X574

HARMONIC OF X187

“Each one of the three circles of Apollonius of the reference triangle cuts the circumcircle in a second point. These three points form a triangle which is perspective with the base triangle through the symmedian point, the co-symmedian triangle. This triangle also possesses a Neuberg cubic, which has three points in common with the cubic of the first triangle in the circumcenter and the two isodynamic centers. Hence the remaining points of intersection must lie on a conic, which is necessarily a circle, since two of these six points are the circular points I and J.” Quoted from (T. W. Moore and J. H. Neelley). The center of the circle is X574. (Bernard Gibert, euclid 4332)
< 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) { 3, 6 }: Brocard Axis { 2, 99 }: Central Line L(2, 99) { 110, 353 }: Central Line L(110, 353) { 232, 378 }: Central Line L(232, 378) { 183, 538 }: Central Line L(183, 538) { 230, 549 }: Central Line L(230, 549) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_2 = CENTROID (0.5374, 0.2000) X_99 = STEINER POINT (0.1109, 0.4322) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_353 = INVERSE-IN-BROCARD-CIRCLE OF X(111) (0.5667, 0.1759) X_232 = -CEVA CONJUGATE OF X(132) (0.6919, 0.7779) X_378 = REFLECTION OF X(22) IN X(3) (0.5857, 0.3262) X_183 = TRILINEAR PRODUCT X(75)X(182) (0.4700, 0.1275) X_538 = DIRECTION OF VECTOR AX + BX + CX, where X = X(39) (1.0053, 0.5542) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_549 = MIDPOINT OF X(2) AND X(3) (0.5187, 0.1510) X_574 = HARMONIC OF X(187) (0.5539, 0.1910) A B C X3 X6 X2 X99 X110 X353 X232 X378 X183 X538 X230 X549 X574 (a, b, c) = (5, 6, 7) | AB = 1

X574 (HARMONIC OF X187) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sin A + \cot D/2 \cos A\), where \(\cot(D/2) = 12 \operatorname{area}/(a^{2} + b^{2} + c^{2})\)
Barycentrics: \(a^{2} (a^{2} - 2 b^{2} - 2 c^{2}) : :\)
Search Value (6, 9, 13): \(2.424876309934\)
▸ Alternate Coordinates5 forms

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

  • Trilinears: \(\sin A + 3 \cos A \tan \omega : :\) (Peter J. C. Moses, 8/22/03)

  • Trilinears: \(2 \sin(A + 2\omega ) - \sin(A - 2\omega ) + \sin A : :\)

  • Trilinears: \(3 \cos A + \sin A \cot \omega : :\)

  • Trilinears: \(\cos(A - \omega ) \sin 2\omega - e^{2} \sin(A + \omega ) : :\)

▸ Linear Combinations1 combination

  • X574 = \(2\) X6 - \(3\) X5034, X7774 + \(3\) X33008, \(5\) X12017 - \(3\) X35429, X14907 - \(3\) X33008, X17131 + \(2\) X31859

X574 lies on the cubics K284, K353, K751, K1125.

▸ Incident Central Lines518 lines

{1, 1571}, {2, 99}, {3, 6}, {4, 1506}, {5, 3055}, {10, 31456}, {11, 9664}, {12, 9651}, {20, 2548}, {22, 9699}, {23, 15302}, {24, 9700}, {25, 15433}, {30, 3815}, {35, 2241}, {36, 2242}, {37, 37599}, {40, 9619}, {42, 9346}, {51, 41275}, {55, 1015}, {56, 1500}, {69, 7810}, {74, 15920}, {76, 7781}, {83, 3552}, {98, 1569}, {100, 16975}, {101, 28563}, {106, 43077}, {109, 41160}, {110, 353}, {112, 14388}, {114, 37242}, {140, 3054}, {141, 6390}, {165, 1572}, {172, 7280}, {183, 538}, {184, 3269}, {186, 10311}, {194, 1078}, {217, 1204}, {230, 549}, {232, 378}, {237, 34417}, {248, 3431}, {262, 11170}, {274, 17684}, {315, 7764}, {316, 7775}, {325, 7761}, {352, 7998}, {376, 6781}, {381, 7603}, {382, 31467}, {384, 7782}, {385, 7757}, {395, 13084}, {396, 13083}, {404, 5283}, {474, 16589}, {493, 34994}, {494, 34995}, {498, 9597}, {499, 9598}, {515, 31398}, {517, 31443}, {548, 9606}, {550, 7745}, {597, 27088}, {598, 9855}, {599, 8575}, {625, 7841}, {626, 7763}, {631, 3767}, {647, 44814}, {691, 9831}, {729, 25424}, {754, 7774}, {755, 43357}, {805, 843}, {842, 34235}, {902, 37586}, {936, 31442}, {940, 16431}, {958, 1574}, {980, 21495}, {988, 3247}, {993, 1575}, {995, 17735}, {999, 31477}, {1003, 7804}, {1007, 32986}, {1083, 11650}, {1092, 22416}, {1100, 37589}, {1107, 25440}, {1147, 9697}, {1153, 8860}, {1180, 5354}, {1194, 7485}, {1196, 7484}, {1285, 19708}, {1296, 15921}, {1334, 9351}, {1348, 19660}, {1349, 19659}, {1352, 14981}, {1376, 1573}, {1383, 7492}, {1385, 31430}, {1420, 31426}, {1478, 31476}, {1495, 3148}, {1511, 14901}, {1593, 3199}, {1597, 33842}, {1600, 8962}, {1656, 39565}, {1657, 11742}, {1724, 37023}, {1879, 14787}, {1914, 5010}, {1968, 3520}, {1971, 11202}, {1975, 3934}, {1992, 8182}, {2023, 8179}, {2070, 44521}, {2071, 22240}, {2176, 24047}, {2177, 2223}, {2207, 3516}, {2269, 34543}, {2273, 22054}, {2291, 41157}, {2491, 44826}, {2502, 5651}, {2794, 9744}, {2896, 7796}, {2936, 9306}, {2996, 32838}, {3016, 4550}, {3070, 31481}, {3087, 37460}, {3096, 7836}, {3097, 11364}, {3100, 9636}, {3117, 3231}, {3124, 9486}, {3146, 31404}, {3229, 30229}, {3230, 37575}, {3288, 8723}, {3291, 20481}, {3314, 7799}, {3329, 3972}, {3357, 32445}, {3523, 5286}, {3524, 5355}, {3526, 11614}, {3528, 12122}, {3530, 5305}, {3533, 12815}, {3534, 14537}, {3541, 27371}, {3549, 15075}, {3576, 9574}, {3589, 8369}, {3618, 32985}, {3619, 6292}, {3620, 3926}, {3629, 3793}, {3630, 7767}, {3631, 3933}, {3723, 37592}, {3730, 21008}, {3735, 17596}, {3785, 7758}, {3788, 6656}, {3818, 37345}, {3830, 40237}, {3849, 11163}, {3981, 39027}, {4048, 10007}, {4188, 5277}, {4191, 21838}, {4218, 30904}, {4293, 31409}, {4297, 31396}, {4299, 9596}, {4302, 9599}, {4317, 31478}, {4383, 16436}, {4414, 5163}, {4426, 5267}, {5025, 7769}, {5054, 37637}, {5055, 39563}, {5077, 11184}, {5108, 45330}, {5152, 8289}, {5217, 16502}, {5248, 16604}, {5275, 16371}, {5276, 13587}, {5291, 17756}, {5304, 15692}, {5306, 12100}, {5318, 41034}, {5319, 15717}, {5321, 41035}, {5337, 14996}, {5346, 15712}, {5368, 10299}, {5438, 31429}, {5471, 10654}, {5472, 10653}, {5476, 37461}, {5477, 11179}, {5523, 37118}, {5569, 22329}, {5587, 31441}, {5640, 13192}, {5646, 33978}, {5650, 33929}, {5691, 31428}, {5881, 31444}, {5888, 8617}, {5926, 39501}, {5968, 17964}, {5970, 32531}, {6032, 10989}, {6118, 32497}, {6119, 32494}, {6179, 7839}, {6184, 8649}, {6198, 9635}, {6284, 9665}, {6460, 31411}, {6502, 31471}, {6560, 31463}, {6644, 9609}, {6655, 7752}, {6680, 7803}, {6683, 7770}, {6704, 16898}, {6748, 37458}, {6749, 37934}, {6776, 10991}, {6787, 14509}, {6914, 34460}, {6995, 39662}, {7117, 14827}, {7354, 9650}, {7496, 9465}, {7527, 15355}, {7694, 39838}, {7697, 13188}, {7734, 40326}, {7750, 7759}, {7754, 7780}, {7760, 7793}, {7768, 7904}, {7773, 7842}, {7776, 7873}, {7778, 7853}, {7779, 7811}, {7784, 7821}, {7785, 7802}, {7787, 33014}, {7788, 7848}, {7789, 7822}, {7792, 35297}, {7797, 7857}, {7806, 7827}, {7807, 7834}, {7809, 7898}, {7812, 14712}, {7814, 7885}, {7823, 7858}, {7828, 7864}, {7829, 32964}, {7832, 7876}, {7838, 20065}, {7840, 7850}, {7845, 9766}, {7846, 33225}, {7849, 7881}, {7851, 7886}, {7852, 32954}, {7859, 7892}, {7860, 7941}, {7861, 7887}, {7866, 7874}, {7868, 7880}, {7870, 7931}, {7871, 7936}, {7875, 33246}, {7877, 13571}, {7878, 33276}, {7879, 7895}, {7883, 7897}, {7884, 16984}, {7889, 14001}, {7890, 11008}, {7893, 7905}, {7899, 7933}, {7901, 7918}, {7909, 7938}, {7911, 7912}, {7917, 7929}, {7922, 7928}, {7923, 7942}, {7924, 7925}, {7926, 11057}, {7930, 7948}, {7943, 14043}, {7944, 7945}, {7987, 9593}, {8149, 37004}, {8150, 32476}, {8176, 8352}, {8290, 10000}, {8318, 8319}, {8334, 8335}, {8370, 34504}, {8373, 32562}, {8374, 32569}, {8542, 9145}, {8556, 14711}, {8623, 9463}, {8627, 46546}, {8666, 20691}, {8667, 22253}, {8703, 9300}, {8715, 17448}, {8716, 9466}, {8719, 22682}, {8724, 11178}, {8743, 35477}, {9112, 41100}, {9113, 41101}, {9331, 37587}, {9464, 10130}, {9468, 33756}, {9481, 14247}, {9515, 19127}, {9516, 42286}, {9575, 35242}, {9594, 37729}, {9604, 22115}, {9680, 19105}, {9709, 31490}, {9741, 42850}, {9745, 39602}, {9771, 37350}, {9876, 39849}, {9971, 21419}, {10097, 34291}, {10104, 32448}, {10204, 33962}, {10298, 10313}, {10304, 37665}, {10312, 21844}, {10334, 10347}, {10358, 11272}, {10545, 37465}, {10546, 37335}, {10601, 35302}, {10766, 14649}, {10837, 10838}, {10902, 10988}, {11004, 20251}, {11059, 26257}, {11060, 40799}, {11112, 37661}, {11132, 34541}, {11133, 34540}, {11159, 42849}, {11164, 14762}, {11257, 37334}, {11284, 40350}, {11291, 32785}, {11292, 32786}, {11317, 32479}, {11318, 31275}, {11326, 23208}, {11410, 14581}, {11464, 13509}, {11488, 37173}, {11489, 37172}, {11643, 15925}, {11937, 11938}, {11939, 11940}, {12038, 23128}, {12040, 22110}, {12093, 38651}, {12157, 21460}, {12203, 32522}, {12367, 30488}, {13006, 14793}, {13233, 44420}, {13367, 39643}, {13596, 33885}, {13857, 30516}, {13858, 32301}, {13859, 32302}, {14041, 17005}, {14148, 16990}, {14246, 41404}, {14357, 34158}, {14482, 15698}, {14568, 17004}, {14614, 46893}, {14685, 35901}, {14880, 32516}, {14997, 21508}, {15018, 35296}, {15031, 33002}, {15066, 30541}, {15080, 37183}, {15107, 37184}, {15693, 39593}, {15696, 31470}, {15720, 44535}, {15820, 31152}, {16041, 34803}, {16187, 20998}, {16308, 37950}, {16788, 20331}, {16808, 37332}, {16809, 37333}, {16814, 25066}, {16953, 39668}, {17008, 34506}, {17222, 28469}, {17277, 21937}, {17549, 33854}, {18374, 37808}, {18570, 19220}, {18992, 31437}, {19382, 19383}, {20859, 32568}, {20965, 41278}, {20977, 30498}, {21166, 35925}, {21448, 33850}, {21509, 37679}, {21511, 37680}, {21539, 37674}, {21736, 23259}, {22355, 46922}, {22467, 26216}, {23210, 33728}, {23302, 37341}, {23303, 37340}, {23698, 37348}, {24855, 30739}, {25497, 27162}, {26233, 31088}, {26864, 42671}, {27318, 33063}, {30747, 31107}, {31417, 33703}, {31483, 44647}, {31860, 41266}, {32006, 33226}, {32563, 32570}, {32816, 33023}, {32819, 32992}, {32826, 32987}, {32827, 33272}, {32829, 32974}, {32832, 33001}, {32835, 33025}, {32839, 32972}, {32851, 34542}, {32911, 35276}, {32953, 39142}, {33036, 36812}, {33184, 44377}, {33190, 37690}, {33228, 37647}, {33846, 33986}, {33853, 37391}, {34098, 34099}, {34161, 40517}, {34758, 38467}, {34864, 38463}, {34865, 38466}, {34883, 41617}, {35268, 37916}, {35485, 41370}, {36709, 42283}, {36714, 42284}, {37342, 42277}, {37343, 42274}, {37775, 38431}, {37776, 38432}, {38750, 44534}, {38865, 41451}, {39227, 39502}, {39228, 39500}, {39477, 39499}, {40921, 42089}, {40922, 42092}, {41437, 41445}, {41438, 41444}, {41490, 44392}, {41491, 44394}, {41939, 45662}, {43809, 44523}, {44218, 46257}, {44525, 45735}, {46053, 46855}, {46054, 46854}

▸ Functional Relations44 relations

  • X574 = midpoint of Xi and Xj for these {i,j}: {183, 31859}, {1296, 34241}, {7774, 14907}, {11163, 35955}, {45564, 45565}

  • X574 = reflection of Xi in Xj for these {i,j}: {182, 39498}, {5475, 3815}, {8722, 3}, {17131, 183}, {35440, 13357}, {44773, 15810}

  • X574 = isogonal conjugate of X598

  • X574 = isotomic conjugate of X40826

  • X574 = complement of X11185

  • X574 = circumcircle-inverse of X5104

  • X574 = Brocard-circle-inverse of X187

  • X574 = 3rd-Lemoine-circle-(Ehrmann)-inverse of X5107

  • X574 = Moses-circle-inverse of X5107

  • X574 = Schoutte-circle-inverse of X182

  • X574 = 2nd-Brocard-circle-inverse of X13330

  • X574 = Ehrmann-circle-inverse of X5107

  • X574 = complement of the isotomic conjugate of X5486

  • X574 = isogonal conjugate of the anticomplement of X15810

  • X574 = isogonal conjugate of the isotomic conjugate of X599

  • X574 = isotomic conjugate of the polar conjugate of X8541

  • X574 = isogonal conjugate of the polar conjugate of X5094

  • X574 = Thomson-isogonal conjugate of X22712

  • X574 = psi-transform of X352

  • X574 = Xi-complementary conjugate of Xj for these (i,j): {31, 8542}, {810, 14672}, {5486, 2887}, {30247, 21259}

  • X574 = Xi-Ceva conjugate of Xj for these (i,j): {2, 8542}, {111, 9872}, {1296, 512}, {5094, 8541}, {9145, 17414}, {9515, 32}, {10130, 599}, {10484, 576}, {15810, 8561}, {19151, 184}, {39389, 6}, {42008, 10510}

  • X574 = X17414-cross conjugate of X9145

  • X574 = Xi-isoconjugate of Xj for these (i,j): {1, 598}, {31, 40826}, {75, 1383}, {82, 23297}, {92, 43697}, {661, 35138}, {662, 8599}, {799, 46001}, {811, 30491}, {896, 18818}, {1577, 11636}, {3112, 30489}, {10511, 16568}, {23287, 36085}

  • X574 = crosspoint of Xi and Xj for these (i,j): {1, 7349}, {2, 5486}, {6, 21448}, {249, 12074}, {599, 5094}

  • X574 = crosssum of Xi and Xj for these (i,j): {1, 6205}, {2, 1992}, {4, 41370}, {6, 1995}, {7, 31599}, {57, 34490}, {115, 12073}, {351, 35507}, {514, 44317}, {523, 8288}, {650, 11936}, {1383, 43697}, {20382, 23287}, {20383, 24976}, {23297, 30489}, {27085, 36415}, {32069, 42365}

  • X574 = crossdifference of every pair of points on line {351, 523}

  • X574 = internal center of similitude of circumcircle and Moses circle

  • X574 = crossdifference of every pair of points on line X351X523

  • X574 = X6-of-2nd-Brocard-triangle

  • X574 = perspector of Lucas Brocard and Lucas(-1) Brocard triangles

  • X574 = perspector of ABC and inverse-in-Brocard-circle of vertices of circumsymmedial triangle

  • X574 = perspector of circumsymmedial triangle and inverse-in-Brocard-circle of A, B, C

  • X574 = pole, wrt Brocard circle, of Lemoine axis

  • X574 = harmonic center of Gallatly circle and Ehrmann circle

  • X574 = homothetic center of circumtangential triangle and unary cofactor triangle of Stammler triangle

  • X574 = perspector of ABC and unary cofactor triangle of Lemoine triangle

  • X574 = X6-of-X3PU(1)

  • X574 = homothetic center of Kosnita triangle and mid-triangle of inner and outer tri-equilateral triangles

  • X574 = {X15,X16}-harmonic conjugate of X182

  • X574 = {X1340,X1341}-harmonic conjugate of X6

  • X574 = Xi-isoconjugate of Xj for these (i,j): {1, 598}, {31, 40826}, {75, 1383}, {82, 23297}, {92, 43697}, {661, 35138}, {662, 8599}, {799, 46001}, {811, 30491}, {896, 18818}, {1577, 11636}, {3112, 30489}, {10511, 16568}, {23287, 36085}

  • X574 = barycentric product Xi*Xj for these {i,j}: {1, 36263}, {3, 5094}, {6, 599}, {32, 9464}, {39, 10130}, {67, 10510}, {69, 8541}, {74, 13857}, {99, 17414}, {106, 4141}, {110, 3906}, {111, 39785}, {187, 42008}, {249, 8288}, {512, 9146}, {513, 3908}, {523, 9145}, {524, 42007}, {690, 32583}, {1177, 19510}, {3917, 32581}, {5467, 23288}, {5486, 8542}, {9872, 34898}, {11165, 21448}, {12074, 17436}, {15810, 39389}

  • X574 = barycentric quotient Xi/Xj for these {i,j}: {2, 40826}, {6, 598}, {32, 1383}, {39, 23297}, {67, 10512}, {110, 35138}, {111, 18818}, {184, 43697}, {351, 23287}, {512, 8599}, {599, 76}, {669, 46001}, {1576, 11636}, {3049, 30491}, {3051, 30489}, {3455, 10511}, {3906, 850}, {3908, 668}, {4141, 3264}, {5094, 264}, {8288, 338}, {8541, 4}, {8542, 11185}, {8586, 8785}, {9145, 99}, {9146, 670}, {9464, 1502}, {9872, 11054}, {10130, 308}, {10510, 316}, {11165, 11059}, {13857, 3260}, {15810, 26235}, {17414, 523}, {19510, 1236}, {32581, 46104}, {32583, 892}, {36263, 75}, {39689, 20380}, {39785, 3266}, {42007, 671}, {42008, 18023}

  • X574 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {1, 31421, 1571}, {2, 99, 3734}, {2, 111, 8585}, {2, 671, 7617}, {2, 2549, 115}, {2, 4045, 7913}, {2, 7618, 2482}, {2, 7790, 7844}, {2, 11648, 18362}, {2, 32480, 671}, {2, 43448, 43620}, {3, 6, 187}, {3, 32, 5206}, {3, 39, 32}, {3, 187, 8588}, {3, 1384, 5210}, {3, 3053, 15513}, {3, 3095, 5171}, {3, 5013, 39}, {3, 5024, 6}, {3, 6421, 39654}, {3, 6422, 39655}, {3, 9605, 3053}, {3, 9737, 30270}, {3, 11171, 182}, {3, 11842, 38225}, {3, 13334, 37479}, {3, 14961, 577}, {3, 15655, 5585}, {3, 15815, 37512}, {3, 21309, 15655}, {3, 22121, 18472}, {3, 22332, 5007}, {3, 30435, 5023}, {3, 32447, 2080}, {3, 35002, 3098}, {3, 37512, 15515}, {4, 31401, 1506}, {4, 31415, 43457}, {6, 187, 32}, {6, 1151, 8375}, {6, 1152, 8376}, {6, 1350, 11173}, {6, 1384, 5008}, {6, 3053, 21309}, {6, 5008, 14075}, {6, 5013, 5024}, {6, 5024, 39}, {6, 5092, 5033}, {6, 5210, 1384}, {6, 5585, 3053}, {6, 6200, 9675}, {6, 6443, 6421}, {6, 6444, 6422}, {6, 8375, 5058}, {6, 8376, 5062}, {6, 8586, 576}, {6, 8589, 8588}, {6, 11173, 5052}, {6, 14806, 10979}, {6, 15602, 15515}, {6, 21309, 5007}, {6, 22246, 5041}, {6, 44453, 8586}, {15, 16, 182}, {20, 2548, 7747}, {20, 31400, 2548}, {20, 31450, 9698}, {32, 39, 7772}, {32, 8588, 187}, {32, 14075, 5008}, {32, 15515, 3}, {35, 2275, 2241}, {35, 16784, 10987}, {36, 2276, 2242}, {39, 187, 6}, {39, 5007, 9605}, {39, 5188, 46305}, {39, 5210, 14075}, {39, 8589, 187}, {39, 15513, 5007}, {39, 15515, 5206}, {39, 15602, 8589}, {39, 15815, 15515}, {39, 21163, 2021}, {39, 31652, 5013}, {39, 35007, 5041}, {39, 37512, 3}, {39, 46283, 3095}, {56, 31448, 1500}, {61, 62, 22234}, {69, 34511, 7813}, {76, 7783, 7781}, {76, 7824, 7815}, {99, 15483, 2482}, {115, 2549, 11648}, {140, 5254, 7746}, {140, 43291, 3054}, {141, 6390, 7801}, {165, 9592, 1572}, {182, 3094, 5028}, {182, 5028, 39764}, {182, 9734, 3}, {184, 3269, 39913}, {187, 5008, 1384}, {187, 8588, 5206}, {187, 8589, 3}, {187, 15513, 15655}, {187, 37512, 8589}, {194, 1078, 7751}, {194, 33004, 1078}, {216, 5065, 46432}, {216, 40349, 3}, {230, 15048, 5309}, {315, 7764, 7903}, {315, 32965, 7830}, {316, 7777, 7775}, {325, 7761, 7818}, {325, 8356, 7761}, {371, 372, 575}, {376, 7736, 7737}, {376, 7737, 6781}, {381, 31489, 7603}, {384, 7786, 7808}, {385, 7757, 7798}, {385, 7798, 41748}, {385, 33273, 7771}, {549, 15048, 230}, {550, 31406, 7745}, {566, 5063, 5158}, {570, 14961, 39}, {575, 43141, 43121}, {575, 43144, 43120}, {575, 44496, 6}, {599, 11165, 39785}, {620, 4045, 2}, {626, 7763, 7888}, {626, 7791, 7935}, {631, 3767, 7749}, {631, 7738, 3767}, {1003, 11174, 7804}, {1030, 5069, 16946}, {1151, 1152, 10541}, {1151, 6421, 5058}, {1151, 6443, 6}, {1151, 8406, 26348}, {1152, 6422, 5062}, {1152, 6444, 6}, {1152, 8414, 26341}, {1340, 1341, 6}, {1342, 1343, 39560}, {1376, 31449, 1573}, {1379, 1380, 5104}, {1384, 5008, 32}, {1384, 5210, 187}, {1478, 31497, 31476}, {1505, 9674, 371}, {1506, 7756, 4}, {1506, 43457, 31415}, {1656, 44518, 39565}, {1664, 1665, 5052}, {1670, 1671, 13330}, {1687, 1688, 11842}, {1689, 1690, 11171}, {1975, 3934, 17130}, {1975, 11285, 3934}, {2021, 18860, 5162}, {2022, 35424, 32}, {2028, 2029, 5107}, {2076, 13331, 5039}, {2080, 32447, 576}, {2092, 36743, 5042}, {2275, 10987, 16784}, {2548, 31400, 9698}, {2548, 31450, 31400}, {2549, 43620, 43448}, {2896, 7796, 7896}, {3053, 5007, 32}, {3053, 5013, 22332}, {3053, 5585, 15655}, {3053, 9605, 5007}, {3053, 15655, 187}, {3053, 22332, 9605}, {3054, 5254, 43291}, {3054, 43291, 7746}, {3094, 5116, 182}, {3094, 11171, 39}, {3094, 15815, 9734}, {3094, 39498, 5034}, {3095, 9301, 37517}, {3096, 7836, 7869}, {3098, 9737, 35002}, {3098, 35002, 30270}, {3314, 7799, 7908}, {3314, 7831, 7865}, {3329, 13586, 3972}, {3520, 39575, 1968}, {3524, 7735, 21843}, {3576, 9574, 9620}, {3589, 32459, 8369}, {3734, 15482, 2}, {3767, 7738, 7765}, {3785, 7758, 7826}, {3788, 6656, 7867}, {3926, 7800, 7794}, {3926, 32990, 7800}, {4255, 5021, 20970}, {4256, 5030, 6}, {4261, 5124, 5019}, {5007, 8589, 5585}, {5007, 15513, 3053}, {5013, 5116, 11171}, {5013, 9734, 5028}, {5013, 15515, 7772}, {5013, 15602, 8588}, {5013, 15815, 3}, {5013, 37512, 32}, {5013, 40349, 5065}, {5023, 5041, 32}, {5023, 30435, 35007}, {5024, 8588, 7772}, {5024, 8589, 32}, {5024, 15655, 9605}, {5024, 15815, 8589}, {5024, 37512, 8588}, {5025, 7769, 7862}, {5025, 7847, 7872}, {5034, 8722, 32}, {5038, 8586, 6}, {5038, 44453, 576}, {5041, 35007, 30435}, {5058, 39654, 32}, {5062, 39655, 32}, {5077, 11184, 31173}, {5111, 38225, 10631}, {5171, 37517, 9301}, {5206, 7772, 32}, {5461, 7619, 2}, {5585, 15513, 8588}, {5585, 15655, 15513}, {5585, 21309, 187}, {5585, 22332, 6}, {6141, 6142, 2502}, {6200, 6396, 5092}, {6292, 7863, 7795}, {6337, 7795, 7863}, {6337, 16043, 7795}, {6390, 8359, 141}, {6421, 6422, 10542}, {6421, 8375, 6}, {6421, 10541, 5062}, {6422, 8376, 6}, {6422, 10541, 5058}, {6655, 7752, 7825}, {6683, 7816, 7770}, {6781, 7753, 7737}, {7354, 31460, 9650}, {7735, 7739, 5355}, {7736, 7737, 7753}, {7739, 21843, 7735}, {7747, 9698, 2548}, {7748, 31455, 5}, {7748, 31457, 31455}, {7749, 7765, 3767}, {7757, 7771, 385}, {7760, 43459, 7793}, {7763, 7791, 626}, {7764, 7830, 315}, {7768, 7906, 7916}, {7769, 7847, 5025}, {7773, 33234, 7842}, {7774, 33008, 14907}, {7777, 7833, 316}, {7778, 11287, 7853}, {7780, 32450, 7754}, {7781, 7815, 76}, {7782, 7786, 384}, {7783, 7824, 76}, {7785, 33260, 7802}, {7789, 8362, 7822}, {7793, 33022, 43459}, {7795, 16043, 6292}, {7797, 33259, 7857}, {7799, 7831, 3314}, {7803, 16925, 6680}, {7804, 32456, 1003}, {7804, 44562, 11174}, {7810, 7813, 69}, {7814, 7910, 7885}, {7828, 7864, 7902}, {7832, 7876, 7914}, {7836, 33021, 3096}, {7851, 33233, 7886}, {7862, 7872, 5025}, {7864, 7907, 7828}, {7865, 7908, 3314}, {7870, 7937, 7931}, {7871, 7936, 7939}, {7876, 7891, 7832}, {7879, 32821, 7895}, {7888, 7935, 626}, {7904, 7906, 7768}, {7918, 7940, 7901}, {7923, 33245, 7942}, {7924, 7925, 7934}, {7928, 7947, 7922}, {8400, 26348, 8406}, {8407, 26341, 8414}, {8589, 15602, 37512}, {8589, 31652, 5024}, {9605, 15513, 32}, {9605, 15655, 21309}, {9605, 21309, 6}, {9605, 22332, 39}, {9619, 31422, 40}, {9651, 31501, 12}, {9885, 9886, 7622}, {9888, 15483, 99}, {10631, 11842, 32}, {10987, 16784, 2241}, {11063, 13337, 33872}, {12055, 35002, 39}, {13335, 43183, 32}, {13351, 15109, 571}, {14482, 15698, 46453}, {14630, 22242, 6}, {14631, 22243, 6}, {15602, 31652, 6}, {15655, 21309, 3053}, {15810, 39785, 599}, {15815, 31652, 32}, {18472, 22121, 10316}, {18860, 21163, 3}, {22246, 30435, 6}, {30435, 35007, 32}, {31401, 43619, 31415}, {31415, 43619, 4}, {31467, 44519, 39590}, {31489, 44526, 381}, {31652, 37512, 39}, {32456, 44562, 7804}, {32568, 32575, 43650}, {33215, 34511, 7810}, {39229, 39230, 6}, {41196, 41197, 5158}, {41406, 41407, 41412}, {43448, 43620, 115}

Notes

Let A', B', C' be the inverse-in-Brocard-circle of A, B, C. Let A″, B″, C″ be the inverse-in-circumcircle of the vertices of 1st Brocard triangle. The lines A'A″, B'B″, C'C″ concur in X574. (Randy Hutson, November 18, 2015)

Let X'Y'Z' be the circumcevian-inversion triangle of X6, as defined in the preamble just before X34864. The A-vertex of this triangle is given by

X' = (2 a^2 - b^2 - c^2) (4 a^2 + b^2 + c^2) : b^2 (-2 a^2 + b^2 - 2 c^2) (5 a^2 + 2 b^2 + 2 c^2) : c^2 (-2 a^2 - 2 b^2 + c^2) (5 a^2 + 2 b^2 + 2 c^2).

The triangles ABC and X'Y'Z' are perspective, and their perspector is X574. (Peter Moses, December 9, 2019)

Let P1' be the tangent, other than U(1), to the Brocard circle from P(2). Let U1' be the tangent, other than P(1), to the Brocard circle from U(2). P1' and U1' are also the Schoute-circle-inverses of P(1) and U(1), resp. Then X574 = P(1)U1'∩U(1)P1'. (Randy Hutson, January 17, 2020)