X371

KENMOTU POINT (CONGRUENT SQUARES POINT)

There exist three congruent squares U, V, W positioned in ABC as follows: U has opposing vertices on segments AB and AC; V has opposing vertices on segments BC and BA; W has opposing vertices on segments CA and CB, and there is a single point common to U, V, W. The common point, X371, may have first been published in Kenmotu’s Collection of Sangaku Problems in 1840, indicating that its first appearance may have been anonymously inscribed on a wooden board hung up in a Japanese shrine or temple. (The Kenmotu configuration uses only half-squares; i.e., isosceles right triangles). Trilinears were found by John Rigby.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5652, 0.2097) A B C X371 (a, b, c) = (5, 6, 7) | AB = 1

X371 (KENMOTU POINT (CONGRUENT SQUARES POINT)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Trilinears: \(\cos(A - \pi /4) : :\)
Barycentrics: \(\sin A \cos(A - \pi /4) : :\)
Search Value (6, 9, 13): \(2.432560433445\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(\cos A + \sin A : :\)

  • Trilinears: \(a(S_{A} + S) : :\)

  • Trilinears: \(a(b^{2} + c^{2} - a^{2} + 2S) : :\)

▸ Linear Combinations4 combinations

  • X371 = \(L_{A}/R_{A} + L_{B}/R_{B} + L_{C}/R_{C} -\) X3 / \(R\), where LA, LB, LC are the centers of the Lucas circles, and \(R_{A}\), \(R_{B}\), \(R_{C}\) their radii (Randy Hutson, July 23, 2015)

  • X371 = \(L_{A}/R_{A} + L_{B}/R_{B} + L_{C}/R_{C} - L_{R}/R_{R}\), where LR and \(R_{R}\) are the center and radius of the Lucas radical circle (Randy Hutson, July 23, 2015)

  • X371 = \(3\) X2 + X43134, \(2\) X639 + X43134, \(3\) X6 - \(4\) X44482, \(3\) X6 - \(2\) X44502, X1152 - \(3\) X12963, \(2\) X1504 + X40274, X6396 - \(4\) X9675, \(2\) X6398 - \(5\) X41410, X11825 - \(4\) X43120, X11825 + \(2\) X45489, \(3\) X35840 - \(8\) X44482, \(3\) X35840 - \(4\) X44502, \(2\) X43120 + X45489, X3070 - \(3\) X32787, \(2\) X3070 - \(3\) X35822, X3070 + \(3\) X41945, \(2\) X3070 + X42266, \(3\) X3070 - X42272, \(5\) X3070 + \(3\) X43210, X3070 + \(6\) X52047, \(2\) X7583 - \(3\) X32787, \(4\) X7583 - X35820, \(4\) X7583 - \(3\) X35822, \(2\) X7583 + \(3\) X41945, \(2\) X7583 + X42258, \(4\) X7583 + X42266, \(6\) X7583 - X42272, \(10\) X7583 + \(3\) X43210, X7583 + \(3\) X52047, \(6\) X32787 - X35820, \(3\) X32787 + X42258, \(6\) X32787 + X42266, \(9\) X32787 - X42272, \(5\) X32787 + X43210, X32787 + \(2\) X52047, X35820 - \(3\) X35822, X35820 + \(6\) X41945, X35820 + \(2\) X42258, \(3\) X35820 - \(2\) X42272, \(5\) X35820 + \(6\) X43210, X35820 + \(12\) X52047, X35822 + \(2\) X41945, \(3\) X35822 + \(2\) X42258, \(3\) X35822 + X42266, \(9\) X35822 - \(2\) X42272, \(5\) X35822 + \(2\) X43210, and many others

  • Floor van Lamoen, Vierkanten in een driehoik: 3. Congruente vierkanten Tony Rothman, with the cooperation of Hidetoshi Fukagawa, Japanese Temple Geometry (feature article in Scientific American)

▸ Functional Relations46 relations

  • X371 = midpoint of Xi and Xj for these {i,j}: {3, 45489}, {4, 26441}, {637, 43134}, {638, 20065}, {1151, 12962}, {3070, 42258}, {7969, 49226}, {12375, 35826}, {12964, 49250}, {32787, 41945}, {35610, 35641}, {35820, 42266}, {35824, 35878}, {35828, 35880}, {35856, 35882}, {35949, 45420}, {48700, 48714}, {49212, 49266}, {49216, 49268}, {49218, 49270}

  • X371 = reflection of Xi in Xj for these {i,j}: {3, 43120}, {315, 640}, {372, 32}, {637, 639}, {3070, 7583}, {10962, 42866}, {11825, 3}, {12375, 49268}, {12965, 49256}, {35610, 49226}, {35641, 7969}, {35698, 49214}, {35753, 49208}, {35820, 3070}, {35822, 32787}, {35824, 49212}, {35826, 49216}, {35828, 49218}, {35830, 49220}, {35832, 44647}, {35834, 49222}, {35836, 49224}, {35838, 49230}, {35840, 6}, {35842, 49232}, {35844, 49234}, {35846, 49236}, {35848, 49238}, {35850, 49210}, {35852, 49240}, {35854, 49242}, {35856, 48700}, {35858, 49244}, {35860, 49246}, {35862, 49248}, {35864, 49250}, {35866, 49252}, {35868, 49254}, {35870, 49258}, {35872, 49260}, {35874, 49262}, {35876, 49264}, {35878, 49266}, {35880, 49270}, {35882, 48714}, {39893, 49228}, {41945, 52047}, {42009, 641}, {42266, 42258}, {44486, 575}, {44502, 44482}, {49222, 46688}, {49226, 31439}, {49601, 13883}

  • X371 = isogonal conjugate of X485

  • X371 = isotomic conjugate of X34391

  • X371 = complement of X637

  • X371 = anticomplement of X639

  • X371 = circumcircle-inverse of X2459

  • X371 = Brocard-circle-inverse of X372

  • X371 = 1st-Lemoine-circle-inverse of X2461

  • X371 = Schoutte-circle-inverse of X6200

  • X371 = 2nd-Brocard-circle-inverse of X3103

  • X371 = Lucas-circles-radical-circle-inverse of X2460

  • X371 = isogonal conjugate of the anticomplement of X641

  • X371 = isogonal conjugate of the complement of X488

  • X371 = isogonal conjugate of the isotomic conjugate of X492

  • X371 = isotomic conjugate of the polar conjugate of X5413

  • X371 = isogonal conjugate of the polar conjugate of X1585

  • X371 = polar conjugate of the isotomic conjugate of X5408

  • X371 = polar conjugate of the isogonal conjugate of X8911

  • X371 = Thomson-isogonal conjugate of X13712

  • X371 = orthic-isogonal conjugate of X372

  • X371 = psi-transform of X7599

  • X371 = X31-complementary conjugate of X10962

  • X371 = Xi-Ceva conjugate of Xj for these (i,j): {2, 10962}, {4, 372}, {492, 5408}, {588, 6}, {1585, 5413}, {3068, 8854}, {5412, 8855}, {5417, 6420}, {18819, 8577}, {26912, 23248}

  • X371 = Xi-cross conjugate of Xj for these (i,j): {1147, 372}, {8911, 5408}, {12239, 4}

  • X371 = Xi-isoconjugate of Xj for these (i,j): {1, 485}, {7, 13455}, {19, 11090}, {31, 34391}, {63, 41515}, {75, 8577}, {91, 372}, {92, 6413}, {486, 3377}, {1577, 39383}, {1586, 1820}, {1953, 16032}, {8769, 8944}, {19218, 24246}

  • X371 = Xi-Dao conjugate of Xj for these (i,j): {2, 10962}, {2, 34391}, {3, 485}, {6, 11090}, {69, 5409}, {206, 8577}, {371, 637}, {372, 34116}, {577, 5409}, {1504, 42009}, {3162, 41515}, {5392, 24245}, {5408, 13441}, {6413, 22391}, {10960, 13439}

  • X371 = cevapoint of Xi and Xj for these (i,j): {3, 8909}, {6, 10132}, {3311, 8826}

  • X371 = crosspoint of Xi and Xj for these (i,j): {4, 13429}, {492, 1585}, {3387, 3388}

  • X371 = crosssum of Xi and Xj for these (i,j): {3, 10665}, {6, 3155}, {590, 8035}, {3385, 3386}, {6413, 8577}

  • X371 = crossdifference of every pair of points on line {523, 17431}

  • X371 = perspector of ABC and the Lucas tangents triangle

  • X371 = perspector of the Lucas central triangle and the anticevian triangle of X6

  • X371 = perspector of the Lucas inner triangle and Lucas(-1:1) tangents triangle

  • X371 = perspector of the Lucas(4:3) central triangle and the circumcevian triangle of X6

  • X371 = perspector of the Lucas central triangle and the cevian triangle of X588

  • X371 = radical center of the Lucas(2:1) circles

  • X371 = X481-of-Lucas-central-triangle

  • X371 = perspector of ABC and 2nd Lucas(-1) secondary tangents triangle

  • X371 = perspector of tangential triangle and Lucas secondary central triangle

  • X371 = perspector of Lucas inner tangential triangle and Lucas(-1) central triangle

  • X371 = inverse-in-Lucas-radical-circle of X2460

  • X371 = exsimilicenter of circumcircle and Lucas radical circle

  • X371 = barycentric product Xi*Xj for these {i,j}: {3, 1585}, {4, 5408}, {6, 492}, {24, 11091}, {32, 45805}, {52, 16037}, {69, 5413}, {264, 8911}, {317, 6414}, {372, 13428}, {485, 1599}, {486, 1993}, {491, 44193}, {493, 39387}, {571, 34392}, {588, 641}, {1124, 13457}, {1306, 14325}, {1586, 10666}, {5409, 13429}, {5412, 13430}, {6563, 39384}, {7763, 8576}, {9723, 41516}, {11547, 26922}

  • X371 = barycentric quotient Xi/Xj for these {i,j}: {2, 34391}, {3, 11090}, {6, 485}, {24, 1586}, {25, 41515}, {32, 8577}, {41, 13455}, {54, 16032}, {184, 6413}, {372, 13439}, {486, 5392}, {492, 76}, {571, 372}, {1147, 5409}, {1576, 39383}, {1585, 264}, {1599, 492}, {1993, 491}, {3053, 8944}, {5062, 8035}, {5408, 69}, {5409, 13441}, {5411, 1321}, {5412, 13440}, {5413, 4}, {6414, 68}, {7763, 45806}, {8576, 2165}, {8911, 3}, {8950, 493}, {10132, 24246}, {10666, 11091}, {10962, 637}, {11091, 20563}, {13428, 34392}, {16037, 34385}, {26920, 10665}, {32568, 45472}, {39384, 925}, {41411, 21463}, {41516, 847}, {44077, 5412}, {44193, 486}, {45805, 1502}

  • X371 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {1, 1702, 35775}, {1, 2066, 35808}, {1, 2067, 35768}, {1, 6204, 8953}, {1, 35642, 35811}, {1, 35775, 35642}, {1, 45527, 45587}, {1, 45529, 45585}, {1, 45531, 45716}, {1, 45641, 35819}, {1, 45643, 35817}, {1, 45716, 45573}, {2, 486, 10577}, {2, 637, 639}, {2, 1588, 486}, {2, 9540, 5418}, {2, 43134, 637}, {2, 45509, 642}, {3, 6, 372}, {3, 182, 45552}, {3, 372, 6396}, {3, 1151, 6200}, {3, 1160, 12305}, {3, 1161, 1350}, {3, 1351, 9733}, {3, 1504, 35840}, {3, 3102, 45565}, {3, 3103, 40275}, {3, 3311, 6}, {3, 3312, 1152}, {3, 3592, 6419}, {3, 5050, 43118}, {3, 5093, 12314}, {3, 6199, 3311}, {3, 6221, 1151}, {3, 6395, 6450}, {3, 6398, 6410}, {3, 6407, 6449}, {3, 6408, 6452}, {3, 6417, 3312}, {3, 6418, 6398}, {3, 6419, 6420}, {3, 6420, 6454}, {3, 6422, 3103}, {3, 6423, 2459}, {3, 6425, 6453}, {3, 6427, 3594}, {3, 6428, 6426}, {3, 6429, 6484}, {3, 6431, 35770}, {3, 6432, 6481}, {3, 6445, 6455}, {3, 6446, 6497}, {3, 6447, 6425}, {3, 6449, 6409}, {3, 6450, 6412}, {3, 6455, 6411}, {3, 6474, 9690}, {3, 6480, 6486}, {3, 6500, 6395}, {3, 9691, 6445}, {3, 9732, 11824}, {3, 9733, 45498}, {3, 9738, 45499}, {3, 10897, 11513}, {3, 11916, 1160}, {3, 12313, 9732}, {3, 18457, 10897}, {3, 26348, 5085}, {3, 30435, 39679}, {3, 43119, 45553}, {3, 45410, 43121}, {3, 45411, 182}, {3, 45488, 9739}, {3, 45578, 43141}, {3, 45579, 9738}, {4, 485, 6564}, {4, 3068, 485}, {4, 5412, 35764}, {4, 6459, 6561}, {4, 6561, 35821}, {4, 6564, 35786}, {4, 10880, 5412}, {4, 13886, 31412}, {4, 31412, 42269}, {4, 42228, 42232}, {4, 42230, 42234}, {4, 45511, 48467}, {4, 45524, 48735}, {4, 46621, 6413}, {5, 590, 10576}, {5, 3071, 6565}, {5, 8981, 590}, {5, 31454, 35812}, {5, 42215, 3071}, {5, 48773, 45555}, {6, 39, 45513}, {6, 372, 6420}, {6, 1151, 3}, {6, 1152, 3312}, {6, 1160, 35794}, {6, 1161, 35792}, {6, 3053, 6423}, {6, 3311, 6419}, {6, 3312, 35770}, {6, 3592, 3311}, {6, 3594, 6418}, {6, 5013, 6421}, {6, 5058, 45514}, {6, 5085, 19146}, {6, 5210, 8376}, {6, 6200, 6396}, {6, 6221, 6200}, {6, 6395, 6436}, {6, 6409, 1152}, {6, 6410, 3594}, {6, 6411, 6398}, {6, 6412, 6395}, {6, 6419, 35771}, {6, 6422, 45512}, {6, 6423, 45515}, {6, 6425, 1151}, {6, 6426, 6432}, {6, 6429, 6409}, {6, 6431, 6417}, {6, 6432, 6428}, {6, 6433, 6412}, {6, 6437, 6221}, {6, 6451, 6481}, {6, 6468, 6411}, {6, 6469, 6442}, {6, 6470, 6431}, {6, 6488, 6497}, {6, 6567, 45499}, {6, 8414, 43119}, {6, 9601, 15815}, {6, 9675, 41410}, {6, 9732, 3103}, {6, 9733, 45462}, {6, 11477, 9974}, {6, 12962, 1504}, {6, 12963, 32}, {6, 12968, 5062}, {6, 45579, 45564}, {10, 48765, 45547}, {10, 49602, 35789}, {11, 18965, 9661}, {12, 13901, 9646}, {14, 35731, 18586}, {15, 16, 6200}, {15, 61, 372}, {15, 62, 3389}, {15, 3364, 3}, {15, 35739, 5238}, {16, 61, 3364}, {16, 62, 372}, {16, 3389, 3}, {20, 1587, 6560}, {20, 6560, 42267}, {20, 7585, 1587}, {20, 9541, 42260}, {20, 42522, 7585}, {20, 43512, 9541}, {25, 493, 8956}, {25, 3092, 35765}, {32, 1504, 6}, {32, 9675, 12963}, {32, 12963, 41410}, {32, 40274, 6396}, {32, 45489, 35840}, {35, 3301, 5414}, {36, 3299, 6502}, {39, 182, 372}, {39, 1151, 45499}, {39, 5058, 6}, {39, 6567, 9738}, and many others

Notes

The edgelength of the three squares is \(2^{1/2}\)abc/(\(a^{2}\) + \(b^{2}\) + \(c^{2}\) + 4σ), where σ = area(ABC). (Edward Brisse, 2/12/2000)

X371 is the internal center of similitude of the circumcircle and the 2nd Lemoine circle (cosine circle) (Peter J. C. Moses, 5/9/03). Also, X371 is the internal center of similitude of the Gallatly circle (defined just before X2007) and the 1st Lemoine circle (Peter J. C. Moses, 9/10/2003).

X371 is the perspector of several pairs of triangles associated with Lucas circles. Some of these triangles are defined at MathWorld; e.g. Lucas Central Triangle, and others are Lucas(L:W) configurations. The latter are generalizations of configurations associated with Lucas circles, in which the squares are replaced by rectangles of length-to-width ratio L:W, with length on the corresponding sideline of ABC. A negative ratio indicates that the rectangles are directed inward; e.g. Lucas(-1:1) indicates inward-directed squares, whereas Lucas(1:1) indicates the classical case of outward-directed squares. These generalizations and the following properties of X371 were contributed by Randy Hutson, 9/23/2011. See also X372 and X1084.

Hidetoshi Fukagawa and John F. Rigby, Traditional Japanese Mathematics Problems of the 18th and 19th Centuries, SCT Publishing, Singapore, 2002. Reviewed, together with the Fukagawa and Pedoe book, Japanese Tempe Geometry Problems: San Gaku, by Clark Kimberling in The Mathematical Intelligencer 28, no. 1 (Winter 2006) 61-63.

Let A'B'C' be the Lucas central triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X371. (Randy Hutson, July 23, 2015)

X371 X371 lies on the cubics K006, K233, K250, K337, K415, K424a, K633, K897, K1197, K1257, the curve Q168, and these lines: {1, 1702}, {2, 486}, {3, 6}, {4, 485}, {5, 590}, {8, 35843}, {10, 13912}, {11, 9661}, {12, 9646}, {13, 3391}, {14, 3366}, {17, 3367}, {18, 3392}, {20, 1587}, {22, 9683}, {24, 5413}, {25, 493}, {30, 3070}, {35, 3301}, {36, 3299}, {40, 5415}, {46, 2362}, {51, 3156}, {54, 6414}, {55, 1335}, {56, 1124}, {64, 6415}, {67, 19397}, {68, 35837}, {69, 11292}, {74, 19111}, {76, 35867}, {79, 35855}, {80, 35853}, {81, 16441}, {83, 22718}, {84, 35845}, {86, 21909}, {90, 7133}, {98, 13885}, {99, 19056}, {100, 19082}, {104, 19113}, {110, 7598}, {112, 19094}, {113, 8998}, {114, 6289}, {115, 8980}, {119, 13922}, {125, 8994}, {127, 13918}, {132, 13923}, {140, 615}, {155, 8909}, {157, 30398}, {165, 1703}, {184, 3155}, {185, 21640}, {186, 10881}, {193, 488}, {194, 22716}, {212, 3077}, {230, 49029}, {235, 13884}, {238, 8225}, {254, 41516}, {256, 8331}, {262, 10850}, {265, 35835}, {291, 8347}, {315, 491}, {355, 13911}, {376, 6460}, {378, 11474}, {381, 8976}, {382, 13665}, {394, 1583}, {395, 15765}, {396, 18585}, {397, 14814}, {398, 14813}, {399, 44592}, {402, 35791}, {427, 8280}, {474, 31473}, {489, 7389}, {490, 32421}, {492, 641}, {494, 35805}, {498, 44622}, {499, 44624}, {515, 13883}, {517, 7969}, {542, 32291}, {546, 13925}, {547, 43211}, {549, 13966}, {550, 19117}, {588, 3060}, {601, 606}, {602, 605}, {603, 3076}, {626, 11314}, {631, 3069}, {632, 32790}, {638, 11294}, {671, 35699}, {736, 49352}, {754, 1991}, {760, 45714}, {936, 31438}, {940, 16433}, {944, 19066}, {946, 8983}, {952, 35842}, {958, 1378}, {971, 35844}, {997, 30556}, {999, 3297}, {1092, 9686}, {1131, 1327}, {1132, 5056}, {1147, 9676}, {1154, 12965}, {1181, 19355}, {1296, 11835}, {1297, 19115}, {1328, 3316}, {1344, 2466}, {1345, 2465}, {1352, 37343}, {1376, 1377}, {1385, 7968}, {1478, 13905}, {1479, 13904}, {1482, 35810}, {1490, 19068}, {1498, 17819}, {1503, 36709}, {1511, 49269}, {1584, 5407}, {1585, 8968}, {1589, 11433}, {1590, 11427}, {1593, 3093}, {1594, 26951}, {1598, 13889}, {1599, 1993}, {1600, 5419}, {1614, 26886}, {1656, 8253}, {1657, 18512}, {1658, 11266}, {1672, 2008}, {1673, 2007}, {1676, 2010}, {1677, 2009}, {1680, 2014}, {1681, 2013}, {1699, 13888}, {1700, 2018}, {1701, 2017}, {1707, 6213}, {1708, 13388}, {1742, 31544}, {1743, 32555}, {1843, 19358}, {1870, 26948}, {1916, 8315}, {2015, 2472}, {2016, 2471}, {2041, 40693}, {2042, 40694}, {2043, 10653}, {2044, 10654}, {2045, 42149}, {2046, 42152}, {2077, 26465}, {2242, 31471}, {2544, 2547}, {2545, 2546}, {2548, 31463}, {2777, 13287}, {2782, 35824}, {2794, 13749}, {3008, 31541}, {3090, 23273}, {3091, 8972}, {3100, 9631}, {3146, 22644}, {3167, 8912}, {3235, 3238}, {3236, 3237}, {3295, 3298}, {3317, 3533}, {3357, 49251}, {3387, 42783}, {3388, 42784}, {3425, 8989}, {3428, 19014}, {3515, 5411}, {3522, 9543}, {3523, 7586}, {3524, 19053}, {3525, 13939}, {3526, 8252}, {3528, 9693}, {3529, 23267}, {3530, 35256}, {3534, 43339}, {3563, 39384}, {3564, 32497}, {3567, 26894}, {3574, 8995}, {3576, 9615}, {3579, 49227}, {3590, 12819}, {3591, 43559}, {3618, 11291}, {3627, 42225}, {3628, 18762}, {3629, 40288}, {3664, 31540}, {3788, 11315}, {3832, 23263}, {3843, 45384}, {3845, 42417}, {3859, 41948}, {3972, 35939}, {4293, 31408}, {4297, 49548}, {4317, 31475}, {4383, 16432}, {4769, 45445}, {4845, 30335}, {5054, 13847}, {5066, 42606}, {5204, 18995}, {5217, 19037}, {5254, 12960}, {5286, 21737}, {5318, 42176}, {5321, 35740}, {5334, 35732}, {5335, 42172}, {5339, 42279}, {5340, 42278}, {5393, 31562}, {5416, 10902}, {5417, 5446}, {5432, 9648}, {5433, 9663}, {5461, 13663}, {5473, 19074}, {5474, 19076}, {5475, 31481}, {5478, 13917}, {5479, 13916}, {5480, 13910}, {5587, 13893}, {5591, 7375}, {5597, 35781}, {5598, 35779}, {5603, 13902}, {5640, 35300}, {5657, 19065}, {5663, 12375}, {5690, 49233}, {5860, 6216}, {5861, 26619}, {5870, 7374}, {5881, 31440}, {5889, 26912}, {5890, 6458}, {5989, 8304}, {6000, 11241}, {6102, 44612}, {6145, 35859}, {6179, 6312}, {6197, 26952}, {6198, 9632}, {6201, 8975}, {6202, 7000}, {6222, 9756}, {6228, 44365}, {6241, 11462}, {6245, 8987}, {6246, 8988}, {6247, 8991}, {6248, 8992}, {6249, 8993}, {6251, 13921}, {6281, 13650}, {6284, 9660}, {6304, 33352}, {6305, 33350}, {6401, 11977}, {6402, 11976}, {6406, 45840}, {6416, 14528}, {6515, 11090}, {6525, 22838}, {6642, 10963}, {6680, 11316}, {6684, 13936}, {6699, 46689}, {6750, 8955}, {6759, 10533}, {6771, 49209}, {6774, 49211}, {6776, 21736}, {6811, 9744}, {6813, 9753}, {7160, 35863}, {7354, 9647}, {7387, 35776}, {7488, 11418}, {7494, 18290}, {7592, 26920}, {7599, 7601}, {7691, 19096}, {7709, 32471}, {7737, 31411}, {7738, 12123}, {7759, 51395}, {7762, 35685}, {7773, 32432}, {7834, 11313}, {7851, 32433}, {7887, 32435}, {7981, 45719}, {7987, 9585}, {8149, 49351}, {8196, 13890}, {8203, 13891}, {8212, 13899}, {8213, 13900}, {8277, 19005}, {8301, 8336}, {8305, 8308}, {8306, 8317}, {8320, 8326}, {8321, 8324}, {8322, 8333}, {8337, 8340}, {8338, 8349}, {8703, 41946}, {8825, 10237}, {8884, 19183}, {8908, 10282}, {8943, 15805}, {8946, 14248}, {8967, 12147}, {8978, 13389}, {9441, 31545}, {9584, 16192}, {9593, 31427}, {9618, 35242}, {9620, 31437}, {9649, 15326}, {9662, 15338}, {9692, 15717}, {9694, 17928}, {9695, 10323}, {9709, 31485}, {9751, 19091}, {9752, 9757}, {9880, 13908}, {9922, 45400}, {9927, 13909}, {9932, 12425}, {9993, 13892}, {10109, 43435}, {10113, 13915}, {10133, 11402}, {10164, 13975}, {10165, 13971}, {10246, 44636}, {10266, 35871}, {10267, 19049}, {10269, 19047}, {10299, 43510}, {10303, 13941}, {10304, 43511}, {10310, 19000}, {10525, 35796}, {10526, 35798}, {10531, 13906}, {10532, 13907}, {10606, 19087}, {10610, 49257}, {10665, 13754}, {10669, 12979}, {10673, 35806}, {10679, 35816}, {10680, 35818}, {10799, 12840}, {10817, 15055}, {10820, 15035}, {10840, 10843}, {10841, 10852}, {10893, 13895}, {10894, 13896}, {10895, 13897}, {10896, 13898}, {11001, 42414}, {11012, 26464}, {11202, 11242}, {11248, 19048}, {11249, 19050}, {11251, 35790}, {11252, 35778}, {11253, 35780}, {11257, 19090}, {11414, 19006}, {11447, 12111}, {11448, 11449}, {11463, 11464}, {11496, 13887}, {11539, 43887}, {11542, 42251}, {11543, 35738}, {11812, 43212}, {11822, 19008}, {11823, 19010}, {11826, 19024}, {11827, 19026}, {11828, 19032}, {11829, 19034}, {11833, 38698}, {11836, 38716}, {11897, 13894}, {11941, 11967}, {11942, 11963}, {11943, 11970}, {11944, 11965}, {11959, 11981}, {11961, 11980}, {12041, 49217}, {12042, 49213}, {12100, 52046}, {12110, 22722}, {12117, 19058}, {12118, 19062}, {12119, 19078}, {12120, 19086}, {12121, 19052}, {12122, 19092}, {12124, 19103}, {12160, 51946}, {12164, 19492}, {12256, 14912}, {12297, 22646}, {12359, 49225}, {12424, 35836}, {12514, 30557}, {12556, 19098}, {12584, 32292}, {12590, 19588}, {12599, 13914}, {12600, 13919}, {12601, 13881}, {12619, 49241}, {12788, 49078}, {12818, 14241}, {12835, 12841}, {12836, 12838}, {12837, 12839}, {12891, 17702}, {12892, 12893}, {12961, 13045}, {12966, 12972}, {12967, 12973}, {13047, 13049}, {13048, 13050}, {13134, 26518}, {13135, 26517}, {13288, 13289}, {13367, 21641}, {13630, 43867}, {13644, 45440}, {13666, 22541}, {13674, 45023}, {13687, 13920}, {13712, 26288}, {13748, 36656}, {13758, 45577}, {13786, 19100}, {13807, 13848}, {13835, 26620}, {13938, 22723}, {13943, 19409}, {13947, 31423}, {13958, 31500}, {13961, 15720}, {13967, 38737}, {13969, 38727}, {13973, 26446}, {13976, 38133}, {13977, 21154}, {13979, 34128}, {13980, 23328}, {13983, 15819}, {13989, 38748}, {13990, 38793}, {13991, 38760}, {14226, 43343}, {14233, 15846}, {14561, 37342}, {14893, 41952}, {14996, 21565}, {14997, 21568}, {15022, 43792}, {15061, 19051}, {15141, 19384}, {15640, 43342}, {15686, 43209}, {15690, 42418}, {15692, 42523}, {15702, 43255}, {15704, 42226}, {15712, 41961}, {15716, 43258}, {15723, 42579}, {15764, 43229}, {15803, 51842}, {16029, 16035}, {16030, 16034}, {16113, 19080}, {16125, 16148}, {16202, 44646}, {16203, 44644}, {16419, 34515}, {16440, 32911}, {16809, 35730}, {16962, 36469}, {16963, 36452}, {16964, 42238}, {16965, 42237}, {17277, 21992}, {17809, 21097}, {17818, 19022}, {17820, 17821}, {18400, 32384}, {18539, 22625}, {18581, 42246}, {18582, 42247}, {18909, 18923}, {18916, 26873}, {18924, 18925}, {18980, 19436}, {18981, 19438}, {19042, 31987}, {19055, 34473}, {19064, 22676}, {19067, 52027}, {19070, 22890}, {19072, 22843}, {19073, 21156}, {19075, 21157}, {19081, 38693}, {19084, 22951}, {19089, 22712}, {19093, 38717}, {19102, 26516}, {19108, 21166}, {19112, 34474}, {19114, 38699}, {19128, 26925}, {19184, 19185}, {19356, 19357}, {19360, 44589}, {19385, 19388}, {19386, 19399}, {19437, 19441}, {19439, 19440}, {19442, 42022}, {19548, 36586}, {19708, 42524}, {19925, 49618}, {21445, 33370}, {21492, 37680}, {21545, 37682}, {21547, 37674}, {21548, 37679}, {21553, 37633}, {21555, 37687}, {21566, 37685}, {22466, 35861}, {22521, 33435}, {22537, 22554}, {22591, 45078}, {22682, 22720}, {22753, 22763}, {22831, 22876}, {22832, 22921}, {22833, 22976}, {22960, 35860}, {22961, 22962}, {23046, 42639}, {23269, 31414}, {23311, 32491}, {23358, 32385}, {23698, 39823}, {24244, 49390}, {24813, 24819}, {24828, 24842}, {25406, 39875}, {26290, 26385}, {26291, 26409}, {26292, 26460}, {26293, 26461}, {26294, 26462}, {26295, 26463}, {26326, 45365}, {26327, 45368}, {26328, 45607}, {26329, 45605}, {26331, 49027}, {26332, 45650}, {26333, 45652}, {26398, 44583}, {26422, 44585}, {26451, 44611}, {26456, 46453}, {26459, 37561}, {26487, 44621}, {26492, 44619}, {26498, 45596}, {26507, 45598}, {26521, 44597}, {26877, 26930}, {26878, 26940}, {26879, 26950}, {30428, 45841}, {31687, 31689}, {31688, 31691}, {32170, 32171}, {32233, 32253}, {32274, 32303}, {32330, 32343}, {32369, 32399}, {32494, 38110}, {33346, 33452}, {33348, 33454}, {33366, 33438}, {33368, 33436}, {33371, 37334}, {33372, 33430}, {33393, 34508}, {33394, 34509}, {33699, 43340}, {33813, 49267}, {33814, 48715}, {34089, 43571}, {34091, 43569}, {34200, 52048}, {34862, 49235}, {35404, 42572}, {35434, 43385}, {35698, 49214}, {35753, 35848}, {35828, 35880}, {35838, 35868}, {35846, 35850}, {35852, 49240}, {35854, 49242}, {35862, 49248}, {35870, 49258}, {35872, 49260}, {35874, 49262}, {36436, 36446}, {36437, 36450}, {36438, 41944}, {36454, 36464}, {36455, 36468}, {36456, 41943}, {36474, 36492}, {36477, 36555}, {36489, 36553}, {36510, 36585}, {36526, 36549}, {36557, 36581}, {36655, 45441}, {36711, 36990}, {36998, 45407}, {37567, 38235}, {38602, 48701}, {38608, 49271}, {38624, 49219}, {39824, 39825}, {39853, 39854}, {40108, 49231}, {41490, 45421}, {41947, 42640}, {41957, 43409}, {41959, 41970}, {41965, 43317}, {42085, 42249}, {42086, 42248}, {42093, 42206}, {42094, 42204}, {42101, 42210}, {42102, 42208}, {42117, 42253}, {42118, 42252}, {42133, 42186}, {42134, 42184}, {42155, 51925}, {42159, 42187}, {42160, 42190}, {42161, 42189}, {42162, 42188}, {42163, 42211}, {42164, 42214}, {42165, 42213}, {42166, 42212}, {42168, 42201}, {42170, 42199}, {42227, 42999}, {42229, 42998}, {42557, 45385}, {42569, 43318}, {42573, 47599}, {42575, 43791}, {43321, 44245}, {43336, 43788}, {43337, 43383}, {43376, 43507}, {43377, 46935}, {43412, 43517}, {43432, 50690}, {43504, 43520}, {43505, 43513}, {43508, 43516}, {43560, 43570}, {43561, 43568}, {43601, 43826}, {43831, 43863}, {44394, 49356}, {44582, 45357}, {44584, 45359}, {44601, 45620}, {44603, 45621}, {44628, 45623}, {44630, 45624}, {44648, 49103}, {45375, 45542}, {45422, 45586}, {45424, 45584}, {45426, 45715}, {45438, 48660}, {45544, 45574}, {45546, 49347}, {45554, 49355}, {45572, 45713}, {45595, 45599}, {45597, 45601}, {45860, 45870}, {48906, 49229}, {49087, 49318}, {49102, 49215}, {49105, 49237}, {49106, 49239}, {49107, 49243}, {49108, 49245}, {49109, 49247}, {49110, 49249}, {49111, 49253}, {49112, 49255}, {49113, 49259}, {49114, 49261}, {49115, 49263}, {49116, 49265}, {49947, 51924}