420}, {40528, 23617}, {40608, 2295}, {40615, 17093}, {40621, 6049}, {40626, 51612}, {42067, 608}, {42069, 281}, {42337, 25268}, {42454, 17494}, {42455, 4391}, {42462, 522}, {42753, 1465}, {42754, 22464}, {43728, 13136}, {43921, 1462}, {43923, 32714}, {43924, 1461}, {43929, 32735}, {43932, 4617}, {44311, 32939}, {44426, 6335}, {46101, 3035}, {46103, 18020}, {46107, 46404}, {46384, 3738}, {47694, 14612}, {48278, 4568}, {50333, 42720}, {50512, 36075}, {51402, 51583}, {51442, 30566}

X(11) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5, 12}, {1, 12, 15888}, {1, 119, 10956}, {1, 355, 10944}, {1, 496, 37722}, {1, 1837, 10950}, {1, 2006, 45946}, {1, 5219, 17718}, {1, 5400, 4551}, {1, 5443, 37737}, {1, 5531, 37736}, {1, 5587, 5252}, {1, 5727, 37740}, {1, 5881, 37738}, {1, 5886, 15950}, {1, 6264, 20586}, {1, 6326, 12739}, {1, 7173, 3614}, {1, 7741, 5}, {1, 7951, 495}, {1, 7972, 12735}, {1, 7988, 5219}, {1, 7989, 9578}, {1, 8227, 11375}, {1, 9581, 1837}, {1, 9897, 7972}, {1, 10593, 7173}, {1, 10826, 355}, {1, 10943, 10949}, {1, 10950, 37734}, {1, 10958, 10955}, {1, 16173, 1387}, {1, 17717, 5718}, {1, 17718, 37703}, {1, 17719, 17724}, {1, 17720, 17602}, {1, 17722, 17726}, {1, 23708, 5886}, {1, 26470, 10957}, {1, 26475, 10959}, {1, 26476, 10958}, {1, 37692, 11374}, {1, 37701, 5719}, {1, 37702, 37730}, {1, 37706, 1483}, {1, 37711, 37727}, {1, 37714, 37709}, {1, 37717, 5724}, {1, 37718, 80}, {1, 37720, 496}, {1, 37721, 37739}, {1, 37732, 2594}, {1, 37735, 5901}, {1, 39692, 119}, {1, 50443, 11376}, {1, 50444, 50443}, {2, 55, 5432}, {2, 100, 3035}, {2, 149, 100}, {2, 390, 5218}, {2, 497, 55}, {2, 673, 26007}, {2, 1621, 6690}, {2, 2550, 4413}, {2, 2886, 3925}, {2, 3035, 31235}, {2, 3058, 4995}, {2, 3434, 1376}, {2, 4366, 26629}, {2, 5274, 497}, {2, 5432, 5326}, {2, 11235, 34612}, {2, 11238, 3058}, {2, 11680, 2886}, {2, 23541, 25882}, {2, 26007, 31192}, {2, 26105, 4423}, {2, 26139, 25531}, {2, 26795, 27072}, {2, 26846, 27009}, {2, 31272, 6667}, {2, 33108, 3826}, {3, 499, 5433}, {3, 1479, 6284}, {3, 6284, 15338}, {3, 6713, 21154}, {3, 9668, 4302}, {3, 9669, 1479}, {3, 10525, 11826}, {3, 11928, 10525}, {3, 51517, 10738}, {4, 56, 7354}, {4, 3086, 56}, {4, 4293, 12943}, {4, 10591, 10896}, {4, 10598, 10893}, {4, 10785, 12114}, {4, 12248, 10728}, {4, 47743, 3086}, {5, 12, 3614}, {5, 495, 7951}, {5, 496, 1}, {5, 1837, 10958}, {5, 5533, 1317}, {5, 7741, 7173}, {5, 8227, 7958}, {5, 10593, 7741}, {5, 10943, 355}, {5, 10948, 10944}, {5, 10959, 10955}, {5, 26475, 10950}, {5, 32214, 10942}, {5, 37720, 37722}, {5, 37722, 15888}, {5, 37726, 37725}, {6, 12589, 39873}, {6, 44618, 19024}, {6, 44619, 19023}, {6, 44623, 19030}, {6, 44624, 19029}, {7, 7678, 42356}, {8, 1329, 21031}, {8, 4193, 1329}, {8, 5233, 4023}, {10, 3825, 4187}, {10, 6702, 34122}, {10, 11814, 24003}, {10, 12053, 3057}, {10, 24387, 24390}, {10, 49600, 10914}, {12, 1317, 10956}, {12, 7173, 5}, {12, 10949, 10944}, {12, 10950, 10955}, {12, 10959, 37734}, {12, 37722, 1}, {20, 5225, 12953}, {20, 7288, 5204}, {20, 38693, 38759}, {31, 29662, 37646}, {35, 4857, 15171}, {36, 3582, 15325}, {36, 15325, 5298}, {38, 7069, 24431}, {40, 6922, 50031}, {40, 9614, 12701}, {55, 497, 3058}, {55, 5432, 4995}, {55, 11238, 497}, {56, 10896, 4}, {56, 12943, 4293}, {57, 1699, 1836}, {57, 1836, 11246}, {65, 20118, 12832}, {75, 21580, 21404}, {79, 3337, 24470}, {80, 1387, 1317}, {80, 2006, 14204}, {80, 5443, 45764}, {80, 5533, 37726}, {80, 7741, 39692}, {80, 7972, 9897}, {80, 8068, 119}, {80, 10057, 355}, {80, 16173, 1}, {80, 37718, 12019}, {80, 37720, 5533}, {90, 17437, 24467}, {100, 3035, 6174}, {100, 6667, 31235}, {100, 10707, 149}, {100, 31272, 2}, {104, 10728, 12248}, {104, 13273, 7354}, {116, 17761, 4904}, {119, 23513, 5}, {124, 34589, 26932}, {125, 3270, 26956}, {137, 3327, 14101}, {140, 15171, 35}, {140, 33814, 38760}, {145, 5154, 11681}, {145, 11681, 12607}, {149, 497, 13274}, {149, 3035, 6154}, {149, 3434, 13271}, {149, 6667, 6174}, {149, 31272, 3035}, {149, 45310, 31235}, {153, 388, 12763}, {181, 2051, 10406}, {200, 24392, 4863}, {214, 1125, 34123}, {214, 32557, 1125}, {226, 3817, 17605}, {226, 11019, 354}, {238, 1936, 2361}, {238, 33140, 35466}, {244, 1647, 3756}, {244, 2310, 7004}, {244, 3120, 1086}, {312, 3703, 6057}, {312, 3705, 3703}, {354, 17604, 1864}, {354, 17605, 226}, {354, 17660, 5083}, {355, 10523, 12}, {355, 10948, 10949}, {355, 11373, 1}, {371, 9661, 18965}, {381, 999, 1478}, {381, 10072, 5434}, {381, 12773, 10742}, {381, 18519, 18516}, {388, 3091, 10895}, {388, 14986, 3304}, {390, 5218, 55}, {390, 45043, 20119}, {405, 26363, 24953}, {485, 1124, 19028}, {486, 1335, 19027}, {495, 7951, 12}, {496, 1484, 5533}, {496, 1837, 10959}, {496, 7173, 15888}, {496, 7741, 12}, {496, 10593, 5}, {496, 10826, 10949}, {496, 10943, 10948}, {496, 12019, 37726}, {496, 26476, 10950}, {496, 39692, 1317}, {497, 3434, 10947}, {497, 5218, 390}, {497, 5274, 11238}, {497, 10589, 2}, {499, 1479, 3}, {499, 9669, 6284}, {499, 10058, 6713}, {546, 18990, 3585}, {547, 15170, 3584}, {590, 2066, 13901}, {590, 48714, 13922}, {613, 1352, 39897}, {615, 5414, 13958}, {615, 48715, 13991}, {631, 4294, 5217}, {631, 13199, 34474}, {650, 38347, 14936}, {693, 40619, 23989}, {936, 25522, 24954}, {938, 6828, 15844}, {942, 9955, 12047}, {942, 12047, 3649}, {944, 6941, 18242}, {946, 1210, 65}, {946, 12616, 12672}, {946, 16174, 38038}, {950, 1125, 2646}, {950, 2646, 10543}, {960, 6734, 21677}, {962, 1788, 37567}, {962, 5704, 1788}, {982, 3944, 3782}, {999, 1478, 5434}, {999, 12773, 10074}, {1056, 3545, 10590}, {1056, 10590, 11237}, {1058, 3085, 3303}, {1058, 3090, 3085}, {1086, 3756, 244}, {1111, 1565, 1358}, {1125, 17647, 17614}, {1125, 25639, 442}, {1125, 33709, 32557}, {1145, 34122, 10}, {1146, 2170, 4534}, {1146, 6506, 5514}, {1146, 13609, 3119}, {1210, 10265, 20118}, {1317, 23513, 3614}, {1329, 3813, 8}, {1329, 3847, 4193}, {1376, 3434, 34612}, {1376, 11235, 3434}, {1376, 13205, 100}, {1387, 5533, 37722}, {1387, 10593, 23513}, {1387, 39692, 12}, {1421, 2006, 15253}, {1478, 10072, 999}, {1479, 4302, 9668}, {1479, 5433, 15338}, {1484, 8068, 1317}, {1484, 10057, 10949}, {1484, 10593, 39692}, {1484, 16173, 37722}, {1484, 23513, 37725}, {1500, 1506, 31460}, {1537, 38038, 946}, {1647, 3120, 244}, {1656, 3295, 498}, {1656, 12331, 38752}, {1699, 1768, 34789}, {1699, 8727, 7965}, {1699, 17728, 11246}, {1738, 5121, 16610}, {1768, 11219, 13226}, {1836, 17728, 57}, {1837, 11376, 1}, {1837, 37740, 5727}, {2170, 3119, 38375}, {2170, 21044, 1146}, {2170, 33573, 43960}, {2310, 3120, 38357}, {2310, 4459, 24840}, {2476, 3616, 25466}, {2478, 10527, 958}, {2607, 21381, 14985}, {2886, 3816, 2}, {2886, 3826, 33108}, {2886, 3829, 11680}, {2886, 15842, 1376}, {2886, 25652, 27692}, {2968, 24026, 4081}, {3006, 4358, 3932}, {3035, 6667, 2}, {3035, 45310, 6667}, {3057, 17606, 10}, {3058, 5432, 55}, {3073, 3075, 1399}, {3086, 10591, 4}, {3086, 10598, 18961}, {3086, 10896, 7354}, {3090, 38665, 20400}, {3091, 14986, 388}, {3091, 38669, 38757}, {3100, 9629, 5160}, {3119, 33573, 13609}, {3149, 48482, 6253}, {3295, 12331, 10087}, {3297, 42265, 31472}, {3298, 42262, 44622}, {3304, 10895, 388}, {3333, 9612, 10404}, {3434, 10584, 2}, {3436, 10529, 12513}, {3452, 4847, 210}, {3452, 24386, 4847}, {3486, 3616, 34471}, {3582, 3583, 36}, {3583, 15325, 15326}, {3585, 5563, 18990}, {3600, 3832, 5229}, {3600, 5229, 9657}, {3614, 15888, 12}, {3628, 51525, 38763}, {3666, 24210, 4854}, {3673, 17181, 3665}, {3685, 32851, 3712}, {3687, 3706, 4046}, {3705, 20545, 20487}, {3720, 33105, 17056}, {3741, 3846, 1211}, {3742, 3838, 5249}, {3813, 3847, 1329}, {3813, 4193, 21031}, {3816, 3829, 2886}, {3816, 11680, 3925}, {3816, 15845, 55}, {3817, 11019, 226}, {3825, 24387, 10}, {3826, 33108, 3925}, {3841, 19862, 17529}, {3851, 7373, 9654}, {3912, 20544, 20486}, {3936, 29824, 4966}, {4187, 24390, 10}, {4202, 26094, 25914}, {4293, 12943, 7354}, {4302, 9668, 6284}, {4304, 10165, 37600}, {4413, 31140, 2550}, {4423, 31245, 2}, {4514, 7081, 4030}, {4551, 5400, 45885}, {4858, 4939, 24026}, {4871, 21241, 3836}, {4995, 5326, 5432}, {5055, 6767, 31479}, {5083, 18240, 354}, {5083, 21635, 12831}, {5187, 10529, 3436}, {5204, 9671, 12953}, {5204, 12953, 20}, {5211, 37759, 32922}, {5217, 9670, 4294}, {5225, 7288, 20}, {5226, 10580, 3475}, {5272, 17064, 24789}, {5274, 10589, 55}, {5274, 11680, 15845}, {5274, 45043, 10707}, {5281, 10385, 55}, {5298, 15326, 36}, {5433, 6284, 3}, {5435, 9812, 3474}, {5531, 7988, 15017}, {5531, 15017, 5660}, {5531, 37736, 41701}, {5533, 7741, 119}, {5533, 8068, 1}, {5533, 23513, 10956}, {5587, 6264, 12751}, {5587, 37704, 1}, {5603, 6830, 7680}, {5603, 12247, 10698}, {5603, 18391, 2099}, {5697, 15079, 18395}, {5697, 18395, 5690}, {5722, 5886, 1}, {5722, 23708, 15950}, {5727, 37740, 10950}, {5745, 40998, 3683}, {5748, 36845, 25568}, {5886, 6265, 11729}, {5901, 8070, 12}, {5901, 37730, 1}, {5902, 18393, 39542}, {5927, 17626, 17625}, {6154, 6174, 100}, {6154, 31235, 6174}, {6174, 31235, 3035}, {6224, 32558, 3616}, {6667, 10707, 6154}, {6667, 45310, 31272}, {6684, 10624, 37568}, {6690, 49736, 1621}, {6702, 21630, 1145}, {6713, 10738, 24466}, {6734, 41012, 960}, {6736, 21627, 3893}, {6767, 31479, 10056}, {6833, 10531, 11496}, {6834, 12116, 11500}, {6928, 11249, 11827}, {7004, 35015, 38357}, {7173, 37722, 12}, {7191, 33133, 17061}, {7491, 26286, 30264}, {7741, 8068, 23513}, {7741, 9581, 26476}, {7741, 16173, 8068}, {7741, 26475, 10958}, {7741, 37702, 8070}, {7741, 37720, 1}, {7741, 37722, 3614}, {7951, 9897, 11698}, {7956, 8727, 1699}, 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