X647

CROSSDIFFERENCE OF X2 AND X3

X647 is the point whose trilinears are coefficients for the Euler line.
< 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) { 187, 237 }: Central Line L(187, 237) { 230, 231 }: Central Line L(230, 231) { 441, 525 }: Central Line L(441, 525) { 50, 654 }: Central Line L(50, 654) { 2, 850 }: Central Line L(2, 850) { 111, 842 }: Central Line L(111, 842) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (1.1986, 1.2559) X_237 = -LINE CONJUGATE OF X(2) (0.9894, 1.3825) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_231 = -CEVA CONJUGATE OF X(128) (1.0375, 0.6458) X_441 = COMPLEMENT OF X(297) (0.3841, -0.2011) X_525 = ISOGONAL CONJUGATE OF X(112) (-0.0468, 0.2556) X_50 = -CEVA CONJUGATE OF X(184) (2.1559, 2.8368) X_654 = CROSSDIFFERENCE OF X(1) AND X(5) (2.8235, 1.7291) X_2 = CENTROID (0.5374, 0.2000) X_850 = BARYCENTRIC MULTIPLIER FOR KIEPERT HYPERBOLA (-6.8003, 1.7304) X_111 = PARRY POINT (0.9885, -0.0457) X_842 = ISOGONAL CONJUGATE OF X(542) (-0.0101, 0.1156) X_647 = CROSSDIFFERENCE OF X(2) AND X(3) (4.2063, -0.5653) A B C X187 X237 X230 X231 X441 X525 X50 X654 X2 X850 X111 X842 X647 (a, b, c) = (5, 6, 7) | AB = 1

X647 (CROSSDIFFERENCE OF X2 AND X3) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a(b^{2} - c^{2})(b^{2} + c^{2} - a^{2}) : :\)
Barycentrics: \(a^{2} (b^{2} - c^{2}) S_{A} : :\)
Search Value (6, 9, 13): \(-5.076856435422\)
▸ Alternate Coordinates3 forms

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

  • Trilinears: \((\sin A)(\tan B - \tan C) : :\)

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

▸ Linear Combinations1 combination

  • X647 = \(9\) X2 - \(5\) X31072, \(6\) X2 - \(5\) X31277, \(3\) X2 + X31296, \(3\) X2 + \(2\) X41300, \(3\) X351 - X669, \(3\) X351 + X3005, \(6\) X351 - X3804, \(3\) X351 - \(2\) X8651, \(9\) X351 - X8664, \(9\) X351 + X8665, \(2\) X669 - \(3\) X8644, \(3\) X669 - X8664, \(3\) X669 + X8665, X669 + \(3\) X17414, \(3\) X850 - \(5\) X31072, \(2\) X850 - \(3\) X31174, \(2\) X850 - \(5\) X31277, X850 + \(3\) X36900, X850 + \(2\) X41300, X850 - \(6\) X44560, \(3\) X1636 + X17434, \(3\) X1636 - X32320, \(3\) X1637 - \(2\) X2501, \(3\) X1637 - \(4\) X6587, \(3\) X1637 - X12077, \(9\) X1649 - X2528, \(3\) X2485 - \(2\) X2492, \(3\) X2489 - \(4\) X2492, X2501 - \(3\) X9209, X2525 - \(3\) X14417, \(2\) X3005 + X3804, \(2\) X3005 + \(3\) X8644, X3005 + \(2\) X8651, \(3\) X3005 + X8664, \(3\) X3005 - X8665, X3005 - \(3\) X17414, \(3\) X3050 + X39232, \(2\) X3265 - \(3\) X14417, X3288 + \(3\) X9210, X3569 - \(3\) X9210, X3804 - \(3\) X8644, X3804 - \(4\) X8651, \(3\) X3804 - \(2\) X8664, \(3\) X3804 + \(2\) X8665, X3804 + \(6\) X17414, X3806 + \(9\) X9125, \(3\) X5996 - X44445, \(2\) X6587 - \(3\) X9209, \(4\) X6587 - X12077, \(3\) X8644 - \(4\) X8651, \(9\) X8644 - \(2\) X8664, \(9\) X8644 + \(2\) X8665, X8644 + \(2\) X17414, \(6\) X8651 - X8664, \(6\) X8651 + X8665, \(2\) X8651 + \(3\) X17414, X8664 + \(9\) X17414, X8665 - \(9\) X17414, \(3\) X9147 + X44445, \(3\) X9148 - \(5\) X31279, \(6\) X9209 - X12077, X9409 + \(3\) X15451, X9409 - \(3\) X39201, \(2\) X9409 - \(3\) X42658, \(3\) X9979 + X41298, \(3\) X11176 - \(2\) X44451, \(2\) X15451 + X42658, \(6\) X30476 - \(5\) X31072, \(4\) X30476 - \(3\) X31174, \(4\) X30476 - \(5\) X31277, \(2\) X30476 + X31296, \(2\) X30476 + \(3\) X36900, X30476 - \(3\) X44560, \(10\) X31072 - \(9\) X31174, \(2\) X31072 - \(3\) X31277, \(5\) X31072 + \(3\) X31296, \(5\) X31072 + \(9\) X36900, \(5\) X31072 + \(6\) X41300, \(5\) X31072 - \(18\) X44560, \(3\) X31174 - \(5\) X31277, \(3\) X31174 + \(2\) X31296, X31174 + \(2\) X36900, \(3\) X31174 + \(4\) X41300, X31174 - \(4\) X44560, \(5\) X31277 + \(2\) X31296, \(5\) X31277 + \(6\) X36900, \(5\) X31277 + \(4\) X41300, \(5\) X31277 - \(12\) X44560, X31296 - \(3\) X36900, X31296 + \(6\) X44560, \(3\) X36900 - \(2\) X41300, X36900 + \(2\) X44560, X41300 + \(3\) X44560

X647 lies on the Yff parabola.

▸ Incident Central Lines161 lines

{1, 1021}, {2, 850}, {3, 10097}, {6, 2433}, {10, 21719}, {23, 13114}, {25, 34212}, {32, 44895}, {37, 3700}, {39, 1640}, {42, 4524}, {50, 654}, {75, 21437}, {99, 9091}, {110, 2715}, {111, 842}, {112, 1304}, {114, 38975}, {115, 3258}, {122, 33504}, {125, 22264}, {136, 38970}, {141, 9030}, {184, 878}, {185, 9242}, {187, 237}, {216, 14401}, {230, 231}, {248, 35909}, {352, 9138}, {441, 525}, {511, 40283}, {513, 43060}, {514, 23723}, {520, 652}, {570, 8562}, {574, 44814}, {612, 4477}, {656, 8611}, {657, 2451}, {661, 3709}, {688, 2513}, {804, 23301}, {810, 20754}, {826, 7651}, {879, 43718}, {881, 38237}, {906, 23084}, {1015, 35090}, {1084, 6791}, {1125, 19948}, {1194, 10190}, {1196, 11123}, {1214, 17094}, {1301, 32687}, {1331, 23139}, {1499, 11615}, {1555, 19912}, {1560, 42426}, {1562, 35071}, {1570, 30219}, {1577, 27731}, {1624, 35325}, {1635, 4139}, {1962, 6608}, {1995, 33752}, {2394, 43530}, {2422, 5651}, {2430, 14642}, {2436, 18320}, {2516, 4145}, {2605, 7252}, {2611, 14936}, {2780, 32231}, {2799, 6563}, {2966, 9514}, {2972, 38356}, {3117, 11183}, {3124, 38987}, {3221, 17415}, {3261, 25511}, {3267, 4580}, {3666, 17069}, {3708, 7117}, {3766, 27293}, {3768, 23751}, {4132, 4394}, {4151, 4765}, {4391, 25902}, {4458, 16612}, {4467, 16751}, {4789, 24900}, {5466, 7607}, {5642, 11672}, {5972, 23584}, {5996, 9147}, {6331, 43188}, {6368, 14345}, {7630, 30474}, {7631, 9148}, {7656, 25925}, {7746, 23105}, {8045, 14838}, {8371, 37742}, {8673, 10317}, {8704, 9185}, {9033, 44892}, {9044, 9188}, {9137, 11580}, {9168, 9465}, {9171, 22111}, {9175, 20481}, {9178, 21448}, {9216, 32583}, {9306, 22391}, {9426, 19558}, {9517, 30491}, {9979, 41298}, {10192, 40588}, {10766, 14919}, {11063, 39180}, {11124, 40973}, {11424, 14456}, {13006, 23993}, {13558, 20998}, {14165, 14618}, {14208, 24459}, {14321, 21894}, {14341, 44817}, {14376, 23107}, {14773, 23292}, {14966, 15329}, {15328, 46262}, {15411, 15420}, {15820, 18313}, {16186, 41172}, {16598, 23988}, {16599, 23585}, {18675, 40628}, {18877, 32663}, {21053, 27711}, {21225, 26114}, {21761, 23567}, {21789, 43925}, {21837, 24290}, {22055, 22375}, {22089, 22091}, {22121, 22155}, {23786, 23887}, {23963, 34947}, {25423, 45333}, {25861, 25862}, {27648, 45746}, {30230, 40115}, {32120, 35282}, {32661, 32662}, {33843, 42733}, {39013, 39021}, {39482, 39601}, {39503, 39799}, {41336, 44896}

▸ Functional Relations57 relations

  • X647 = radical center of the circumcircle, nine-point circle, and Brocard circle (Wilson Stothers, 3/13/2003)

  • X647 is the perspector of triangle ABC and the tangential triangle of the Jerabek hyperbola. (Randy Hutson, 9/23/2011)

  • X647 = midpoint of Xi and Xj for these {i,j}: {2, 36900}, {351, 17414}, {649, 42664}, {669, 3005}, {850, 31296}, {2966, 46245}, {3267, 4580}, {3288, 3569}, {5996, 9147}, {6137, 6138}, {6563, 33294}, {8639, 42661}, {8664, 8665}, {14270, 18117}, {14618, 15412}, {15451, 39201}, {17431, 17432}, {17434, 32320}, {21731, 42660}, {30476, 41300}

  • X647 = reflection of Xi in Xj for these {i,j}: {2, 44560}, {125, 22264}, {669, 8651}, {850, 30476}, {1495, 42654}, {1637, 9209}, {2489, 2485}, {2501, 6587}, {2525, 3265}, {3804, 669}, {6753, 16040}, {8644, 351}, {12077, 2501}, {31174, 2}, {31296, 41300}, {42658, 39201}, {45907, 2491}

  • X647 = isogonal conjugate of X648

  • X647 = isotomic conjugate of X6331

  • X647 = complement of X850

  • X647 = anticomplement of X30476

  • X647 = circumcircle-inverse of X35901

  • X647 = Parry-circle-inverse of X3569

  • X647 = Dao-Moses-Telv-circle inverse of X6130

  • X647 = orthoptic-circle-of-Steiner-inellipe-inverse of X34235

  • X647 = Moses-Parry-circle-inverse of X1637

  • X647 = Parry-isodynamic-circle inverse of X5191

  • X647 = polar conjugate of X6528

  • X647 = orthic-isogonal conjugate of X34980

  • X647 = center of the Moses radical circle

  • X647 = tripolar centroid for these (i,j): {41518, 41519}

  • X647 = trilinear pole of line {3269, 9409}

  • X647 = crossdifference of every pair of points on line {2, 3}

  • X647 = perspector of the Jerabek hyperbola

  • X647 = orthojoin of X125

  • X647 = perspector of ABC and the side-triangle of the cevian triangles of X3 and X6

  • X647 = PU(4)-harmonic conjugate of X232

  • X647 = bicentric difference of PU(i) for these i: 17, 145, 157

  • X647 = PU(17)-harmonic conjugate of X185

  • X647 = crossdifference of PU(30)

  • X647 = barycentric product of PU(75)

  • X647 = trilinear pole of PU(109) (line X3269X9409)

  • X647 = center of circumconic that is locus of trilinear poles of lines passing through X125

  • X647 = intersection of trilinear polars of any 2 points on the Jerabek hyperbola

  • X647 = X187-of-2nd-Parry-triangle

  • X647 = X187-of-3rd-Parry-triangle

  • X647 = intersection of the Lemoine axes of ABC and the 5th Euler triangle

  • X647 = intersection of orthic and Lemoine axes (trilinear polars of X4 and X6)

  • X647 = X92-isoconjugate of X110

  • X647 = pole wrt polar circle of trilinear polar of X6528

  • X647 = X48-isoconjugate (polar conjugate) of X6528

  • X647 = midpoint of PU(145)

  • X647 = PU(157)-harmonic conjugate of X51

  • X647 = X649-of-orthic-triangle if ABC is acute

  • X647 = trilinear product of Jerabek hyperbola intercepts of antiorthic axis

  • X647 = excentral-to-ABC functional image of X649

  • X647 = orthoptic-circle-of-Steiner-inellipse-inverse of X34235

  • X647 = Xi-anticomplementary conjugate of Xj for these (i,j): {9255, 13219}, {9258, 3448}, {9292, 21221}, {9307, 21294}, {43188, 6327}

  • X647 = Xi-complementary conjugate of Xj for these (i,j): {6, 21253}, {31, 125}, {32, 8287}, {48, 127}, {58, 21252}, {99, 21235}, {101, 21245}, {110, 2887}, {112, 20305}, {162, 21243}, {163, 141}, {184, 34846}, {249, 42327}, {250, 21259}, {560, 115}, {662, 626}, {669, 24040}, {691, 21256}, {692, 3454}, {799, 40379}, {827, 21238}, {922, 5099}, {1101, 512}, {1110, 31946}, {1333, 116}, {1397, 8286}, {1408, 17059}, {1414, 17047}, {1415, 17052}, {1501, 16592}, {1576, 10}, {1755, 36471}, {1917, 1084}, {1923, 15449}, {1924, 23991}, {1933, 2679}, {2194, 124}, {2205, 6627}, {2206, 11}, {4556, 21240}, {4565, 17046}, {4567, 21262}, {4570, 21260}, {4575, 1368}, {4590, 21263}, {4602, 40380}, {4630, 1215}, {5546, 21244}, {9247, 15526}, {9417, 35088}, {14574, 37}, {14575, 16573}, {14585, 16595}, {14586, 21231}, {16947, 4904}, {23357, 4369}, {23963, 14838}, {23990, 4129}, {23995, 523}, {24041, 23301}, {32656, 21530}, {32660, 18642}, {32661, 18589}, {32676, 5}, {32678, 34827}, {32729, 4892}, {32734, 34825}, {32739, 1211}, {34072, 3934}, {36084, 21531}, {36134, 3819}, {36142, 625}, {41280, 16613}, {46289, 7668}

  • X647 = Xi-Ceva conjugate of Xj for these (i,j): {1, 3270}, {2, 125}, {3, 20975}, {4, 34980}, {6, 3269}, {25, 38356}, {48, 7117}, {63, 3937}, {99, 6467}, {101, 44093}, {107, 51}, {108, 40952}, {110, 184}, {112, 6}, {184, 38352}, {222, 22094}, {248, 41172}, {254, 8754}, {393, 1562}, {520, 42658}, {523, 512}, {525, 520}, {648, 185}, {650, 661}, {652, 822}, {691, 21639}, {827, 21637}, {879, 39469}, {905, 656}, {906, 18591}, {907, 19459}, {933, 13366}, {935, 2393}, {1020, 1425}, {1073, 2972}, {1113, 44125}, {1114, 44126}, {1214, 18210}, {1289, 1843}, {1296, 10602}, {1301, 25}, {1304, 1495}, {1331, 22080}, {1332, 22076}, {1415, 2092}, {1459, 810}, {1576, 39}, {1942, 35236}, {2164, 20982}, {2165, 115}, {2218, 3271}, {2394, 526}, {2407, 974}, {2433, 686}, {2489, 2519}, {2623, 30451}, {2715, 8779}, {2966, 511}, {3049, 2524}, {3267, 8673}, {3459, 24862}, {3463, 41212}, {3504, 23216}, {3563, 44114}, {3903, 23526}, {3952, 43218}, {4551, 3611}, {4557, 23620}, {4558, 3}, {4559, 1409}, {4563, 3917}, {4574, 71}, {4580, 525}, {5649, 5622}, {6331, 1899}, {7612, 6784}, {8770, 3124}, {9064, 34417}, {10423, 44102}, {14376, 15526}, {14380, 9409}, {14570, 6146}, {14586, 570}, {14618, 924}, {14910, 2088}, {14919, 16186}, {14977, 9517}, {15352, 4}, {15412, 523}, {15420, 521}, {15421, 9033}, {15459, 6000}, {16806, 21647}, {16807, 21648}, {16813, 389}, {18315, 13367}, {18890, 35071}, {22239, 44084}, {23067, 228}, {23086, 22373}, {23181, 23195}, {23286, 39201}, {26705, 40954}, {30247, 40673}, {30249, 44079}, {30441, 26937}, {32640, 3003}, {32641, 2245}, {32661, 216}, {32662, 3284}, {32734, 39643}, {33640, 44110}, {34212, 3569}, {34429, 20974}, {34568, 74}, {35178, 15073}, {36067, 44113}, {39180, 15451}, {39181, 23286}, {39183, 1510}, {39290, 21650}, {39291, 287}, {39382, 8541}, {39383, 21640}, {39384, 21641}, {39417, 1974}, {40097, 44092}, {40116, 39690}, {40323, 6388}, {40347, 1648}, {40428, 15630}, {41210, 41204}, {41676, 26926}, {41677, 32377}, {41678, 1885}, {41769, 127}, {42401, 42400}, {43188, 2}, {43679, 7668}, {44060, 154}, {44769, 21663}, {46087, 34982}

  • X647 = Xi-cross conjugate of Xj for these (i,j): {686, 14582}, {1084, 40319}, {3049, 512}, {3124, 2351}, {3269, 6}, {3708, 2197}, {9409, 14380}, {15451, 523}, {20975, 3}, {34980, 4}, {35071, 14642}, {35236, 1942}, {38974, 232}, {39201, 520}, {39469, 879}, {41172, 248}, {42293, 39201}

  • X647 = Xi-isoconjugate of Xj for these (i,j): {1, 648}, {2, 162}, {3, 823}, {4, 662}, {6, 811}, {19, 99}, {21, 653}, {25, 799}, {27, 100}, {28, 190}, {29, 651}, {31, 6331}, {33, 4573}, {34, 645}, {38, 42396}, {47, 30450}, {48, 6528}, {57, 36797}, {58, 6335}, {63, 107}, {69, 24019}, {74, 24001}, {75, 112}, {76, 32676}, {81, 1897}, {82, 41676}, {86, 1783}, {91, 41679}, {92, 110}, {101, 286}, {108, 333}, {109, 31623}, {158, 4558}, {163, 264}, {186, 32680}, {204, 44326}, {225, 4612}, {230, 36105}, {232, 36036}, {240, 2966}, {242, 4584}, {243, 41206}, {249, 24006}, {250, 1577}, {255, 15352}, {270, 4552}, {273, 5546}, {274, 8750}, {275, 2617}, {278, 643}, {281, 1414}, {284, 18026}, {297, 36084}, {304, 32713}, {314, 32674}, {317, 36145}, {318, 4565}, {323, 36129}, {324, 36134}, {325, 36104}, {326, 6529}, {340, 32678}, {393, 4592}, {394, 36126}, {419, 37134}, {423, 37135}, {427, 4599}, {441, 36092}, {468, 36085}, {512, 46254}, {514, 5379}, {525, 24000}, {607, 4625}, {608, 7257}, {647, 23999}, {655, 17515}, {656, 23582}, {658, 4183}, {660, 31905}, {661, 18020}, {664, 1172}, {668, 1474}, {670, 1973}, {673, 4238}, {685, 1959}, {687, 1725}, {692, 44129}, {765, 17925}, {775, 41678}, {827, 20883}, {858, 36095}, {860, 37140}, {877, 1910}, {897, 4235}, {925, 1748}, {931, 5307}, {933, 14213}, {934, 2322}, {935, 16568}, {1010, 36099}, {1019, 15742}, {1043, 32714}, {1096, 4563}, {1099, 34568}, {1101, 14618}, {1113, 2581}, {1114, 2580}, {1119, 7259}, {1235, 34072}, {1289, 1760}, {1301, 18750}, {1303, 9252}, {1304, 14206}, {1332, 8747}, {1396, 3699}, {1398, 7258}, {1415, 44130}, {1435, 7256}, {1492, 31909}, {1576, 1969}, {1625, 40440}, {1633, 40411}, {1733, 32697}, {1755, 22456}, {1784, 44769}, {1812, 36127}, {1813, 1896}, {1821, 4230}, {1824, 4610}, {1839, 4596}, {1843, 4593}, {1936, 41207}, {1953, 18831}, {1957, 43188}, {1974, 4602}, {1978, 2203}, {1981, 37142}, {1995, 37217}, {2052, 4575}, {2166, 14590}, {2167, 35360}, {2173, 16077}, {2190, 14570}, {2201, 4589}, {2204, 4572}, {2216, 41677}, {2287, 36118}, {2299, 4554}, {2313, 41208}, {2326, 4566}, {2328, 13149}, {2332, 4569}, {2333, 4623}, {2349, 4240}, {2355, 4632}, {2407, 36119}, {2421, 36120}, {2489, 24037}, {2501, 24041}, {2576, 15165}, {2577, 15164}, {2586, 8116}, {2587, 8115}, {2588, 39298}, {2589, 39299}, {2631, 42308}, {2633, 15351}, {2715, 40703}, {2964, 38342}, {3112, 35325}, {3194, 44327}, {3257, 37168}, {3260, 36131}, {3580, 36114}, {3658, 37203}, {3737, 46102}, {4143, 24022}, {4206, 37215}, {4222, 37205}, {4232, 37216}, {4233, 37206}, {4241, 36101}, {4242, 24624}, {4246, 34234}, {4248, 27834}, {4249, 37130}, {4551, 46103}, {4556, 41013}, {4560, 7012}, {4561, 5317}, {4567, 7649}, {4570, 17924}, {4577, 17442}, {4591, 38462}, {4594, 7119}, {4600, 6591}, {4603, 7009}, {4606, 31903}, {4616, 7079}, {4620, 18344}, {4622, 8756}, {4627, 5342}, {4628, 16747}, {4633, 5338}, {4635, 7071}, {4636, 40149}, {4637, 7046}, {5271, 36077}, {5468, 36128}, {6516, 8748}, {7035, 43925}, {7045, 17926}, {7115, 18155}, {7128, 7253}, {7452, 36100}, {7480, 36102}, {8753, 24039}, {8767, 34211}, {8822, 40117}, {9390, 39062}, {10423, 20884}, {11107, 38340}, {13138, 41083}, {14006, 37137}, {14165, 36061}, {14208, 23964}, {14212, 33640}, {14543, 40431}, {15149, 36086}, {15411, 24033}, {16081, 23997}, {16237, 36053}, {16813, 44706}, {17438, 33513}, {17569, 37207}, {17983, 23889}, {21582, 39417}, {23090, 24032}, {23347, 33805}, {23353, 35145}, {24018, 32230}, {27369, 37204}, {30528, 36063}, {30737, 36046}, {31900, 37212}, {31902, 37211}, {31926, 37138}, {32002, 36148}, {32696, 46238}, {32715, 46234}, {34055, 46151}, {34085, 37908}, {35201, 39290}, {36034, 46106}, {36043, 44436}, {36142, 44146}, {36149, 44134}

  • X647 = cevapoint of Xi and Xj for these (i,j): {3, 22143}, {3049, 39201}, {15451, 42293}

  • X647 = crosspoint of Xi and Xj for these (i,j): {1, 1020}, {2, 110}, {3, 4558}, {4, 15352}, {6, 112}, {37, 4559}, {71, 4574}, {74, 34568}, {99, 40405}, {100, 40406}, {101, 2983}, {107, 275}, {108, 2982}, {109, 40407}, {288, 933}, {523, 525}, {648, 1105}, {650, 652}, {694, 39291}, {905, 1459}, {1073, 1301}, {1113, 8116}, {1114, 8115}, {1214, 23067}, {1289, 40404}, {1304, 40384}, {1331, 1796}, {1332, 1791}, {1576, 10547}, {1799, 4563}, {1813, 40442}, {5994, 41893}, {5995, 41892}, {6331, 34405}, {10419, 43755}, {10423, 41511}, {11079, 32662}, {14533, 32661}, {15412, 23286}, {39180, 39181}

  • X647 = crosssum of Xi and Xj for these (i,j): {1, 1021}, {2, 525}, {3, 32320}, {4, 2501}, {6, 523}, {21, 23090}, {27, 17925}, {28, 43925}, {29, 17926}, {30, 14401}, {52, 6753}, {81, 4560}, {110, 112}, {185, 647}, {216, 520}, {230, 38359}, {233, 6368}, {323, 5664}, {324, 14618}, {394, 20580}, {512, 1196}, {513, 40941}, {514, 40940}, {521, 40937}, {522, 40942}, {524, 18311}, {650, 1858}, {651, 653}, {850, 1235}, {924, 40939}, {1249, 8057}, {1625, 35360}, {1636, 40948}, {1637, 13202}, {1783, 1897}, {1829, 6591}, {1839, 7649}, {1843, 2489}, {1994, 20577}, {2322, 7253}, {2409, 15639}, {2451, 9308}, {2492, 40949}, {2574, 8106}, {2575, 8105}, {2592, 2593}, {2799, 15595}, {2967, 3569}, {3049, 40947}, {3064, 40950}, {3163, 9033}, {3288, 9755}, {3574, 12077}, {3737, 40979}, {3800, 40179}, {5095, 14273}, {5895, 6587}, {7252, 40980}, {8673, 40938}, {8743, 33294}, {14091, 30211}, {14920, 44427}, {15412, 19170}, {17434, 42441}, {31296, 41334}, {35311, 35318}, {35325, 41676}, {36054, 40946}, {41392, 41512}

  • X647 = trilinear pole of line {3269, 9409}

  • X647 = crossdifference of every pair of points on line {2, 3}

  • X647 = barycentric product Xi*Xj for these {i,j}: {1, 656}, {3, 523}, {4, 520}, {5, 23286}, {6, 525}, {10, 1459}, {11, 23067}, {12, 23189}, {19, 24018}, {25, 3265}, {30, 14380}, {31, 14208}, {32, 3267}, {37, 905}, {39, 4580}, {42, 4025}, {48, 1577}, {50, 14592}, {54, 6368}, {55, 17094}, {57, 8611}, {58, 4064}, {63, 661}, {64, 8057}, {65, 521}, {66, 8673}, {67, 9517}, {68, 924}, {69, 512}, {71, 514}, {72, 513}, {73, 522}, {74, 9033}, {75, 810}, {76, 3049}, {77, 4041}, {78, 4017}, {92, 822}, {93, 37084}, {95, 15451}, {97, 12077}, {98, 684}, {99, 20975}, {100, 18210}, {101, 4466}, {106, 14429}, {107, 2972}, {110, 125}, {111, 14417}, {112, 15526}, {115, 4558}, {122, 1301}, {140, 39180}, {162, 2632}, {163, 20902}, {184, 850}, {186, 43083}, {187, 14977}, {201, 3737}, {212, 4077}, {213, 15413}, {216, 15412}, {219, 7178}, {222, 3700}, {226, 652}, {228, 693}, {233, 39181}, {248, 2799}, {250, 5489}, {251, 2525}, {253, 42658}, {255, 24006}, {264, 39201}, {265, 526}, {275, 17434}, {276, 42293}, {287, 3569}, {288, 35441}, {290, 39469}, {292, 24459}, {295, 4010}, {304, 798}, {305, 669}, {306, 649}, {307, 663}, {321, 22383}, {323, 14582}, {324, 46088}, {325, 878}, {328, 14270}, {337, 4455}, {338, 32661}, {339, 1576}, {343, 2623}, {345, 7180}, {348, 3709}, {351, 30786}, {394, 2501}, {441, 34212}, {476, 16186}, {493, 17431}, {494, 17432}, {511, 879}, {518, 10099}, {524, 10097}, {542, 35909}, {577, 14618}, {594, 7254}, {599, 30491}, {603, 4086}, {648, 3269}, {650, 1214}, {662, 3708}, {667, 20336}, {686, 2986}, {690, 895}, {694, 24284}, {804, 36214}, {823, 37754}, {826, 1176}, {868, 43754}, {882, 12215}, {906, 16732}, {912, 3657}, {933, 35442}, {1018, 3942}, {1019, 3949}, {1020, 34591}, {1021, 37755}, {1073, 6587}, {1086, 4574}, {1109, 4575}, {1231, 3063}, {1245, 23874}, {1265, 7250}, {1304, 1650}, {1331, 3120}, {1332, 3125}, {1400, 6332}, {1402, 35518}, {1409, 4391}, {1410, 4397}, {1425, 7253}, {1437, 4036}, {1439, 3900}, {1441, 1946}, {1444, 4705}, {1494, 9409}, {1495, 34767}, {1500, 15419}, {1503, 2435}, {1510, 3519}, {1565, 4557}, {1636, 16080}, {1637, 14919}, {1649, 15398}, {1790, 4024}, {1794, 23752}, {1796, 4988}, {1797, 4120}, {1799, 3005}, {1813, 21044}, {1814, 24290}, {1824, 4131}, {1826, 4091}, {1919, 40071}, {1924, 40364}, {1942, 2797}, {1976, 6333}, {1989, 8552}, {2052, 32320}, {2092, 15420}, {2169, 2618}, {2197, 4560}, {2200, 3261}, {2207, 4143}, {2318, 3676}, {2333, 30805}, {2349, 2631}, {2351, 6563}, {2359, 21124}, {2373, 42665}, {2394, 3284}, {2395, 36212}, {2419, 42671}, {2422, 6393}, {2433, 11064}, {2451, 9289}, {2485, 14376}, {2489, 3926}, {2492, 34897}, {2519, 6339}, {2524, 2998}, {2533, 7015}, {2574, 2575}, {2578, 2583}, {2579, 2582}, {2584, 2589}, {2585, 2588}, {2616, 44706}, {2629, 9392}, {2643, 4592}, {2774, 38535}, {2806, 43723}, {2850, 10693}, {2966, 41172}, {3003, 15421}, {3050, 36952}, {3064, 40152}, {3122, 4561}, {3124, 4563}, {3172, 14638}, {3199, 15414}, {3221, 43714}, {3270, 4566}, {3288, 42313}, {3289, 43665}, {3292, 5466}, {3426, 9007}, {3504, 23301}, {3564, 35364}, {3566, 6391}, {3669, 3694}, {3682, 7649}, {3690, 7192}, {3692, 7216}, {3695, 3733}, {3710, 43924}, {3800, 34817}, {3906, 43697}, {3933, 18105}, {3937, 3952}, {3990, 17924}, {3998, 6591}, {4020, 18070}, {4049, 22356}, {4055, 46107}, {4079, 17206}, {4080, 22086}, {4088, 36057}, {4171, 7177}, {4516, 6516}, {4524, 7056}, {4551, 7004}, {4552, 7117}, {4556, 21046}, {4559, 26932}, {4570, 21134}, {4581, 22076}, {4608, 22080}, {4609, 23216}, {4846, 8675}, {5027, 40708}, {5392, 30451}, {5486, 30209}, {5663, 14220}, {5664, 11079}, {6000, 43701}, {6003, 43708}, {6103, 35911}, {6130, 14941}, {6137, 40709}, {6138, 40710}, {6334, 14910}, {6340, 8651}, {6354, 23090}, {6356, 21789}, {6390, 9178}, {6394, 17994}, {6528, 34980}, {6664, 22159}, {6728, 7591}, {6742, 22094}, {6757, 23226}, {7019, 7234}, {7123, 21107}, {7193, 35352}, {7252, 26942}, {7332, 23084}, {7927, 41435}, {8061, 34055}, {8562, 15392}, {8672, 34259}, {8676, 28786}, {8677, 38955}, {8749, 41077}, {8750, 17216}, {8779, 43673}, {8808, 10397}, {8901, 23181}, {9003, 34802}, {9247, 20948}, {9255, 17478}, {9293, 22143}, {9307, 22089}, {9391, 37142}, {9426, 40050}, {10412, 22115}, {10547, 23285}, {11060, 45792}, {11077, 41078}, {11794, 38352}, {13418, 30210}, {13481, 39228}, {13611, 44060}, {13754, 15328}, {14060, 36955}, {14295, 17970}, {14314, 18316}, {14401, 40384}, {14533, 18314}, {14575, 44173}, {14837, 41087}, {14908, 35522}, {15065, 22379}, {15313, 28787}, {15352, 35071}, {15377, 21196}, {15394, 44705}, {15453, 17702}, {15470, 39170}, {16230, 17974}, {16606, 25098}, {17879, 32676}, {17898, 19614}, {17932, 44114}, {17971, 18003}, {17972, 18004}, {17973, 18006}, {17975, 18013}, {17976, 18014}, {17977, 18015}, {18019, 42659}, {18877, 41079}, {19210, 23290}, {20188, 34483}, {20563, 34952}, {20578, 44718}, {20579, 44719}, {20580, 41489}, {20906, 22381}, {21051, 23086}, {21207, 32656}, {22052, 39183}, {22054, 31010}, {22085, 42345}, {22090, 42027}, {22092, 27809}, {22096, 27808}, {22154, 40085}, {22339, 42667}, {22340, 42668}, {22341, 44426}, {22352, 31065}, {22384, 43534}, {23107, 41937}, {23224, 41013}, {23616, 23964}, {23870, 36296}, {23871, 36297}, {23872, 32585}, {23873, 32586}, {23878, 43718}, {30211, 43695}, {30465, 38414}, {30468, 38413}, {32112, 35912}, {32641, 42761}, {32740, 45807}, {34568, 39008}, {35200, 36035}, {35372, 38401}, {36054, 40149}, {38279, 45689}, {38356, 44766}, {39473, 43717}, {40161, 43060}, {40715, 42662}, {40995, 46005}, {43704, 45147}, {43709, 44665}

  • X647 = barycentric quotient Xi/Xj for these {i,j}: {1, 811}, {2, 6331}, {3, 99}, {4, 6528}, {6, 648}, {19, 823}, {25, 107}, {31, 162}, {32, 112}, {37, 6335}, {39, 41676}, {42, 1897}, {48, 662}, {50, 14590}, {51, 35360}, {54, 18831}, {55, 36797}, {63, 799}, {65, 18026}, {68, 46134}, {69, 670}, {71, 190}, {72, 668}, {73, 664}, {74, 16077}, {77, 4625}, {78, 7257}, {98, 22456}, {110, 18020}, {112, 23582}, {115, 14618}, {125, 850}, {162, 23999}, {184, 110}, {187, 4235}, {212, 643}, {213, 1783}, {216, 14570}, {217, 1625}, {219, 645}, {222, 4573}, {228, 100}, {237, 4230}, {248, 2966}, {251, 42396}, {255, 4592}, {265, 35139}, {275, 42405}, {287, 43187}, {293, 36036}, {295, 4589}, {304, 4602}, {305, 4609}, {306, 1978}, {307, 4572}, {339, 44173}, {351, 468}, {393, 15352}, {394, 4563}, {418, 23181}, {511, 877}, {512, 4}, {513, 286}, {514, 44129}, {520, 69}, {521, 314}, {522, 44130}, {523, 264}, {525, 76}, {526, 340}, {560, 32676}, {570, 41677}, {571, 41679}, {577, 4558}, {603, 1414}, {649, 27}, {650, 31623}, {652, 333}, {656, 75}, {657, 2322}, {661, 92}, {662, 46254}, {663, 29}, {665, 15149}, {667, 28}, {669, 25}, {684, 325}, {686, 3580}, {688, 1843}, {690, 44146}, {692, 5379}, {798, 19}, {800, 41678}, {804, 17984}, {810, 1}, {822, 63}, {826, 1235}, {850, 18022}, {878, 98}, {879, 290}, {881, 17980}, {895, 892}, {905, 274}, {906, 4567}, {924, 317}, {1015, 17925}, {1042, 36118}, {1073, 44326}, {1084, 2489}, {1096, 36126}, {1173, 33513}, {1176, 4577}, {1214, 4554}, {1260, 7256}, {1298, 41208}, {1301, 44181}, {1304, 42308}, {1331, 4600}, {1332, 4601}, {1400, 653}, {1402, 108}, {1409, 651}, {1410, 934}, {1425, 4566}, {1427, 13149}, {1439, 4569}, {1444, 4623}, {1459, 86}, {1495, 4240}, {1510, 32002}, {1576, 250}, {1577, 1969}, {1636, 11064}, {1637, 46106}, {1649, 34336}, {1790, 4610}, {1796, 4632}, {1797, 4615}, {1799, 689}, {1802, 7259}, {1812, 4631}, {1813, 4620}, {1843, 46151}, {1918, 8750}, {1919, 1474}, {1924, 1973}, {1945, 41207}, {1946, 21}, {1949, 41206}, {1960, 37168}, {1973, 24019}, {1974, 32713}, {1976, 685}, {1977, 43925}, {1980, 2203}, {2081, 14918}, {2084, 17442}, {2088, 44427}, {2165, 30450}, {2173, 24001}, {2193, 4612}, {2196, 4584}, {2197, 4552}, {2200, 101}, {2207, 6529}, {2223, 4238}, {2281, 36099}, {2318, 3699}, {2351, 925}, {2353, 1289}, {2395, 16081}, {2422, 6531}, {2433, 16080}, {2435, 35140}, {2451, 9308}, {2485, 17907}, {2489, 393}, {2491, 232}, {2492, 37765}, {2501, 2052}, {2510, 14568}, {2519, 6392}, {2523, 30599}, {2524, 194}, {2525, 8024}, {2530, 16747}, {2574, 15165}, {2575, 15164}, {2578, 2581}, {2579, 2580}, {2616, 40440}, {2623, 275}, {2631, 14206}, {2632, 14208}, {2643, 24006}, {2799, 44132}, {2963, 38342}, {2966, 41174}, {2972, 3265}, {3003, 16237}, {3005, 427}, {3049, 6}, {3050, 36794}, {3051, 35325}, {3063, 1172}, {3120, 46107}, {3121, 6591}, {3122, 7649}, {3124, 2501}, {3125, 17924}, {3221, 3186}, {3250, 31909}, {3265, 305}, {3267, 1502}, {3269, 525}, {3270, 7253}, {3284, 2407}, {3288, 458}, {3289, 2421}, {3292, 5468}, {3455, 935}, {3457, 36306}, {3458, 36309}, {3504, 3222}, {3519, 46139}, {3569, 297}, {3657, 46133}, {3682, 4561}, {3690, 3952}, {3692, 7258}, {3694, 646}, {3695, 27808}, {3700, 7017}, {3708, 1577}, {3709, 281}, {3724, 4242}, {3804, 6995}, {3917, 4576}, {3937, 7192}, {3942, 7199}, {3949, 4033}, {3990, 1332}, {4010, 40717}, {4017, 273}, {4025, 310}, {4041, 318}, {4055, 1331}, {4064, 313}, {4079, 1826}, {4091, 17206}, {4120, 46109}, {4171, 7101}, {4455, 242}, {4466, 3261}, {4516, 44426}, {4524, 7046}, {4557, 15742}, {4558, 4590}, {4559, 46102}, {4563, 34537}, {4574, 1016}, {4575, 24041}, {4580, 308}, {4592, 24037}, {4705, 41013}, {4730, 38462}, {4822, 5342}, {4834, 31902}, {5027, 419}, {5029, 423}, {5040, 422}, {5075, 415}, {5113, 420}, {5191, 7473}, {5466, 46111}, {5489, 339}, {5504, 18878}, {6041, 6103}, {6130, 16089}, {6137, 470}, {6138, 471}, {6140, 37943}, {6332, 28660}, {6367, 44143}, {6368, 311}, {6377, 17921}, {6391, 35136}, {6529, 34538}, {6587, 15466}, {6753, 11547}, {7004, 18155}, {7015, 4594}, {7053, 4616}, {7099, 4637}, {7116, 4603}, {7117, 4560}, {7177, 4635}, {7178, 331}, {7180, 278}, {7216, 1847}, {7234, 7009}, {7250, 1119}, {7252, 46103}, {7254, 1509}, {7669, 30716}, {7927, 44142}, {8029, 2970}, {8034, 2969}, {8057, 14615}, {8061, 20883}, {8552, 7799}, {8611, 312}, {8618, 4249}, {8632, 31905}, {8636, 17520}, {8638, 37908}, {8639, 4185}, {8641, 4183}, {8642, 4233}, {8643, 4248}, {8644, 4232}, {8646, 4206}, {8648, 17515}, {8651, 6353}, {8653, 461}, {8663, 430}, {8664, 428}, {8665, 5064}, {8673, 315}, {8675, 44134}, {8677, 17139}, {8749, 15459}, {8779, 34211}, {8794, 42401}, {8882, 16813}, {9007, 44133}, {9033, 3260}, {9178, 17983}, {9210, 11331}, {9247, 163}, {9391, 44150}, {9407, 23347}, {9409, 30}, {9426, 1974}, {9491, 11325}, {9494, 27369}, {9517, 316}, {10097, 671}, {10099, 2481}, {10316, 4611}, {10397, 27398}, {10412, 18817}, {10547, 827}, {11079, 39290}, {12077, 324}, {12215, 880}, {13366, 35311}, {14060, 33799}, {14208, 561}, {14270, 186}, {14273, 37778}, {14380, 1494}, {14398, 1990}, {14401, 36789}, {14407, 8756}, {14417, 3266}, {14429, 3264}, {14533, 18315}, {14575, 1576}, {14582, 94}, {14585, 32661}, {14592, 20573}, {14600, 2715}, {14601, 32696}, {14618, 18027}, {14824, 5186}, {14908, 691}, {14910, 687}, {14936, 17926}, {14977, 18023}, {14984, 14221}, {15391, 39291}, {15412, 276}, {15413, 6385}, {15420, 40827}, {15421, 40832}, {15422, 8794}, {15451, 5}, {15475, 6344}, {15526, 3267}, {15905, 36841}, {16186, 3268}, {16573, 17899}, {17094, 6063}, {17414, 5094}, {17434, 343}, {17970, 805}, {17971, 17929}, {17972, 17930}, {17973, 17931}, {17974, 17932}, {17975, 17933}, {17976, 17934}, {17977, 17935}, {17978, 17936}, {17979, 17937}, {17989, 17987}, {17990, 17927}, {17992, 17985}, {17994, 6530}, {18001, 17982}, {18002, 17981}, {18105, 32085}, {18117, 7577}, {18210, 693}, {18344, 1896}, {18877, 44769}, {18890, 30441}, {19627, 14591}, {20336, 6386}, {20727, 33946}, {20775, 1634}, {20902, 20948}, {20975, 523}, {21044, 46110}, {21123, 17171}, {21134, 21207}, {21731, 403}, {21796, 17906}, {21828, 17923}, {21837, 17916}, {21906, 14273}, {22061, 18047}, {22080, 4427}, {22085, 14588}, {22086, 16704}, {22089, 1975}, {22090, 33296}, {22093, 17103}, {22094, 4467}, {22096, 3733}, {22115, 10411}, {22143, 31998}, {22159, 7760}, {22260, 8754}, {22341, 6516}, {22352, 10330}, {22363, 1633}, {22369, 4436}, {22370, 36860}, {22373, 4367}, {22381, 932}, {22383, 81}, {22384, 33295}, {22386, 16695}, {22388, 4184}, {23067, 4998}, {23090, 7058}, {23092, 7304}, {23099, 2971}, {23189, 261}, {23200, 5467}, {23216, 669}, {23220, 859}, {23224, 1444}, {23225, 3286}, {23286, 95}, {23610, 42068}, {23616, 36793}, {23620, 3732}, {23874, 44154}, {23878, 44144}, {24018, 304}, {24284, 3978}, {24290, 46108}, {24459, 1921}, {25098, 31008}, {30209, 11185}, {30451, 1993}, {30491, 598}, {32320, 394}, {32585, 32036}, {32586, 32037}, {32654, 32697}, {32656, 4570}, {32659, 4591}, {32661, 249}, {32662, 39295}, {32663, 30528}, {32676, 24000}, {32713, 32230}, {33581, 1301}, {34055, 4593}, {34212, 6330}, {34449, 39418}, {34952, 24}, {34980, 520}, {35072, 15411}, {35236, 2797}, {35364, 35142}, {35518, 40072}, {35909, 5641}, {36051, 36105}, {36054, 1812}, {36058, 4622}, {36060, 36085}, {36212, 2396}, {36214, 18829}, {36296, 23895}, {36297, 23896}, {37084, 44180}, {37754, 24018}, {38352, 31296}, {38356, 33294}, {39005, 38380}, {39013, 15423}, {39180, 40410}, {39181, 31617}, {39201, 3}, {39228, 7782}, {39469, 511}, {39665, 2479}, {39666, 2480}, {39687, 23090}, {40016, 42395}, {40319, 3565}, {40352, 1304}, {40353, 34568}, {40354, 32695}, {40373, 14574}, {40947, 1632}, {41087, 44327}, {41172, 2799}, {41221, 23290}, {41435, 35137}, {42065, 10425}, {42293, 216}, {42651, 41203}, {42653, 451}, {42658, 20}, {42659, 23}, {42660, 378}, {42661, 429}, {42662, 447}, {42663, 460}, {42664, 469}, {42665, 858}, {42666, 860}, {42667, 1113}, {42668, 1114}, {42669, 1981}, {42670, 2074}, {42671, 2409}, {42702, 42717}, {42752, 39534}, {43083, 328}, {43693, 35169}, {43697, 35138}, {43722, 33514}, {43925, 36419}, {44093, 14543}, {44112, 23353}, {44114, 16230}, {44173, 44161}, {44705, 14249}, {44729, 44721}, {46005, 18848}, {46088, 97}, {46112, 17402}, {46113, 17403}

  • X647 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {2, 850, 30476}, {2, 26080, 24960}, {2, 30476, 31277}, {2, 31296, 850}, {75, 21610, 21437}, {110, 36830, 23357}, {110, 45215, 35319}, {111, 9213, 8430}, {111, 10562, 10561}, {351, 669, 8651}, {351, 3005, 669}, {351, 42665, 42659}, {650, 6129, 6591}, {650, 6589, 3310}, {650, 21348, 6590}, {652, 1459, 22383}, {652, 22383, 22086}, {661, 21828, 7180}, {669, 8651, 8644}, {669, 8665, 8664}, {669, 17414, 3005}, {850, 30476, 31174}, {850, 36900, 31296}, {905, 2522, 2523}, {905, 25098, 4025}, {1636, 17434, 32320}, {1637, 12077, 2501}, {2501, 6587, 1637}, {2501, 6753, 2489}, {2501, 9209, 6587}, {2525, 14417, 3265}, {2623, 23286, 46088}, {2966, 18020, 9514}, {3005, 8651, 3804}, {3005, 8664, 8665}, {3005, 9420, 21646}, {3049, 32320, 30451}, {3288, 9210, 3569}, {3709, 7180, 661}, {3804, 8644, 669}, {5638, 5639, 3569}, {6041, 34291, 10567}, {6129, 6586, 33525}, {6586, 6589, 650}, {6587, 16040, 2485}, {8105, 8106, 1637}, {8644, 42658, 42659}, {9213, 10561, 10562}, {10561, 10562, 8430}, {16751, 28606, 4467}, {17414, 39201, 42665}, {24007, 24008, 6130}, {24622, 25594, 2}, {24782, 25084, 2}, {31174, 31277, 30476}, {31296, 36900, 41300}, {31296, 44560, 31277}, {36900, 44560, 31174}, {39665, 39666, 9409}, {41300, 44560, 30476}, {42667, 42668, 42659}

Notes

Let L be the line at infinity. X647 = X230X231∩X441X525; that is, the polar conjugate of isotomic conjugate of L and the isotomic conjugate of polar conjugate of L. (Randy Hutson, February 10, 2016)

The circumcircle of the anti-orthocentroidal triangle is here named the anti-orthocentroidal circle. The point X647 is the radical center of these three circles: circumcircle, orthocentroidal circle, anti-orthocentroidal circle. (Randy Hutson, December 10, 2016)