X115¶
CENTER OF KIEPERT HYPERBOLA
X115 (CENTER OF KIEPERT HYPERBOLA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶
Constructed Elements
Circle: Nine-Point Circle (ninepointcircle) — Passes through side midpoints M_a, M_b, M_c and altitude feet H_a, H_b, H_c; centered at X5
Related Centers: X5 (NINE-POINT CENTER)
Alternate Coordinates4 forms
Trilinears: \(\cos A - 2 \cos(B - C) + \cot \omega \sin A : :\) (Peter J. C. Moses, 9/12/03)
Trilinears: \(\sin A \sin^{2}(B - C) : :\)
Barycentrics: \((S_{B} - S_{C})^{2} : :\)
Barycentrics: \((S_{A} - S_{\omega}) (S_{B} + S_{C}) + 4 S_{B} S_{C} : :\)
Linear Combinations2 combinations
X115 = X13 + X14(Randy Hutson, July 23, 2015) X115 = \(3\) X2 + X148, \(3\) X2 - \(4\) X6722, \(5\) X2 - X8591, \(7\) X2 + X8596, X2 - \(3\) X9166, \(4\) X2 - \(3\) X9167, \(3\) X2 - \(5\) X14061, \(2\) X2 - \(3\) X14971, \(4\) X2 - X15300, \(9\) X2 - X20094, \(5\) X2 - \(4\) X22247, \(6\) X2 - \(5\) X31274, \(9\) X2 - \(4\) X35022, X3 - \(4\) X20398, \(3\) X4 + X9862, \(3\) X4 - X10722, \(2\) X4 + X10991, X4 + \(2\) X11623, X4 - \(3\) X14639, X4 + \(3\) X14651, \(4\) X5 - X14981, \(3\) X5 - \(4\) X15092, \(2\) X5 - \(3\) X23514, X6 - \(3\) X6034, X13 + \(3\) X5469, X13 - \(3\) X5470, \(3\) X13 + X6777, \(3\) X13 - X6778, X14 - \(3\) X5469, X14 + \(3\) X5470, \(3\) X14 - X6777, \(3\) X14 + X6778, X15 - \(3\) X22510, X16 - \(3\) X22511, X20 - \(3\) X34473, \(3\) X98 - X9862, \(3\) X98 + X10722, X98 + \(3\) X14639, X98 - \(3\) X14651, X99 + \(3\) X671, \(2\) X99 - \(3\) X2482, X99 - \(6\) X5461, X99 - \(4\) X6722, \(5\) X99 - \(3\) X8591, \(7\) X99 + \(3\) X8596, X99 - \(9\) X9166, \(4\) X99 - \(9\) X9167, X99 - \(5\) X14061, \(2\) X99 - \(9\) X14971, \(4\) X99 - \(3\) X15300, \(3\) X99 - X20094, \(5\) X99 - \(12\) X22247, \(2\) X99 - \(5\) X31274
X115 lies on these curves: nine-point circle, Steiner inellipse, bicevian conic of X2 and X98, 3rd Lemoine circle, Moses circle, Moses-Parry circle, the cubics K203, K237, K239, K301, K392, K483, K599, K629, K886, K942.
Incident Central Lines702 lines
{1, 11725}, {2, 99}, {3, 2079}, {4, 32}, {5, 39}, {6, 13}, {8, 7983}, {10, 3029}, {11, 1015}, {12, 1500}, {15, 6771}, {16, 6774}, {17, 11602}, {18, 11603}, {20, 5206}, {25, 3455}, {30, 187}, {33, 9636}, {37, 3822}, {50, 231}, {53, 133}, {54, 9697}, {55, 9664}, {56, 9651}, {61, 16002}, {62, 16001}, {69, 7818}, {75, 21431}, {76, 626}, {83, 4027}, {100, 10769}, {110, 3044}, {116, 1086}, {117, 1901}, {118, 1834}, {119, 2092}, {120, 442}, {121, 1213}, {122, 2797}, {123, 2798}, {124, 1146}, {125, 245}, {127, 338}, {128, 233}, {129, 389}, {130, 24862}, {131, 216}, {135, 2970}, {136, 8901}, {140, 10992}, {141, 5969}, {142, 35074}, {147, 2548}, {149, 21341}, {172, 3585}, {182, 13517}, {183, 7761}, {193, 32827}, {194, 7752}, {213, 21935}, {226, 11608}, {232, 403}, {235, 3199}, {244, 23063}, {274, 17669}, {302, 14904}, {303, 14905}, {312, 34542}, {315, 7751}, {316, 385}, {321, 11611}, {325, 538}, {371, 8980}, {372, 13967}, {373, 27779}, {376, 8588}, {378, 21397}, {382, 3053}, {384, 5152}, {393, 6529}, {395, 530}, {396, 531}, {397, 22832}, {398, 22831}, {427, 1196}, {429, 20621}, {485, 1504}, {486, 1505}, {498, 9598}, {499, 9597}, {511, 15980}, {512, 2679}, {513, 35079}, {514, 35080}, {515, 11710}, {516, 20666}, {517, 5164}, {518, 35084}, {519, 10026}, {522, 35086}, {523, 5099}, {524, 5107}, {525, 34943}, {536, 35089}, {546, 5007}, {547, 3055}, {549, 3054}, {550, 15513}, {567, 9604}, {571, 31723}, {577, 2165}, {590, 6306}, {591, 13927}, {593, 1029}, {594, 4013}, {597, 3363}, {599, 19662}, {609, 18513}, {615, 6302}, {616, 11489}, {617, 11488}, {618, 6670}, {619, 6669}, {623, 31711}, {624, 31712}, {631, 13172}, {645, 26081}, {647, 3258}, {650, 5520}, {656, 35091}, {661, 2170}, {662, 24957}, {694, 34359}, {698, 5031}, {726, 20546}, {732, 5103}, {759, 21004}, {799, 25472}, {804, 1084}, {805, 31513}, {857, 35093}, {858, 3291}, {958, 13181}, {1007, 32984}, {1062, 9635}, {1078, 6655}, {1089, 16886}, {1107, 25639}, {1109, 21833}, {1111, 1577}, {1125, 11711}, {1131, 6568}, {1132, 6569}, {1141, 14586}, {1184, 5064}, {1194, 5133}, {1210, 24472}, {1211, 13466}, {1250, 13076}, {1281, 5051}, {1312, 8106}, {1313, 8105}, {1316, 8429}, {1329, 1574}, {1346, 8427}, {1347, 8426}, {1348, 2033}, {1349, 2034}, {1352, 5028}, {1365, 3708}, {1376, 13173}, {1384, 3830}, {1415, 13273}, {1478, 2242}, {1479, 2241}, {1503, 1692}, {1509, 6625}, {1513, 2021}, {1565, 4403}, {1568, 3289}, {1570, 3564}, {1571, 1698}, {1572, 1699}, {1573, 2886}, {1575, 3814}, {1587, 19055}, {1588, 19056}, {1597, 34809}, {1598, 9861}, {1609, 18534}, {1611, 34609}, {1650, 13179}, {1656, 5013}, {1657, 5023}, {1676, 2035}, {1677, 2036}, {1691, 29012}, {1737, 20461}, {1865, 2791}, {1914, 3583}, {1970, 13403}, {1971, 18400}, {1975, 3788}, {1990, 18809}, {1991, 13850}, {1992, 11161}, {1995, 13233}, {2028, 2039}, {2029, 2040}, {2030, 11645}, {2051, 9561}, {2072, 14961}, {2076, 29317}, {2086, 9427}, {2176, 24045}, {2238, 5526}, {2240, 5990}, {2245, 31841}, {2275, 7741}, {2276, 7951}, {2310, 15607}, {2321, 34528}, {2378, 5618}, {2379, 5619}, {2386, 5140}, {2387, 5167}, {2394, 14223}, {2395, 14998}, {2420, 12295}, {2475, 5277}, {2476, 5283}, {2489, 34978}, {2501, 12079}, {2502, 5642}, {2503, 5540}, {2566, 32481}, {2567, 32482}, {2643, 4092}, {2653, 12047}, {2770, 10415}, {2788, 5511}, {2789, 5510}, {2793, 5512}, {2896, 7911}, {2936, 5020}, {2963, 14367}, {2971, 4516}, {2980, 9233}, {2996, 3926}, {3003, 11799}, {3008, 35115}, {3011, 33329}, {3068, 19109}, {3069, 19108}, {3070, 5062}, {3071, 5058}, {3087, 20774}, {3090, 7738}, {3094, 7697}, {3096, 7933}, {3104, 16630}, {3105, 16631}, {3117, 22735}, {3136, 5513}, {3141, 16614}, {3153, 10313}, {3154, 9209}, {3229, 21531}, {3239, 35122}, {3271, 5509}, {3284, 10297}, {3290, 30447}, {3329, 7827}, {3438, 8742}, {3439, 8741}, {3448, 15342}, {3452, 35130}, {3454, 20532}, {3524, 12117}, {3526, 15815}, {3534, 5210}, {3545, 6054}, {3548, 15075}, {3552, 7857}, {3566, 34953}, {3589, 5026}, {3614, 31460}, {3618, 5182}, {3619, 33223}, {3627, 35007}, {3628, 31652}, {3695, 7230}, {3700, 24040}, {3721, 34829}, {3729, 27688}, {3735, 3944}, {3741, 23917}, {3785, 32982}, {3817, 21636}, {3825, 16604}, {3832, 5319}, {3836, 35118}, {3839, 5304}, {3840, 23918}, {3843, 5346}, {3845, 5008}, {3849, 8352}, {3850, 5041}, {3851, 9605}, {3856, 34571}, {3875, 27556}, {3912, 20337}, {3933, 7821}, {3934, 5976}, {3936, 17310}, {3948, 35126}, {3954, 21029}, {3972, 7806}, {3981, 21243}, {3992, 20483}, {4016, 21018}, {4037, 21057}, {4052, 34899}, {4129, 6547}, {4136, 22036}, {4428, 12326}, {4721, 24995}, {4995, 12354}, {5024, 5055}, {5034, 12177}, {5038, 25555}, {5046, 5985}, {5052, 5480}, {5054, 12355}, {5056, 31400}, {5065, 18537}, {5066, 9300}, {5068, 31404}, {5070, 31457}, {5071, 23234}, {5072, 14692}, {5076, 22331}, {5077, 7610}, {5079, 22332}, {5080, 5291}, {5097, 14160}, {5104, 19924}, {5111, 5965}, {5134, 17734}, {5149, 5989}, {5162, 5999}, {5169, 6032}, {5215, 27088}, {5275, 17532}, {5276, 17577}, {5298, 18969}, {5318, 5478}, {5321, 5479}, {5334, 6770}, {5335, 6773}, {5418, 9674}, {5432, 15452}, {5449, 8571}, {5463, 16645}, {5464, 16644}, {5466, 9180}, {5473, 11481}, {5474, 11480}, {5485, 5503}, {5515, 6377}, {5521, 8735}, {5552, 13189}, {5569, 8860}, {5585, 15688}, {5587, 9620}, {5590, 6320}, {5591, 6319}, {5599, 13176}, {5600, 13177}, {5603, 7970}, {5613, 18582}, {5617, 18581}, {5663, 15535}, {5705, 31442}, {5886, 11724}, {5912, 16092}, {5939, 7792}, {5972, 20998}, {5977, 26601}, {5980, 11304}, {5981, 11303}, {5986, 7394}, {5987, 7533}, {5992, 27040}, {6179, 7823}, {6201, 6226}, {6202, 6227}, {6328, 8574}, {6337, 32969}, {6381, 20541}, {6390, 31275}, {6392, 7758}, {6422, 31481}, {6423, 23251}, {6424, 23261}, {6528, 16081}, {6560, 13790}, {6561, 9675}, {6587, 16177}, {6683, 32992}, {6697, 33324}, {6704, 7859}, {6739, 8649}, {6748, 18402}, {6753, 16178}, {6776, 7694}, {6779, 16963}, {6780, 16962}, {6782, 10612}, {6783, 10611}, {6792, 9140}, {6794, 11005}, {7031, 18514}, {7051, 18975}, {7200, 35131}, {7257, 25685}, {7336, 8061}, {7486, 31450}, {7514, 9609}, {7576, 10985}, {7585, 13640}, {7586, 13760}, {7669, 11641}, {7684, 31701}, {7685, 31702}, {7728, 14849}, {7750, 7780}, {7754, 7759}, {7757, 7777}, {7760, 7785}, {7762, 7805}, {7763, 7781}, {7766, 7812}, {7767, 7873}, {7768, 7885}, {7769, 7783}, {7771, 7833}, {7774, 7775}, {7776, 7855}, {7778, 7801}, {7779, 7809}, {7782, 7907}, {7784, 7854}, {7786, 7864}, {7787, 7856}, {7789, 7874}, {7791, 7815}, {7793, 7802}, {7795, 7867}, {7796, 7912}, {7799, 7925}, {7800, 7935}, {7803, 7808}, {7807, 7816}, {7811, 7898}, {7814, 7906}, {7819, 7852}, {7822, 7866}, {7824, 7847}, {7831, 7924}, {7832, 7901}, {7836, 7899}, {7837, 7926}, {7839, 7858}, {7840, 11054}, {7846, 7932}, {7860, 7893}, {7865, 16990}, {7868, 33219}, {7869, 33283}, {7875, 7884}, {7876, 7918}, {7877, 7900}, {7878, 7920}, {7891, 7940}, {7892, 7942}, {7894, 7921}, {7896, 33290}, {7904, 7910}, {7905, 7941}, {7908, 32833}, {7915, 8363}, {7930, 14065}, {7931, 14046}, {7937, 16986}, {7939, 33289}, {7943, 16895}, {7988, 9592}, {7989, 9593}, {8029, 12076}, {8068, 13006}, {8145, 20387}, {8176, 11163}, {8182, 23055}, {8196, 12179}, {8203, 12180}, {8212, 12186}, {8213, 12187}, {8222, 13184}, {8223, 13185}, {8227, 9619}, {8253, 9600}, {8258, 25607}, {8259, 20394}, {8260, 20395}, {8265, 34845}, {8355, 22110}, {8362, 31239}, {8553, 12083}, {8573, 18535}, {8587, 17503}, {8597, 8859}, {8598, 32479}, {8623, 14957}, {8770, 30771}, {8779, 13851}, {8799, 12233}, {8882, 9378}, {8902, 20625}, {8960, 12962}, {8962, 15234}, {9114, 22578}, {9116, 22577}, {9151, 23301}, {9293, 10278}, {9302, 14492}, {9346, 11269}, {9479, 15449}, {9574, 31441}, {9603, 18350}, {9608, 13861}, {9650, 10895}, {9665, 10896}, {9669, 16781}, {9766, 22253}, {9855, 26613}, {9875, 25055}, {9881, 19875}, {9927, 23128}, {9971, 13249}, {10054, 10072}, {10056, 10070}, {10061, 10078}, {10062, 10077}, {10104, 32152}, {10151, 14581}, {10153, 32532}, {10175, 31398}, {10314, 18420}, {10315, 18406}, {10316, 18404}, {10317, 18403}, {10527, 13190}, {10531, 12189}, {10532, 12190}, {10575, 15575}, {10590, 31409}, {10638, 13075}, {10641, 12141}, {10642, 12142}, {10653, 16635}, {10654, 16634}, {10753, 14853}, {10893, 12182}, {10894, 12183}, {10986, 18559}, {10989, 11580}, {11056, 26257}, {11081, 15929}, {11082, 16807}, {11086, 15930}, {11087, 16806}, {11121, 11129}, {11122, 11128}, {11168, 15810}, {11231, 31443}, {11284, 34013}, {11287, 15271}, {11295, 22575}, {11296, 22576}, {11485, 13102}, {11486, 13103}, {11496, 12178}, {11539, 11614}, {11585, 22401}, {11625, 27550}, {11627, 27551}, {11680, 16975}, {11742, 15689}, {11897, 12181}, {12077, 30460}, {12100, 26614}, {12191, 14712}, {12900, 33512}, {13192, 15360}, {13479, 17983}, {13509, 25739}, {13540, 35111}, {13703, 13705}, {13823, 13825}, {13893, 31437}, {14023, 32006}, {14157, 15340}, {14269, 21309}, {14389, 30685}, {14585, 21659}, {14873, 23537}, {14907, 17008}, {14989, 32640}, {15081, 18331}, {15118, 32740}, {15538, 15545}, {15655, 15681}, {15903, 17056}, {16267, 22997}, {16268, 22998}, {16303, 18487}, {16317, 24855}, {16529, 16960}, {16530, 16961}, {16989, 33016}, {17006, 33273}, {17116, 27707}, {17423, 34952}, {17448, 24387}, {17677, 26244}, {17702, 32661}, {17757, 21956}, {18140, 33841}, {18472, 18564}, {18584, 19709}, {18591, 30445}, {18974, 19373}, {19053, 19058}, {19054, 19057}, {19102, 22502}, {19105, 22501}, {19106, 19780}, {19107, 19781}, {19905, 20423}, {20065, 32996}, {20595, 23912}, {20976, 24981}, {21138, 21253}, {21448, 32216}, {21674, 21816}, {21675, 21810}, {22425, 30449}, {22491, 22579}, {22492, 22580}, {22495, 22571}, {22496, 22572}, {22504, 22753}, {23288, 34206}, {23536, 27555}, {24345, 24711}, {24443, 30436}, {25561, 25562}, {26235, 31107}, {26363, 31456}, {26582, 27076}, {27318, 33061}, {27374, 27375}, {27570, 33932}, {27706, 33933}, {27966, 33940}, {28473, 34959}, {30466, 30469}, {30714, 35324}, {31411, 31412}, {31416, 31418}, {31422, 31423}, {31433, 31434}, {31448, 31501}, {31471, 31472}, {31477, 31479}, {31482, 31484}, {31490, 31493}, {31685, 31686}, {31689, 31691}, {31690, 31692}, {32190, 32476}, {32456, 35297}, {32490, 32494}, {32491, 32497}, {32822, 32955}, {32826, 32973}, {32829, 32988}, {32834, 33200}, {32838, 32990}, {32986, 34229}, {34989, 34990}
Functional Relations93 relations
X115 = midpoint of PU(40)
X115 = reflection of Xi in Xj for these {i,j}: {1, 11725}, {2, 5461}, {3, 6036}, {39, 2023}, {98, 11623}, {99, 620}, {114, 5}, {125, 15359}, {187, 230}, {325, 625}, {599, 19662}, {618, 6670}, {619, 6669}, {620, 6722}, {1569, 39}, {2482, 2}, {4027, 7829}, {5026, 3589}, {5099, 14120}, {5477, 6}, {5976, 3934}, {6036, 20398}, {6292, 9478}, {6781, 187}, {6782, 11543}, {6783, 11542}, {7813, 325}, {8290, 6704}, {9115, 395}, {9117, 396}, {9167, 14971}, {10991, 98}, {10992, 33813}, {11711, 1125}, {12829, 5305}, {14501, 2040}, {14502, 2039}, {14928, 5026}, {14971, 9166}, {14981, 114}, {15300, 2482}, {15357, 125}, {18800, 597}, {20387, 8145}, {22110, 8355}, {22505, 546}, {22566, 5066}, {22848, 22847}, {22892, 22893}, {25486, 8176}, {25559, 20415}, {25560, 20416}, {25562, 25561}, {25607, 8258}, {25608, 8259}, {25609, 8260}, {31274, 14061}, {32135, 25555}, {32458, 626}, {33512, 12900}, {33813, 140}, {34602, 22846}
X115 = isogonal conjugate of X249
X115 = isotomic conjugate of X4590
X115 = complement of X99
X115 = anticomplement of X620
X115 = Stevanovic-circle-inverse of X5520
X115 = Lester-circle-inverse of X10413
X115 = Spieker-radical-circle-inverse of X5213
X115 = orthoptic-circle-of-Steiner-inellipse-inverse of X111
X115 = orthoptic-circle-of-Steiner-circumellipe-inverse of X20099
X115 = polar conjugate of X18020
X115 = antigonal image of X6328
X115 = isotomic conjugate of the anticomplement of X23991
X115 = isotomic conjugate of the isogonal conjugate of X3124
X115 = isotomic conjugate of the polar conjugate of X8754
X115 = polar conjugate of the isogonal conjugate of X20975
X115 = psi-transform of X6792
X115 = tripolar centroid of X5466
X115 = X2-Hirst inverse of X148
X115 = X99-of-medial triangle
X115 = X101-of-orthic triangle if ABC is acute
X115 = projection from Steiner circumellipse to Steiner inellipse of X671
X115 = center of similitude of incentral and Feuerbach triangles
X115 = center of circumconic that is locus of trilinear poles of lines parallel to the orthic axis (i.e. lines that pass through X523)
X115 = perspector of circumconic centered at X523 (parabola {A, B, C,X476,X523,X685})
X115 = inverse-in-{circumcircle, nine-point circle}-inverter of X111
X115 = inverse-in-excircles-radical-circle of X5213
X115 = {X6108,X6109}-harmonic conjugate of X6055
X115 = orthopole of Brocard axis
X115 = perspector of orthic triangle and tangential triangle of hyperbola {A, B, C,X2,X6}}
X115 = similitude center of (equilateral) pedal triangles of X15 and X16
X115 = exsimilicenter of Moses circle and the nine-point circle
X115 = X1101-isoconjugate of X2
X115 = harmonic center of nine-point circle and Gallatly circle
X115 = perspector of medial triangle and Schroeter triangle
X115 = homothetic center of medial triangle and mid-triangle of antipedal triangles of X13 and X14
X115 = homothetic center of Ehrmann vertex-triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles
X115 = centroid of reflection triangle of X187
X115 = centroid of mid-triangle of 1st and 2nd isodynamic-Dao triangles
X115 = centroid of mid-triangle of 3rd and 4th isodynamic-Dao triangles
X115 = point of concurrence of cevian circles of the excenters
X115 = intersection of Fermat axes of outer and inner Vecten triangles
X115 = Schroeter-isogonal conjugate of X35605
X115 = point of concurrence of cevian circles of vertices of anticevian triangle of X4
X115 = cevapoint of Xi and Xj for these (i,j): {512, 8574}, {523, 10278}, {1648, 23992}, {1649, 33906}, {3124, 20975}, {21043, 21833}
X115 = crosspoint of Xi and Xj for these (i,j): {2, 523}, {4, 14618}, {6, 2623}, {10, 31010}, {25, 18105}, {68, 525}, {252, 15412}, {338, 2970}, {393, 2501}, {512, 27375}, {594, 4024}, {661, 2171}, {850, 1502}, {1089, 1577}, {2394, 5627}, {3120, 16732}, {3267, 6340}, {3413, 3414}, {4036, 28654}, {8024, 31067}, {8105, 8106}, {11080, 20578}, {11085, 20579}, {12028, 14592}
X115 = crosssum of Xi and Xj for these (i,j): {2, 14570}, {3, 32661}, {6, 110}, {24, 112}, {69, 4576}, {99, 1078}, {101, 33771}, {143, 1625}, {163, 849}, {394, 4558}, {593, 4556}, {662, 2185}, {1379, 1380}, {1501, 1576}, {1511, 2420}, {1634, 8041}, {1986, 14591}, {4636, 7054}, {5546, 35193}, {8115, 8116}, {11130, 17403}, {11131, 17402}
X115 = trilinear pole of line {1648, 8029}
X115 = crossdifference of every pair of points on line {110, 351}
X115 = Xi-complementary conjugate of Xj for these (i,j): {1, 512}, {4, 21259}, {6, 4369}, {10, 21260}, {19, 30476}, {25, 8062}, {31, 523}, {32, 14838}, {37, 3835}, {42, 513}, {57, 17066}, {65, 17072}, {75, 23301}, {76, 21263}, {110, 21254}, {115, 21253}, {163, 620}, {213, 514}, {228, 20315}, {251, 8060}, {321, 21262}, {351, 16597}, {512, 10}, {513, 3741}, {514, 21240}, {523, 2887}, {560, 647}, {604, 17069}, {647, 18589}, {649, 3739}, {650, 21246}, {656, 1368}, {661, 141}, {663, 960}, {667, 1125}, {669, 37}, {688, 16587}, {741, 4155}, {756, 31946}, {798, 2}, {810, 3}, {822, 6389}, {850, 21235}, {872, 661}, {875, 740}, {881, 18904}, {922, 1649}, {923, 690}, {1015, 17761}, {1018, 27076}, {1042, 3900}, {1084, 16592}, {1096, 520}, {1101, 10190}, {1245, 8678}, {1333, 21196}, {1334, 20317}, {1356, 16613}, {1400, 4885}, {1402, 522}, {1438, 24285}, {1500, 4129}, {1576, 16598}, {1577, 626}, {1824, 20316}, {1910, 24284}, {1911, 9508}, {1918, 650}, {1919, 3666}, {1924, 39}, {1927, 2491}, {1946, 34851}, {1960, 34587}, {1964, 3005}, {1967, 804}, {1973, 525}, {1974, 16612}, {2084, 6292}, {2179, 18314}, {2203, 21187}, {2205, 6586}, {2206, 31947}, {2238, 27854}, {2258, 8672}, {2422, 16609}, {2485, 21247}, {2489, 226}, {2491, 16591}, {2501, 20305}, {2616, 3819}, {2623, 21231}, {2642, 126}, {2643, 125}, {2786, 20548}, {3005, 21249}, {3049, 1214}, {3063, 5745}, {3112, 688}, {3120, 21252}, {3121, 1086}, {3122, 11}, {3124, 8287}, {3125, 116}, {3223, 3221}, {3248, 244}, {3271, 34589}, {3402, 23878}, {3669, 17050}, {3700, 21244}, {3708, 127}, {3709, 3452}, {4010, 20542}, {4017, 2886}, {4024, 21245}, {4041, 1329}, {4069, 3038}, {4077, 17047}, {4079, 1211}, {4117, 1084}, {4455, 17793}, {4516, 124}, {4557, 24003}, {4559, 21232}, {4705, 3454}, {4729, 2885}, {4730, 121}, {4770, 21251}, {5027, 19563}, {5029, 20529}, {5466, 21256}, {6591, 34830}, {7178, 17046}, {7180, 142}, {7216, 21258}, {7250, 11019}, {7252, 21233}, {7257, 3037}, {8061, 21248}, {9178, 4892}, {9406, 5664}, {9426, 16584}, {9508, 20339}, {14407, 16594}, {14574, 23993}, {16606, 21191}, {18105, 1215}, {18344, 34831}, {18793, 6373}, {20975, 34846}, {21759, 31286}, {21832, 20333}, {21834, 21250}, {22260, 24040}, {23493, 4083}, {23503, 6374}, {23894, 625}, {24006, 21243}, {24290, 20540}, {28625, 4932}, {32676, 5972}, {32713, 23998}
X115 = Xi-Ceva conjugate of Xj for these (i,j): {1, 12071}, {2, 523}, {4, 512}, {6, 12077}, {7, 12072}, {8, 12069}, {10, 6367}, {12, 4705}, {76, 826}, {83, 7927}, {99, 12076}, {148, 13187}, {190, 12078}, {264, 12075}, {338, 125}, {393, 2501}, {427, 2514}, {523, 8029}, {594, 4024}, {598, 12073}, {671, 690}, {850, 22260}, {1029, 513}, {1109, 4092}, {1119, 12070}, {1141, 14270}, {1502, 850}, {1916, 2799}, {1989, 1637}, {2165, 647}, {2970, 8754}, {2980, 669}, {2986, 9033}, {2996, 525}, {3120, 2643}, {3613, 3005}, {4024, 21131}, {4080, 6370}, {5392, 6368}, {5395, 3800}, {5466, 33919}, {5485, 3906}, {5627, 15475}, {6504, 520}, {6625, 514}, {7332, 2611}, {8754, 6388}, {8791, 14273}, {8818, 661}, {8901, 20975}, {10415, 9178}, {11080, 20578}, {11082, 6137}, {11085, 20579}, {11087, 6138}, {11121, 23870}, {11122, 23871}, {11538, 20188}, {11606, 804}, {12066, 30}, {12067, 12064}, {13576, 4155}, {13579, 924}, {13580, 2575}, {13581, 2574}, {13582, 526}, {13583, 3900}, {13584, 4083}, {13585, 1510}, {13854, 2489}, {14618, 23105}, {16081, 16230}, {16732, 1109}, {18023, 9134}, {18840, 7950}, {21043, 6627}, {21044, 3708}, {24624, 6089}, {28654, 4036}, {34294, 3124}, {34449, 34952}, {34816, 3806}
X115 = Xi-cross conjugate of Xj for these (i,j): {512, 6328}, {2643, 1365}, {3124, 8754}, {8029, 523}, {14443, 690}, {20975, 125}, {21043, 1109}, {21131, 4024}, {21833, 2643}, {22260, 850}, {23991, 2}, {23992, 1648}, {33919, 5466}
X115 = Xi-isoconjugate of Xj for these (i,j): {1, 249}, {2, 1101}, {6, 24041}, {31, 4590}, {32, 24037}, {41, 7340}, {48, 18020}, {58, 4567}, {59, 2185}, {60, 4564}, {63, 250}, {75, 23357}, {76, 23995}, {81, 4570}, {99, 163}, {100, 4556}, {109, 4612}, {110, 662}, {112, 4592}, {162, 4558}, {255, 23582}, {261, 2149}, {326, 23964}, {394, 24000}, {560, 34537}, {561, 23963}, {577, 23999}, {593, 765}, {604, 6064}, {643, 4565}, {648, 4575}, {651, 4636}, {691, 23889}, {692, 4610}, {757, 1252}, {758, 9273}, {798, 31614}, {799, 1576}, {811, 32661}, {849, 1016}, {873, 23990}, {1098, 1262}, {1110, 1509}, {1333, 4600}, {1414, 5546}, {1634, 4599}, {1725, 18879}, {1790, 5379}, {2150, 4998}, {2194, 4620}, {2206, 4601}, {2617, 18315}, {2966, 23997}, {3936, 9274}, {4563, 32676}, {4576, 34072}, {4602, 14574}, {4623, 32739}, {6507, 32230}, {6578, 35342}, {7045, 7054}, {7058, 24027}, {9268, 30576}, {10411, 32678}, {14213, 14587}, {18042, 27867}, {24039, 32729}, {35049, 35193}
X115 = barycentric product Xi*Xj for these {i,j}: {1, 1109}, {3, 2970}, {4, 125}, {5, 8901}, {6, 338}, {7, 4092}, {8, 1365}, {10, 3120}, {11, 12}, {13, 30465}, {14, 30468}, {19, 20902}, {25, 339}, {27, 21046}, {30, 12079}, {31, 23994}, {32, 23962}, {37, 16732}, {42, 21207}, {68, 136}, {69, 8754}, {75, 2643}, {76, 3124}, {79, 21054}, {86, 21043}, {92, 3708}, {94, 2088}, {98, 868}, {99, 8029}, {107, 5489}, {110, 23105}, {122, 6526}, {127, 13854}, {135, 32132}, {137, 252}, {141, 34294}, {158, 2632}, {181, 34387}, {190, 21131}, {226, 21044}, {244, 1089}, {264, 20975}, {265, 35235}, {274, 21833}, {298, 30452}, {299, 30453}, {305, 2971}, {313, 3122}, {321, 3125}, {327, 6784}, {393, 15526}, {459, 1562}, {477, 6070}, {512, 850}, {513, 4036}, {514, 4024}, {523, 523}, {525, 2501}, {526, 10412}, {594, 1086}, {598, 8288}, {647, 14618}, {656, 24006}, {661, 1577}, {670, 22260}, {671, 1648}, {690, 5466}, {693, 4705}, {756, 1111}, {762, 16727}, {798, 20948}, {879, 16230}, {882, 14295}, {892, 33919}, {1015, 28654}, {1084, 1502}, {1093, 2972}, {1096, 17879}, {1118, 7068}, {1146, 6354}, {1254, 24026}, {1312, 1313}, {1358, 6057}, {1367, 1857}, {1425, 21666}, {1441, 4516}, {1500, 23989}, {1565, 7140}, {1637, 2394}, {1640, 14223}, {1647, 4013}, {1826, 4466}, {1897, 21134}, {1928, 4117}, {2052, 3269}, {2086, 18896}, {2170, 6358}, {2171, 4858}, {2395, 2799}, {2489, 3267}, {2611, 6757}, {2616, 2618}, {2623, 18314}, {2793, 34246}, {2969, 3695}, {2973, 3690}, {2996, 6388}, {3121, 27801}, {3258, 5627}, {3261, 4079}, {3268, 15475}, {3271, 34388}, {3318, 7157}, {3413, 13722}, {3414, 13636}, {3448, 6328}, {3613, 7668}, {3700, 7178}, {3906, 8599}, {3937, 7141}, {4010, 35352}, {4017, 4086}, {4041, 4077}, {4049, 4120}, {4052, 21950}, {4064, 7649}, {4081, 6046}, {4103, 6545}, {4608, 6367}, {4609, 23099}, {4988, 31010}, {5099, 10415}, {5139, 6340}, {5485, 6791}, {5512, 32133}, {5514, 13853}, {5949, 7332}, {6344, 16186}, {6506, 7363}, {6529, 23616}, {6535, 17205}, {6625, 6627}, {7018, 21725}, {7372, 12071}, {7927, 31065}, {8034, 27808}, {8061, 18070}, {8287, 8818}, {8371, 9180}, {8735, 26942}, {8736, 26932}, {8769, 17876}, {9033, 18808}, {9293, 10278}, {10413, 13582}, {10555, 14357}, {11792, 34110}, {12028, 16221}, {12077, 15412}, {14052, 34953}, {14086, 14086}, {14273, 14977}, {14431, 35353}, {14998, 18312}, {15320, 21045}, {18003, 18015}, {18004, 18014}, {18006, 18013}, {18007, 34763}, {18023, 21906}, {18105, 23285}, {20578, 23870}, {20579, 23871}, {21824, 30690}, {23283, 23284}, {23286, 23290}, {23287, 23288}
X115 = barycentric quotient Xi/Xj for these {i,j}: {1, 24041}, {2, 4590}, {4, 18020}, {6, 249}, {7, 7340}, {8, 6064}, {10, 4600}, {11, 261}, {12, 4998}, {25, 250}, {31, 1101}, {32, 23357}, {37, 4567}, {42, 4570}, {75, 24037}, {76, 34537}, {99, 31614}, {125, 69}, {127, 34254}, {136, 317}, {148, 31632}, {158, 23999}, {181, 59}, {226, 4620}, {244, 757}, {321, 4601}, {338, 76}, {339, 305}, {351, 5467}, {393, 23582}, {512, 110}, {514, 4610}, {523, 99}, {525, 4563}, {526, 10411}, {560, 23995}, {594, 1016}, {647, 4558}, {649, 4556}, {650, 4612}, {656, 4592}, {661, 662}, {663, 4636}, {669, 1576}, {690, 5468}, {693, 4623}, {756, 765}, {798, 163}, {804, 17941}, {810, 4575}, {826, 4576}, {850, 670}, {868, 325}, {872, 1110}, {879, 17932}, {881, 17938}, {882, 805}, {888, 5118}, {1015, 593}, {1084, 32}, {1086, 1509}, {1089, 7035}, {1096, 24000}, {1109, 75}, {1111, 873}, {1146, 7058}, {1254, 7045}, {1356, 1397}, {1357, 7341}, {1358, 552}, {1365, 7}, {1367, 7055}, {1500, 1252}, {1501, 23963}, {1577, 799}, {1637, 2407}, {1640, 14999}, {1645, 33875}, {1648, 524}, {1824, 5379}, {2086, 1691}, {2087, 30576}, {2088, 323}, {2170, 2185}, {2171, 4564}, {2207, 23964}, {2310, 1098}, {2395, 2966}, {2422, 2715}, {2485, 4611}, {2489, 112}, {2491, 14966}, {2501, 648}, {2610, 4585}, {2623, 18315}, {2632, 326}, {2642, 23889}, {2643, 1}, {2682, 5642}, {2793, 34245}, {2799, 2396}, {2970, 264}, {2971, 25}, {2972, 3964}, {3005, 1634}, {3022, 6061}, {3049, 32661}, {3120, 86}, {3121, 1333}, {3122, 58}, {3124, 6}, {3125, 81}, {3248, 849}, {3258, 6148}, {3269, 394}, {3271, 60}, {3569, 2421}, {3700, 645}, {3708, 63}, {3709, 5546}, {3906, 9146}, {4017, 1414}, {4024, 190}, {4036, 668}, {4041, 643}, {4049, 4615}, {4064, 4561}, {4077, 4625}, {4079, 101}, {4086, 7257}, {4092, 8}, {4103, 6632}, {4117, 560}, {4155, 3573}, {4171, 7259}, {4391, 4631}, {4466, 17206}, {4516, 21}, {4530, 30606}, {4705, 100}, {5099, 7664}, {5139, 6353}, {5466, 892}, {5489, 3265}, {6057, 4076}, {6137, 17402}, {6138, 17403}, {6328, 13485}, {6354, 1275}, {6367, 4427}, {6388, 193}, {6524, 32230}, {6627, 1654}, {6784, 182}, {6791, 1992}, {7063, 2175}, {7064, 6065}, {7068, 1264}, {7109, 23990}, {7140, 15742}, {7143, 7339}, {7178, 4573}, {7180, 4565}, {7216, 4637}, {7668, 1078}, {7669, 14366}, {7927, 10330}, {8029, 523}, {8034, 3733}, {8287, 34016}, {8288, 599}, {8371, 9182}, {8599, 35138}, {8663, 35327}, {8754, 4}, {8901, 95}, {9148, 23342}, {9171, 9181}, {9178, 691}, {9180, 9170}, {9426, 14574}, {9427, 1501}, {10278, 31998}, {10412, 35139}, {10630, 34539}, {11123, 14588}, {11182, 14607}, {12071, 6758}, {12073, 35356}, {12077, 14570}, {12079, 1494}, {13636, 6189}, {13722, 6190}, {14086, 14089}, {14223, 6035}, {14273, 4235}, {14295, 880}, {14398, 2420}, {14423, 33921}, {14443, 1649}, {14444, 8030}, {14618, 6331}, {14910, 18879}, {14936, 7054}, {14998, 5649}, {15328, 18878}, {15422, 16813}, {15449, 7794}, {15451, 23181}, {15475, 476}, {15525, 439}, {15526, 3926}, {15630, 1976}, {16230, 877}, {16726, 763}, {16732, 274}, {17205, 6628}, {17414, 9145}, {17876, 18156}, {17989, 17944}, {17990, 17943}, {17992, 17942}, {17993, 23348}, {17994, 4230}, {18001, 17940}, {18002, 17939}, {18003, 17935}, {18004, 17934}, {18006, 17933}, {18007, 34760}, {18012, 17937}, {18013, 17931}, {18014, 17930}, {18015, 17929}, {18070, 4593}, {18105, 827}, {18210, 1444}, {18808, 16077}, {19610, 9217}, {20382, 20380}, {20578, 23895}, {20579, 23896}, {20902, 304}, {20948, 4602}, {20975, 3}, {21043, 10}, {21044, 333}, {21045, 33297}, {21046, 306}, {21054, 319}, {21131, 514}, {21134, 4025}, {21138, 7304}, {21141, 16755}, {21207, 310}, {21709, 21085}, {21723, 21081}, {21725, 171}, {21731, 15329}, {21823, 172}, {21824, 3219}, {21833, 37}, {21859, 31615}, {21906, 187}, {21944, 3879}, {21961, 33116}, {22260, 512}, {23099, 669}, {23105, 850}, {23216, 14575}, {23282, 33948}, {23610, 9426}, {23616, 4143}, {23943, 32004}, {23962, 1502}, {23991, 620}, {23992, 2482}, {23994, 561}, {24006, 811}, {27375, 27867}, {28654, 31625}, {30452, 13}, {30453, 14}, {30465, 298}, {30468, 299}, {31010, 4632}, {31065, 35137}, {31644, 14061}, {33919, 690}, {34079, 9273}, {34294, 83}, {34387, 18021}, {34980, 1092}, {34981, 7752}, {34982, 13198}, {35078, 4027}, {35088, 32458}, {35235, 340}, {35352, 4589}, {35364, 10425}
X115 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {2, 99, 620}, {2, 111, 10418}, {2, 148, 99}, {2, 620, 31274}, {2, 2482, 9167}, {2, 2549, 574}, {2, 3734, 7820}, {2, 5461, 14971}, {2, 7790, 4045}, {2, 9166, 5461}, {2, 11185, 3734}, {2, 14061, 6722}, {2, 26138, 25530}, {2, 26794, 27071}, {2, 26845, 27008}, {3, 7746, 7749}, {3, 7748, 7756}, {3, 13881, 7746}, {4, 32, 7747}, {4, 3767, 32}, {4, 7735, 7737}, {4, 9862, 10722}, {4, 11623, 10991}, {4, 14651, 98}, {5, 39, 1506}, {5, 3815, 7603}, {5, 5254, 39}, {5, 7765, 9698}, {5, 15048, 3815}, {6, 13, 5472}, {6, 14, 5471}, {6, 381, 5475}, {6, 1989, 3018}, {6, 3018, 3163}, {6, 5309, 5355}, {6, 5475, 7753}, {6, 18353, 1879}, {10, 4115, 21711}, {10, 24070, 4115}, {13, 5469, 14}, {13, 6777, 6778}, {14, 5470, 13}, {14, 6778, 6777}, {32, 3767, 7755}, {39, 1506, 9698}, {39, 5254, 7765}, {39, 7603, 3815}, {53, 1596, 33842}, {75, 21604, 21431}, {76, 626, 7794}, {76, 5025, 626}, {76, 7934, 3314}, {83, 7797, 7829}, {83, 15031, 16044}, {98, 10722, 9862}, {98, 14639, 4}, {98, 14651, 11623}, {99, 620, 2482}, {99, 671, 148}, {99, 6722, 31274}, {99, 9166, 14061}, {99, 14061, 2}, {114, 23514, 5}, {125, 1562, 3269}, {125, 3124, 6388}, {125, 6791, 1648}, {127, 339, 15526}, {141, 33184, 7853}, {148, 5461, 31274}, {148, 6722, 2482}, {148, 9166, 6722}, {148, 14061, 620}, {148, 31274, 15300}, {183, 7761, 7810}, {183, 7841, 7761}, {194, 7752, 7764}, {194, 32966, 7752}, {235, 27376, 3199}, {315, 7751, 7826}, {315, 14063, 7825}, {316, 14568, 385}, {325, 33228, 625}, {376, 21843, 8588}, {381, 5309, 7753}, {381, 12188, 6033}, {384, 7828, 6680}, {385, 14041, 316}, {403, 5523, 232}, {498, 9598, 31451}, {546, 5305, 7745}, {574, 11648, 2549}, {597, 20112, 3363}, {620, 5461, 6722}, {620, 6722, 2}, {620, 31274, 9167}, {631, 13172, 21166}, {661, 2170, 20982}, {671, 5461, 2482}, {671, 7919, 14931}, {671, 9166, 2}, {671, 14061, 99}, {671, 14971, 15300}, {1078, 6655, 7830}, {1086, 8287, 17058}, {1107, 25639, 31488}, {1506, 7765, 39}, {1648, 3124, 6791}, {1648, 6791, 6388}, {1648, 8288, 125}, {1656, 5013, 31455}, {1656, 13188, 15561}, {1656, 15561, 6721}, {1975, 3788, 7863}, {1975, 7887, 3788}, {1989, 6128, 3163}, {2079, 34866, 3}, {2276, 7951, 31476}, {2482, 14971, 2}, {2482, 31274, 620}, {2548, 5286, 7772}, {2643, 21043, 4092}, {2996, 32972, 3926}, {3018, 6128, 6}, {3090, 7738, 31401}, {3090, 23235, 20399}, {3091, 5286, 2548}, {3120, 21044, 3125}, {3124, 8288, 1648}, {3314, 5025, 7934}, {3314, 7934, 626}, {3545, 7736, 31415}, {3545, 12243, 6054}, {3734, 7844, 2}, {3734, 8178, 99}, {3734, 18546, 11185}, {3767, 7737, 7735}, {3815, 5254, 15048}, {3815, 7603, 1506}, {3815, 15048, 39}, {3845, 5306, 14537}, {3934, 6656, 6292}, {3934, 7861, 6656}, {5007, 5305, 5368}, {5008, 14537, 18907}, {5024, 5055, 31489}, {5080, 17737, 5291}, {5134, 17734, 17735}, {5139, 8754, 2971}, {5305, 7745, 5007}, {5306, 18907, 5008}, {5309, 5475, 6}, {5309, 18424, 5475}, {5355, 7753, 6}, {5461, 6722, 14061}, {5471, 5472, 5477}, {5475, 18424, 381}, {5989, 7770, 5149}, {6033, 11632, 12188}, {6034, 11632, 5309}, {6034, 11646, 6}, {6108, 6109, 6055}, {6114, 6115, 114}, {6392, 32816, 7758}, {6392, 32980, 32816}, {6564, 6565, 3818}, {6722, 14061, 14971}, {7668, 34294, 1084}, {7735, 7737, 32}, {7739, 31415, 7736}, {7746, 7748, 3}, {7747, 7755, 32}, {7748, 13881, 7749}, {7749, 7756, 3}, {7750, 33229, 7842}, {7751, 7825, 315}, {7754, 7759, 7890}, {7754, 7773, 7759}, {7758, 32816, 7903}, {7760, 7785, 7838}, {7763, 32961, 7862}, {7770, 7834, 7889}, {7770, 7851, 7834}, {7771, 17004, 34506}, {7775, 7798, 7774}, {7780, 7842, 7750}, {7781, 7862, 7763}, {7783, 32967, 7769}, {7789, 8361, 7874}, {7791, 32832, 7815}, {7792, 8370, 7804}, {7793, 33019, 7802}, {7795, 14064, 7867}, {7797, 16044, 83}, {7797, 32528, 4027}, {7800, 32974, 7935}, {7803, 16924, 7808}, {7804, 7817, 7792}, {7805, 7843, 7762}, {7806, 11361, 3972}, {7807, 32819, 7816}, {7808, 7902, 7803}, {7815, 7872, 7791}, {7816, 7886, 7807}, {7818, 17131, 69}, {7833, 17004, 7771}, {7844, 11185, 7820}, {7844, 18546, 3734}, {7853, 9466, 141}, {7864, 16921, 7786}, {7867, 17130, 7795}, {7885, 17129, 7768}, {7901, 17128, 7832}, {7912, 20081, 7796}, {7933, 31276, 3096}, {8591, 22247, 2482}, {9167, 15300, 2482}, {9293, 10278, 13187}, {9607, 31406, 39}, {10723, 34473, 20}, {10896, 16502, 9665}, {11318, 34505, 7801}, {11485, 13102, 23013}, {11486, 13103, 23006}, {11606, 16044, 32528}, {11648, 18362, 2}, {13636, 13722, 1648}, {13653, 13773, 2}, {13831, 13832, 1992}, {13873, 13926, 6036}, {14045, 17129, 7885}, {14136, 20252, 10611}, {14137, 20253, 10612}, {14644, 22265, 11005}, {15359, 16278, 15357}, {16092, 34169, 17964}, {16592, 16613, 1015}, {16960, 25236, 16529}, {16961, 25235, 16530}, {16962, 25166, 6780}, {16963, 25156, 6779}, {17008, 33017, 14907}, {20337, 23947, 3912}, {21043, 23938, 21709}, {22510, 23004, 15}, {22511, 23005, 16}, {25468, 25683, 2}, {27133, 27189, 2}, {30465, 30468, 125}, {31709, 31710, 9880}, {31862, 31863, 6}, {32457, 33228, 7813}, {32828, 32974, 7800}, {33813, 34127, 140}
Notes¶
The circumcircle of the incentral triangle intersects the nine-point circle at 2 points, X11 and X115, and X115 lies on the incentral circle and the cevian circle of every point on the Kiepert hyperbola. Let HaHbHc be the orthic triangle. The Brocard axes of AHbHc, BHcHa, CHaHb concur in X115. Let P be a point on the Brocard circle, and let L be the line tangent to the Brocard circle at P. Let P' be the trilinear pole of L, and let P″ be the isotomic conjugate of P'. As P varies, P″ traces an ellipse with center at X115. (Randy Hutson, July 23, 2015)
Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. 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 the Brocard axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X115; see Hyacinthos #16741/16782, Sep 2008.
Let F be the Feuerbach point, X11, and FaFbFc be the Feuerbach triangle (the extraversion triangle of X11). Let A' be the barycentric product F*Fa, and define B', C' cyclically. The lines AA', BB', CC' concur in X115. (Randy Hutson, January 29, 2018)
Roland H. Eddy and R. Fritsch, “The conics of Ludwig Kiepert: a comprehensive lesson in the geometry of the triangle,” Mathematics Magazine 67 (1994) 188-205.
For a construction of X115 as intersection point of four conics see Peter Moses, euclid 5328.
p> X115 = midpoint of Xi and Xj for these {i,j}: {1, 13178}, {2, 671}, {3, 6321}, {4, 98}, {6, 11646}, {8, 7983}, {10, 11599}, {13, 14}, {15, 23004}, {16, 23005}, {17, 11602}, {18, 11603}, {20, 10723}, {69, 10754}, {76, 1916}, {83, 11606}, {99, 148}, {100, 10769}, {125, 16278}, {226, 11608}, {265, 18332}, {316, 385}, {321, 11611}, {381, 11632}, {625, 32457}, {805, 31513}, {1131, 6568}, {1132, 6569}, {1650, 13179}, {1992, 11161}, {2009, 2010}, {2394, 14223}, {2679, 6071}, {2996, 8781}, {3448, 15342}, {3830, 14830}, {4052, 34899}, {5466, 9180}, {5469, 5470}, {5485, 5503}, {5912, 34169}, {6033, 12188}, {6054, 12243}, {6055, 9880}, {6108, 31710}, {6109, 31709}, {6772, 6775}, {6777, 6778}, {6779, 25156}, {6780, 25166}, {6782, 33517}, {6783, 33518}, {7809, 19570}, {7840, 11054}, {8029, 14443}, {8352, 22329}, {8587, 17503}, {9114, 22578}, {9116, 22577}, {9140, 9144}, {9293, 13187}, {9302, 14492}, {9862, 10722}, {10153, 32532}, {10418, 14832}, {11005, 22265}, {12042, 22515}, {13173, 13180}, {13181, 22514}, {14041, 14568}, {14639, 14651}, {16001, 25560}, {16002, 25559}, {19905, 20423}, {22512, 22513}, {24345, 24711}