X₁₃₁₉

BEVAN-SCHRÖDER POINT

X₁₃₁₉ = circumcircle-inverse of X₅₆

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₃₁₉ (BEVAN-SCHRÖDER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {11, 515}

• Related Centers: X₁ (INCENTER), X₃ (CIRCUMCENTER), X₁₁ (FEUERBACH POINT), X₅₁₅ (ISOGONAL CONJUGATE OF X₁₀₂)

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

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

Search Value (6, 9, 13): \(-0.442489138601\)

▸ Alternate Coordinates1 form

• Trilinears: \(2 - 2 \cos A - \cos B - \cos C\)

▸ Linear Combinations2 combinations

• X₁₃₁₉ = \((R-r)\) X₁ - \(r\) X₃

• X₁₃₁₉ = \(3\) X₁ + X₄₈₄, \(2\) X₁ + X₁₁₅₅, \(5\) X₁ + X₃₂₄₅, \(3\) X₁ + \(2\) X₅₁₂₂, X₁ + \(2\) X₅₁₂₆, \(5\) X₁ + \(3\) X₅₁₃₁, \(4\) X₁ + X₅₁₈₃, \(3\) X₃ - X₃₅₄₆₀, \(3\) X₃₆ - X₄₈₄, \(5\) X₃₆ - X₃₂₄₅, \(2\) X₃₆ + X₅₀₄₈, \(3\) X₃₆ - \(2\) X₅₁₂₂, \(5\) X₃₆ - \(3\) X₅₁₃₁, \(4\) X₃₆ - X₅₁₈₃, X₃₆ + \(2\) X₂₅₄₀₅, X₅₇ - \(3\) X₅₁₉₃, X₆₅ - \(4\) X₃₆₆₀, \(3\) X₃₅₄ - \(2\) X₅₅₇₀, , and many others

X₁₃₁₉ lies on the curves K338, K826, K1160, and Q071.

▸ Incident Central Lines563 lines

{1, 3}, {2, 3476}, {4, 7704}, {5, 45287}, {7, 2320}, {8, 6049}, {10, 5433}, {11, 515}, {12, 1125}, {19, 37519}, {20, 12701}, {21, 1408}, {30, 1387}, {33, 37391}, {34, 1878}, {37, 604}, {41, 40133}, {42, 1450}, {44, 1404}, {48, 1108}, {59, 518}, {63, 5289}, {72, 8666}, {73, 1104}, {77, 1122}, {78, 12513}, {79, 31776}, {80, 3582}, {85, 24805}, {100, 3880}, {101, 2348}, {104, 2720}, {105, 14733}, {106, 1168}, {108, 953}, {109, 2718}, {140, 10039}, {145, 1788}, {198, 3554}, {210, 956}, {214, 519}, {221, 1616}, {222, 16483}, {223, 16485}, {225, 23675}, {226, 535}, {227, 3924}, {238, 9282}, {269, 35227}, {278, 5146}, {279, 24796}, {329, 34610}, {348, 24798}, {355, 499}, {376, 30305}, {381, 23708}, {388, 2478}, {392, 993}, {404, 4861}, {405, 9850}, {442, 10957}, {474, 3698}, {495, 38028}, {496, 10572}, {497, 5731}, {513, 663}, {516, 15326}, {528, 30379}, {529, 908}, {536, 43135}, {553, 51103}, {572, 38855}, {595, 1399}, {603, 3915}, {611, 38029}, {614, 37366}, {651, 3246}, {664, 1447}, {672, 6603}, {679, 1318}, {751, 1386}, {758, 5083}, {759, 34921}, {840, 934}, {859, 18191}, {901, 1417}, {909, 2161}, {910, 1055}, {912, 6265}, {936, 3983}, {938, 6962}, {943, 15179}, {944, 1837}, {946, 4311}, {950, 37722}, {952, 1737}, {958, 3305}, {959, 25417}, {960, 2975}, {961, 1255}, {1000, 3524}, {1001, 8545}, {1006, 44085}, {1014, 7269}, {1015, 43039}, {1042, 20615}, {1056, 6947}, {1058, 4305}, {1071, 7335}, {1100, 1400}, {1191, 34046}, {1193, 2594}, {1210, 5882}, {1212, 9310}, {1254, 46190}, {1308, 1477}, {1320, 13587}, {1323, 1358}, {1329, 26482}, {1354, 31522}, {1357, 11717}, {1359, 3326}, {1360, 3328}, {1361, 3025}, {1362, 11712}, {1364, 11713}, {1365, 31524}, {1376, 3872}, {1389, 37518}, {1401, 49480}, {1405, 16666}, {1406, 34040}, {1407, 16486}, {1409, 16685}, {1412, 4653}, {1415, 1914}, {1418, 7225}, {1419, 16487}, {1434, 4955}, {1442, 24471}, {1445, 42871}, {1452, 11396}, {1471, 49478}, {1478, 5886}, {1479, 11373}, {1519, 2829}, {1538, 12764}, {1575, 21859}, {1612, 26888}, {1621, 10179}, {1647, 14584}, {1656, 10827}, {1698, 37709}, {1699, 12943}, {1723, 20818}, {1727, 48667}, {1743, 38296}, {1770, 22791}, {1836, 4293}, {1841, 7120}, {1846, 1877}, {1858, 12675}, {1864, 18446}, {1876, 3446}, {1887, 6198}, {1898, 6261}, {1935, 9363}, {1960, 6550}, {2003, 5315}, {2067, 7968}, {2082, 3207}, {2087, 2251}, {2136, 15347}, {2171, 3723}, {2178, 2262}, {2208, 2218}, {2260, 17438}, {2263, 15306}, {2285, 16777}, {2329, 36476}, {2360, 40980}, {2362, 44635}, {2390, 3937}, {2551, 24954}, {2650, 42443}, {2703, 35108}, {2716, 8059}, {2745, 30239}, {2771, 47379}, {2800, 17010}, {2802, 35271}, {3011, 6075}, {3021, 39760}, {3022, 11714}, {3023, 11710}, {3024, 11709}, {3027, 11711}, {3028, 11720}, {3035, 6735}, {3058, 4304}, {3085, 6967}, {3146, 18220}, {3160, 7195}, {3218, 44663}, {3230, 4559}, {3241, 5435}, {3243, 41712}, {3244, 4848}, {3286, 43947}, {3290, 9259}, {3306, 40726}, {3320, 11722}, {3324, 11718}, {3325, 11721}, {3419, 45700}, {3436, 25681}, {3452, 34606}, {3475, 6992}, {3485, 3600}, {3486, 6838}, {3487, 6936}, {3555, 22836}, {3583, 7743}, {3584, 5444}, {3585, 9955}, {3586, 11238}, {3624, 9578}, {3634, 7294}, {3636, 3649}, {3638, 49538}, {3639, 49540}, {3653, 10056}, {3655, 5722}, {3671, 4114}, {3674, 7198}, {3679, 31231}, {3720, 40109}, {3752, 49487}, {3769, 20037}, {3812, 5253}, {3827, 38863}, {3877, 4640}, {3878, 3916}, {3881, 15556}, {3884, 5267}, {3890, 4189}, {3893, 5687}, {3895, 4421}, {3912, 43053}, {3913, 4855}, {3947, 15808}, {3962, 5730}, {4059, 7176}, {4124, 47043}, {4188, 14923}, {4225, 18178}, {4292, 13464}, {4295, 10595}, {4296, 35998}, {4297, 6284}, {4299, 7702}, {4316, 28146}, {4318, 40577}, {4321, 38316}, {4333, 48661}, {4345, 9778}, {4351, 23152}, {4390, 44798}, {4413, 4731}, {4432, 24816}, {4487, 17780}, {4534, 8074}, {4551, 49997}, {4654, 51105}, {4702, 4742}, {4719, 17016}, {4853, 5438}, {4863, 34625}, {4868, 26740}, {4906, 17080}, {4915, 46917}, {4995, 50828}, {4999, 24987}, {5044, 5258}, {5088, 24203}, {5089, 32674}, {5219, 11237}, {5220, 30318}, {5248, 12709}, {5261, 46934}, {5270, 5443}, {5288, 34790}, {5432, 10165}, {5439, 30147}, {5441, 31795}, {5450, 12672}, {5453, 13391}, {5541, 41702}, {5550, 10588}, {5552, 32049}, {5572, 30284}, {5577, 33902}, {5691, 10896}, {5718, 6176}, {5724, 24239}, {5745, 31157}, {5784, 42842}, {5790, 37708}, {5794, 10527}, {5844, 12735}, {5853, 41555}, {5854, 51433}, {5887, 32153}, {5901, 12047}, {5905, 34647}, {6020, 12265}, {6224, 37797}, {6285, 12262}, {6502, 7969}, {6604, 17081}, {6647, 20335}, {6691, 24982}, {6700, 21031}, {6880, 7967}, {6882, 38032}, {6906, 45776}, {7051, 7052}, {7117, 8608}, {7173, 19925}, {7175, 16484}, {7178, 48328}, {7181, 9436}, {7191, 35996}, {7223, 40719}, {7235, 39766}, {7247, 17084}, {7284, 28444}, {7292, 14513}, {7340, 47376}, {7355, 40658}, {7491, 39599}, {7672, 15570}, {7741, 18480}, {7951, 11230}, {7972, 22935}, {8077, 10506}, {8227, 9613}, {8283, 21147}, {8286, 36195}, {8607, 47434}, {8649, 49758}, {8983, 19028}, {9318, 44664}, {9552, 19858}, {9579, 11522}, {9583, 19038}, {9612, 9624}, {9614, 12953}, {9615, 31432}, {9654, 37692}, {9655, 18493}, {9661, 49601}, {9708, 35272}, {9848, 10884}, {9956, 37710}, {10016, 34182}, {10081, 11670}, {10090, 12737}, {10283, 39542}, {10320, 26492}, {10391, 18444}, {10401, 17321}, {10428, 34051}, {10459, 28385}, {10483, 22793}, {10501, 18456}, {10502, 18454}, {10503, 18448}, {10573, 37727}, {10609, 41552}, {10624, 15338}, {10826, 18525}, {10866, 12520}, {10914, 22837}, {10934, 18621}, {10949, 17647}, {10956, 17757}, {11236, 30852}, {11281, 33961}, {11346, 28997}, {11364, 12835}, {11365, 18954}, {11368, 18957}, {11370, 18959}, {11371, 18960}, {11374, 18962}, {11377, 18963}, {11378, 18964}, {11496, 17634}, {11570, 14988}, {11699, 19470}, {11705, 18974}, {11706, 18975}, {11726, 34929}, {11735, 46683}, {11739, 18973}, {11740, 18972}, {11831, 18958}, {12019, 28224}, {12081, 20718}, {12114, 12688}, {12119, 13274}, {12258, 18969}, {12259, 18970}, {12260, 18979}, {12261, 18968}, {12263, 18982}, {12264, 18983}, {12266, 18984}, {12267, 18985}, {12268, 18989}, {12269, 18988}, {12607, 27385}, {12647, 26446}, {12749, 38752}, {12758, 38602}, {12848, 51099}, {13182, 38220}, {13374, 45977}, {13407, 37737}, {13411, 15888}, {13607, 37734}, {13667, 18986}, {13787, 18987}, {13883, 18965}, {13902, 31408}, {13936, 18966}, {13971, 19027}, {14100, 42884}, {14204, 40437}, {14257, 38295}, {14597, 21770}, {14872, 45770}, {15015, 48696}, {15175, 15180}, {15228, 28198}, {15298, 38031}, {15500, 37305}, {15558, 17613}, {15726, 18450}, {15746, 47006}, {15999, 17114}, {16174, 24042}, {16232, 44636}, {16417, 40587}, {16609, 50023}, {16826, 41245}, {16858, 29007}, {17044, 51400}, {17053, 40590}, {17221, 17863}, {17284, 31230}, {17448, 41526}, {17566, 32537}, {17604, 30283}, {17611, 30285}, {17637, 33858}, {17768, 51423}, {18242, 26476}, {18395, 37707}, {18458, 30375}, {18460, 30376}, {18469, 30377}, {18471, 30378}, {18514, 33697}, {18526, 37711}, {18593, 24201}, {18654, 24993}, {18971, 22475}, {18978, 22476}, {18991, 18995}, {18992, 18996}, {19365, 44547}, {19373, 33655}, {19860, 25524}, {21008, 41015}, {21214, 37694}, {21342, 49454}, {21871, 36743}, {22344, 23844}, {22345, 23846}, {23154, 41682}, {23205, 23845}, {23711, 51359}, {24029, 34230}, {24036, 41391}, {24216, 43056}, {24331, 36487}, {24465, 28174}, {24541, 25466}, {24583, 26581}, {24612, 30812}, {24871, 25026}, {25904, 25914}, {26015, 41557}, {26115, 26126}, {26321, 31937}, {26367, 26433}, {26368, 26434}, {26369, 26435}, {26370, 26436}, {26690, 30618}, {26959, 28771}, {28027, 28036}, {28077, 28082}, {28461, 41695}, {28969, 33839}, {29604, 31221}, {29660, 36493}, {29817, 35989}, {30275, 38053}, {30827, 31141}, {31053, 34605}, {31225, 36534}, {31394, 49537}, {32238, 32243}, {32331, 32336}, {32426, 35023}, {33337, 33598}, {33595, 51071}, {33646, 51361}, {33667, 39778}, {34195, 41697}, {34522, 40131}, {34586, 38984}, {34772, 34791}, {35762, 35768}, {35763, 35769}, {36123, 45766}, {36444, 36451}, {36505, 36513}, {36926, 37758}, {36944, 38462}, {37006, 37718}, {37139, 39308}, {37168, 37790}, {37728, 50824}, {37815, 45272}, {39870, 39873}, {39897, 49511}, {41389, 51506}, {43052, 48344}, {43054, 49768}, {43820, 43822}, {45398, 45404}, {45399, 45405}, {45500, 45506}, {45501, 45507}, {48894, 49745}, {50148, 51345}, {50594, 50604}, {51625, 51628}, {51627, 51629}

▸ Functional Relations29 relations

• X₁₃₁₉ = intersection of X₁X₃ and the Helman line (the radical axis of the incircle and circumcircle, i.e., X₅₁₃X₆₆₃; see the preamble just before X₅₁₆₄₀).

• X₁₃₁₉ = midpoint of X_i and X_j for these {i,j}: {1, 36}, {100, 38460}, {1155, 5048}, {1317, 40663}, {2718, 47622}, {3583, 36975}, {4318, 40577}, {5057, 20067}, {5126, 25405}, {5541, 41702}, {7972, 41684}, {10222, 10225}, {14151, 37787}, {21578, 30384}, {31524, 39751}

• X₁₃₁₉ = reflection of X_i in X_j for these {i,j}: {1, 25405}, {3, 18857}, {4, 22835}, {10, 6681}, {11, 44675}, {36, 5126}, {65, 18838}, {484, 5122}, {1155, 36}, {1737, 15325}, {3326, 51616}, {3583, 7743}, {3689, 5440}, {3814, 1125}, {5048, 1}, {5080, 5087}, {5176, 5123}, {5183, 1155}, {5440, 214}, {6735, 3035}, {13528, 3}, {18838, 3660}, {23960, 33179}, {24042, 16174}, {30384, 1387}, {36920, 40663}, {40663, 3911}, {41542, 5427}, {41698, 1538}, {44784, 1145}

• X₁₃₁₉ = isogonal conjugate of X₁₃₂₀

• X₁₃₁₉ = complement of X₅₁₇₆

• X₁₃₁₉ = anticomplement of X₅₁₂₃

• X₁₃₁₉ = circumcircle-inverse of X₅₆

• X₁₃₁₉ = incircle-inverse of X₆₅

• X₁₃₁₉ = Bevan-circle-inverse of X₅₁₂₈

• X₁₃₁₉ = Conway-circle-inverse of X₁₂₄₃₅

• X₁₃₁₉ = isogonal conjugate of the anticomplement of X₁₁₄₅

• X₁₃₁₉ = isogonal conjugate of the polar conjugate of X₃₇₇₉₀

• X₁₃₁₉ = X_i-Ceva conjugate of X_j for these (i,j): {8, 13539}, {1411, 65}, {1443, 1465}, {1465, 2182}, {2222, 513}, {3911, 44}, {8686, 56}, {24029, 654}, {34051, 6}, {36944, 51422}, {37136, 650}, {37168, 1877}

• X₁₃₁₉ = X_i-cross conjugate of X_j for these (i,j): {902, 44}, {2087, 30725}

• X₁₃₁₉ = cevapoint of X_i and X_j for these (i,j): {902, 1404}, {1647, 30572}, {1960, 2087}

• X₁₃₁₉ = crosspoint of X_i and X_j for these (i,j): {1, 104}, {7, 2006}, {59, 2720}, {7052, 33655}

• X₁₃₁₉ = crosssum of X_i and X_j for these (i,j): {1, 517}, {9, 3689}, {11, 2804}, {55, 2323}, {521, 35014}, {2170, 4895}, {3880, 45247}, {4867, 40587}, {5239, 5240}

• X₁₃₁₉ = trilinear pole of line {1635, 20972}

• X₁₃₁₉ = crossdifference of every pair of points on line {9, 650}

• X₁₃₁₉ = X₁₈₆-of-intouch-triangle

• X₁₃₁₉ = X₂₀₇₇-of-Mandart-incircle- triangle

• X₁₃₁₉ = homothetic center of intangents triangle and reflection of extangents triangle in X₂₀₇₇

• X₁₃₁₉ = orthocenter of pedal triangle of X₃₆

• X₁₃₁₉ = perspector of ABC and reflection of extangents triangle in antiorthic axis

• X₁₃₁₉ = X_i-isoconjugate of X_j for these (i,j): {1, 1320}, {2, 2316}, {6, 4997}, {8, 106}, {9, 88}, {11, 9268}, {21, 4674}, {36, 36590}, {41, 20568}, {55, 903}, {60, 4013}, {78, 36125}, {100, 23838}, {219, 6336}, {281, 1797}, {284, 4080}, {312, 9456}, {318, 36058}, {341, 1417}, {345, 8752}, {514, 5548}, {519, 1318}, {522, 901}, {644, 1022}, {649, 4582}, {650, 3257}, {663, 4555}, {679, 3689}, {999, 36596}, {1120, 45247}, {1168, 4511}, {1639, 4638}, {2170, 5376}, {2226, 2325}, {2320, 4792}, {2364, 4945}, {3699, 23345}, {3700, 4591}, {3709, 4615}, {3939, 6548}, {4041, 4622}, {4049, 5546}, {4076, 43922}, {4391, 32665}, {4543, 39414}, {4618, 4895}, {4845, 36887}, {5549, 23598}, {6065, 6549}, {6551, 21132}, {6735, 10428}, {7017, 32659}, {7077, 27922}, {8851, 36814}, {14260, 51565}, {14942, 34230}, {32719, 35519}, {36910, 40215}

• X₁₃₁₉ = X_i-Dao conjugate of X_j for these (i,j): {3, 1320}, {8, 214}, {9, 4997}, {44, 32851}, {88, 478}, {223, 903}, {312, 4370}, {318, 20619}, {519, 4723}, {522, 38979}, {1319, 5176}, {1647, 4768}, {2316, 32664}, {3160, 20568}, {3262, 3911}, {4080, 40590}, {4391, 35092}, {4397, 51402}, {4582, 5375}, {4674, 40611}, {4858, 6544}, {6548, 40617}, {8054, 23838}, {15898, 36590}

• X₁₃₁₉ = barycentric product X_i*X_j for these {i,j}: {1, 3911}, {3, 37790}, {7, 44}, {12, 30576}, {34, 3977}, {36, 14628}, {56, 4358}, {57, 519}, {63, 1877}, {65, 16704}, {75, 1404}, {77, 8756}, {81, 40663}, {85, 902}, {88, 1317}, {89, 36920}, {100, 30725}, {109, 3762}, {214, 2006}, {222, 38462}, {269, 2325}, {273, 22356}, {278, 5440}, {279, 3689}, {331, 23202}, {514, 23703}, {517, 40218}, {603, 46109}, {604, 3264}, {651, 900}, {658, 4895}, {662, 30572}, {664, 1635}, {934, 1639}, {1014, 3943}, {1023, 3676}, {1145, 34051}, {1170, 51463}, {1214, 37168}, {1254, 30606}, {1255, 5298}, {1400, 30939}, {1407, 4723}, {1411, 51583}, {1412, 3992}, {1414, 4120}, {1417, 36791}, {1432, 4434}, {1434, 21805}, {1441, 3285}, {1461, 4768}, {1465, 36944}, {1476, 51415}, {1647, 4564}, {1960, 4554}, {2087, 4998}, {2161, 41801}, {2251, 6063}, {2990, 12832}, {3218, 14584}, {3257, 39771}, {3669, 17780}, {4169, 7203}, {4487, 40151}, {4528, 4617}, {4530, 7045}, {4573, 4730}, {4625, 14407}, {4626, 14427}, {4702, 42290}, {4922, 37137}, {5376, 14027}, {5723, 14191}, {6174, 34056}, {6550, 31615}, {6630, 14122}, {7052, 36668}, {8686, 16594}, {9459, 20567}, {13462, 36915}, {14418, 36118}, {17078, 40172}, {17455, 18815}, {18026, 22086}, {20332, 24816}, {21907, 41541}, {23344, 24002}, {23757, 37136}, {24004, 43924}, {30573, 37139}, {30731, 43932}, {31011, 32636}, {33655, 36669}, {34578, 41553}, {36100, 51422}, {39155, 45273}, {43736, 51406}

• X₁₃₁₉ = barycentric quotient X_i/X_j for these {i,j}: {1, 4997}, {6, 1320}, {7, 20568}, {31, 2316}, {34, 6336}, {44, 8}, {56, 88}, {57, 903}, {59, 5376}, {65, 4080}, {100, 4582}, {109, 3257}, {214, 32851}, {519, 312}, {603, 1797}, {604, 106}, {608, 36125}, {649, 23838}, {651, 4555}, {678, 2325}, {692, 5548}, {900, 4391}, {902, 9}, {1017, 3689}, {1023, 3699}, {1317, 4358}, {1395, 8752}, {1397, 9456}, {1400, 4674}, {1404, 1}, {1405, 4792}, {1414, 4615}, {1415, 901}, {1417, 2226}, {1420, 31227}, {1429, 27922}, {1635, 522}, {1639, 4397}, {1647, 4858}, {1877, 92}, {1960, 650}, {2087, 11}, {2099, 4945}, {2149, 9268}, {2161, 36590}, {2171, 4013}, {2251, 55}, {2325, 341}, {2429, 31343}, {3251, 1639}, {3264, 28659}, {3285, 21}, {3669, 6548}, {3689, 346}, {3762, 35519}, {3911, 75}, {3943, 3701}, {3977, 3718}, {3992, 30713}, {4017, 4049}, {4120, 4086}, {4358, 3596}, {4370, 4723}, {4432, 3975}, {4434, 17787}, {4487, 44723}, {4530, 24026}, {4565, 4622}, {4573, 4634}, {4700, 4673}, {4702, 28809}, {4730, 3700}, {4773, 4811}, {4819, 42712}, {4895, 3239}, {4969, 3702}, {4984, 4985}, {5298, 4359}, {5440, 345}, {6544, 4768}, {6550, 40166}, {6610, 36887}, {8756, 318}, {9456, 1318}, {9459, 41}, {12832, 48380}, {14122, 4440}, {14407, 4041}, {14408, 4147}, {14427, 4163}, {14437, 14430}, {14439, 3717}, {14584, 18359}, {14628, 20566}, {16704, 314}, {17455, 4511}, {17780, 646}, {20972, 3880}, {21805, 2321}, {22086, 521}, {22356, 78}, {23202, 219}, {23344, 644}, {23703, 190}, {30572, 1577}, {30576, 261}, {30725, 693}, {30939, 28660}, {31615, 6635}, {36920, 4671}, {36944, 36795}, {37168, 31623}, {37790, 264}, {38462, 7017}, {39251, 3883}, {39771, 3762}, {40172, 36910}, {40218, 18816}, {40663, 321}, {41541, 32849}, {41553, 17264}, {41554, 37758}, {41556, 37788}, {41801, 20924}, {42084, 4530}, {43924, 1022}, {47420, 35014}, {47425, 45269}, {51415, 20895}, {51463, 1229}

• X₁₃₁₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 3, 3057}, {1, 35, 9957}, {1, 40, 2098}, {1, 46, 1482}, {1, 55, 5919}, {1, 56, 65}, {1, 57, 2099}, {1, 65, 11011}, {1, 165, 7962}, {1, 260, 10508}, {1, 354, 44840}, {1, 988, 37614}, {1, 999, 354}, {1, 1381, 2447}, {1, 1382, 2446}, {1, 1385, 2646}, {1, 1420, 56}, {1, 1482, 33176}, {1, 2093, 16200}, {1, 2646, 37080}, {1, 3304, 17609}, {1, 3336, 11009}, {1, 3361, 3340}, {1, 3428, 17642}, {1, 3576, 55}, {1, 3601, 3303}, {1, 3612, 3295}, {1, 3746, 31792}, {1, 5126, 1155}, {1, 5193, 18838}, {1, 5563, 942}, {1, 5902, 50194}, {1, 5903, 10222}, {1, 7280, 5697}, {1, 7742, 14110}, {1, 7987, 1697}, {1, 10389, 8162}, {1, 10882, 10480}, {1, 11009, 33179}, {1, 13370, 13601}, {1, 13462, 57}, {1, 15803, 7982}, {1, 21842, 1385}, {1, 24928, 20323}, {1, 24929, 3748}, {1, 25415, 10247}, {1, 30282, 31393}, {1, 30389, 3601}, {1, 30392, 13384}, {1, 32612, 25414}, {1, 37524, 11278}, {1, 37525, 24929}, {1, 37552, 37542}, {1, 37587, 5902}, {1, 37602, 5049}, {1, 37605, 37568}, {1, 37616, 3746}, {1, 37617, 3666}, {1, 37618, 3}, {1, 37620, 21334}, {1, 41426, 37566}, {2, 3476, 5252}, {2, 5176, 5123}, {3, 56, 34880}, {3, 999, 22767}, {3, 3057, 37568}, {3, 10680, 40255}, {3, 34489, 37566}, {3, 37618, 37605}, {3, 41426, 56}, {8, 6921, 37828}, {8, 7288, 24914}, {8, 37828, 37829}, {36, 484, 5122}, {36, 2078, 5172}, {36, 3245, 5131}, {36, 5048, 5183}, {36, 5193, 56}, {36, 25405, 5048}, {36, 32760, 3}, {48, 1108, 2264}, {55, 56, 1470}, {55, 3576, 37600}, {56, 65, 32636}, {56, 1388, 1}, {56, 2099, 57}, {56, 3303, 1466}, {56, 3304, 26437}, {56, 5172, 36}, {56, 5221, 3361}, {56, 11510, 3}, {57, 1420, 13462}, {57, 2099, 65}, {57, 13462, 56}, {63, 5289, 31165}, {65, 13751, 942}, {101, 43065, 2348}, {104, 12740, 17638}, {108, 1870, 1875}, {145, 1788, 41687}, {145, 5265, 1788}, {214, 1317, 41541}, {214, 41554, 1317}, {226, 551, 15950}, {226, 4315, 5434}, {226, 15950, 4870}, {348, 30617, 24798}, {355, 499, 17606}, {388, 3616, 11375}, {392, 993, 3683}, {404, 4861, 5836}, {484, 5119, 35460}, {484, 5122, 1155}, {484, 30282, 2077}, {496, 34773, 10572}, {551, 4315, 226}, {551, 5434, 4870}, {559, 1082, 3666}, {664, 1447, 43037}, {672, 17439, 6603}, {934, 38459, 34855}, {942, 15178, 1}, {944, 3086, 1837}, {946, 4311, 7354}, {956, 997, 210}, {958, 19861, 25917}, {999, 1617, 56}, {999, 10246, 1}, {999, 41345, 22765}, {1055, 2170, 910}, {1125, 10106, 12}, {1149, 1458, 1457}, {1155, 2646, 50371}, {1201, 4322, 73}, {1201, 28386, 28389}, {1210, 5882, 10950}, {1279, 1458, 1456}, {1284, 1458, 1463}, {1317, 3911, 36920}, {1317, 5298, 40663}, {1381, 1382, 56}, {1385, 10222, 26287}, {1385, 15178, 24299}, {1385, 20323, 37080}, {1385, 24928, 1}, {1385, 24929, 37525}, {1388, 1420, 65}, {1403, 24806, 65}, {1420, 2078, 5126}, {1420, 34489, 41426}, {1455, 1457, 1456}, {1457, 1458, 1464}, {1467, 3303, 65}, {1467, 3601, 1466}, {1478, 5886, 17605}, {1697, 7987, 5217}, {2078, 3660, 1155}, {2078, 5193, 36}, {2098, 5204, 40}, {2446, 2447, 65}, {2448, 2449, 5128}, {2646, 3748, 24929}, {2646, 20323, 1}, {3057, 37566, 65}, {3057, 37605, 3}, {3304, 34471, 1}, {3336, 11009, 50193}, {3340, 3361, 5221}, {3340, 5221, 65}, {3485, 3600, 10404}, {3513, 3514, 1155}, {3576, 31393, 30282}, {3583, 16173, 7743}, {3585, 37735, 9955}, {3586, 37704, 11238}, {3600, 3622, 3485}, {3616, 4308, 388}, {3660, 5126, 56}, {3748, 24929, 37080}, {3872, 35262, 1376}, {4293, 5603, 1836}, {4297, 12053, 6284}, {4413, 9623, 4731}, {4855, 36846, 3913}, {4881, 38460, 100}, {5172, 18838, 1155}, {5289, 11194, 63}, {5298, 40663, 3911}, {5425, 14798, 5119}, {5433, 10944, 10}, {5434, 15950, 226}, {5537, 11575, 1155}, {5552, 36977, 32049}, {5563, 37583, 56}, {5570, 41345, 1155}, {5597, 5598, 37541}, {5597, 26404, 65}, {5598, 26380, 65}, {5691, 50443, 10896}, {5697, 7280, 3579}, {5901, 18990, 12047}, {5919, 37600, 55}, {6049, 7288, 37738}, {6261, 22760, 1898}, {6265, 10074, 17660}, {7113, 8609, 2182}, {7677, 14151, 37787}, {7967, 18391, 37740}, {7982, 15803, 37567}, {8227, 9613, 10895}, {8666, 30144, 72}, {9957, 13624, 35}, {10090, 12737, 17636}, {10165, 31397, 5432}, {10222, 26287, 33596}, {10222, 37582, 5903}, {10247, 36279, 25415}, {10857, 30389, 3576}, {11011, 32636, 65}, {11014, 37561, 31788}, {11373, 18481, 1479}, {11510, 34489, 65}, {11510, 37566, 37568}, {11510, 41426, 34880}, {13388, 13389, 17595}, {13751, 37583, 32636}, {14792, 14795, 33862}, {15934, 41345, 484}, {16173, 36975, 3583}, {17642, 37578, 7964}, {17728, 37740, 18391}, {18839, 41341, 1155}, {21842, 24928, 2646}, {22765, 41345, 36}, {22837, 25440, 10914}, {24914, 37738, 8}, {24929, 37525, 2646}, {26365, 26366, 37606}, {30282, 31393, 55}, {32760, 34489, 18838}, {32760, 37618, 18857}, {33179, 50193, 11009}, {33812, 41558, 1317}, {34489, 37618, 56}, {34880, 37566, 32636}, {36920, 41541, 3689}, {37566, 37605, 34880}, {37704, 50811, 3586}, {38013, 38014, 50371}, {41556, 50843, 1317}

Notes

Let X'Y'Z' be the pedal triangle of the Bevan point, W = X₄₀; then X₁₃₁₉ is the point, other than W, in which the circles AWX’, BWY’, CWZ’ concur. (Floor van Lamoen, Hyacinthos #6321, 6352).

1. The Helman line is the radical axis of the Zaslavsky pencil, containing the circumcircle and incenter (A. Zaslavsky, August, 2022)

2. the isogonal image of the Helman line is the I-centered circumellipse having semiaxes of lengths 2R(R-r+d) and 2R(R-r-d), where d = |OI. Accordingly, over Chapple’s porism (with fixed R an r), the I-circumellipse rotates rigidly about its center. (D. Reznik et al., August, 2022). See GeoGebra sketch

3. Over Chapple’s porism, the locus of each point on the Helman line is a circle, called a Skutin circle, of radius R. (D. Reznik, August, 2022). This is a special case of a result in A. Skutin, “On Rotation of Isogonal Point”, J. Classical Geom., vol 2, 2013.

4. The set of Skutin circles corresponding to all the points on a given circle in the Zaslavsky pencil all have the same radius. The locus of the center is a circle. (D. Reznik, August, 2022). See GeoGebra sketch