X549

MIDPOINT OF X2 AND X3

X549 is the midpoint of X2 and X3.
< 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) { 2, 3 }: Euler Line { 15, 395 }: Central Line L(15, 395) { 16, 396 }: Central Line L(16, 396) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_395 = MIDPOINT OF X(14) AND X(16) (1.3794, 1.1048) X_16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X_396 = MIDPOINT OF X(13) AND X(15) (0.5699, 0.2349) X_549 = MIDPOINT OF X(2) AND X(3) (0.5187, 0.1510) A B C X2 X3 X15 X395 X16 X396 X549 (a, b, c) = (5, 6, 7) | AB = 1

X549 (MIDPOINT OF X2 AND X3) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cos(B - C) + 4 \cos A : :\)
Barycentrics: \(4a^{4} - 5a^{2}(b^{2} + c^{2}) + (b^{2} - c^{2})^{2} : :\)
Search Value (6, 9, 13): \(4.705865843343\)
▸ Alternate Coordinates4 forms

  • Trilinears: \(5 \cos A + 2 \cos B \cos C : 5 \cos B + 2 \cos C \cos A : 5 \cos C + 2 \cos A \cos B\)

  • Trilinears: \(3 \cos A + 2 \sin B \sin C : 3 \cos B + 2 \sin C \sin A : 3 \cos C + 2 \sin A \sin B\)

  • Trilinears: \(2 \sec A + 5 \sec B \sec C : 2 \sec B + 5 \sec C \sec A : 2 \sec C + 5 \sec A \sec B\)

  • Barycentrics: \(2 S^{2} + 3 S_{A} (S_{B} + S_{C}) : :\)

▸ Linear Combinations1 combination

  • X549 = X1 - \(3\) X3653, \(3\) X3653 + X3654, \(5\) X2 - X4, \(7\) X2 + X20, \(3\) X2 + X376, \(11\) X2 - X382, \(7\) X2 - \(2\) X546, \(5\) X2 + \(2\) X548, \(4\) X2 + X550, X2 - \(5\) X631, \(4\) X2 - \(5\) X632, \(7\) X2 - \(5\) X1656, \(13\) X2 + X1657, \(11\) X2 - \(7\) X3090, \(13\) X2 - \(5\) X3091, \(17\) X2 - X3146, \(11\) X2 + \(5\) X3522, X2 + \(7\) X3523, X2 + \(3\) X3524, \(7\) X2 - \(11\) X3525, \(5\) X2 - \(7\) X3526, \(13\) X2 + \(7\) X3528, \(19\) X2 + X3529, X2 + \(4\) X3530, \(13\) X2 - \(17\) X3533, \(5\) X2 + X3534, \(9\) X2 - X3543, \(37\) X2 - \(17\) X3544, \(7\) X2 - \(3\) X3545, \(8\) X2 - X3627, \(5\) X2 - \(4\) X3628, \(7\) X2 - X3830, \(23\) X2 - \(7\) X3832, \(11\) X2 - \(3\) X3839, \(19\) X2 - \(5\) X3843, \(4\) X2 - X3845, \(11\) X2 - \(4\) X3850, \(17\) X2 - \(7\) X3851, \(13\) X2 - \(2\) X3853, \(49\) X2 - \(17\) X3854, \(31\) X2 - \(11\) X3855, \(25\) X2 - \(8\) X3856, \(20\) X2 - \(7\) X3857, \(16\) X2 - \(5\) X3858, \(29\) X2 - \(10\) X3859, \(13\) X2 - \(4\) X3860, \(17\) X2 - \(4\) X3861, X2 - \(3\) X5054, \(5\) X2 - \(3\) X5055, \(19\) X2 - \(11\) X5056, \(31\) X2 + X5059, \(5\) X2 - \(2\) X5066, \(17\) X2 - \(13\) X5067, \(29\) X2 - \(13\) X5068, \(13\) X2 - \(11\) X5070, \(9\) X2 - \(5\) X5071, \(25\) X2 - \(11\) X5072, \(23\) X2 - X5073, \(31\) X2 - \(5\) X5076, \(23\) X2 - \(13\) X5079, \(25\) X2 - \(17\) X7486, \(2\) X2 + X8703, \(7\) X2 - \(4\) X10109, \(3\) X2 - \(4\) X10124, \(7\) X2 + \(13\) X10299, \(5\) X2 - \(13\) X10303, \(5\) X2 + \(3\) X10304, \(11\) X2 + X11001, \(2\) X2 - \(3\) X11539, \(5\) X2 - \(8\) X11540, \(53\) X2 - X11541, \(9\) X2 - \(4\) X11737, X2 - \(4\) X11812, X2 + \(2\) X12100, \(11\) X2 - \(2\) X12101, \(23\) X2 - \(4\) X12102, \(11\) X2 + \(2\) X12103, X2 - \(8\) X12108, \(19\) X2 - \(8\) X12811, \(17\) X2 - \(10\) X12812, \(9\) X2 + \(5\) X14093, \(13\) X2 - \(3\) X14269, \(2\) X2 - \(7\) X14869, \(5\) X2 - \(12\) X14890, \(3\) X2 + \(4\) X14891, \(13\) X2 - \(6\) X14892, \(9\) X2 - \(2\) X14893, \(35\) X2 - \(19\) X15022, \(25\) X2 - X15640, \(9\) X2 + X15681, \(13\) X2 - X15682, \(15\) X2 + X15683, \(15\) X2 - X15684, \(17\) X2 + X15685, \(6\) X2 + X15686, \(6\) X2 - X15687, \(7\) X2 + \(3\) X15688, \(11\) X2 + \(3\) X15689, \(7\) X2 + \(2\) X15690, \(9\) X2 + \(2\) X15691, \(3\) X2 + \(5\) X15692, X2 + \(5\) X15693, \(3\) X2 - \(5\) X15694, \(13\) X2 + \(5\) X15695, \(17\) X2 + \(5\) X15696, \(19\) X2 + \(5\) X15697, \(5\) X2 + \(7\) X15698, \(4\) X2 - \(3\) X15699, \(3\) X2 + \(7\) X15700, X2 - \(7\) X15701, \(3\) X2 - \(7\) X15702, \(9\) X2 - \(7\) X15703, \(10\) X2 + X15704, \(7\) X2 + \(9\) X15705, \(5\) X2 + \(9\) X15706, X2 + \(9\) X15707, X2 - \(9\) X15708, \(5\) X2 - \(9\) X15709, \(11\) X2 + \(9\) X15710, \(4\) X2 + \(5\) X15711, \(2\) X2 + \(5\) X15712, \(2\) X2 - \(5\) X15713, \(6\) X2 + \(5\) X15714, \(9\) X2 + \(11\) X15715, \(7\) X2 + \(11\) X15716, \(5\) X2 + \(11\) X15717, \(3\) X2 + \(11\) X15718, X2 + \(11\) X15719, X2 - \(11\) X15720, \(3\) X2 - \(11\) X15721, X2 + \(17\) X15722, \(9\) X2 - \(11\) X15723, \(5\) X2 + \(4\) X15759, \(7\) X2 - \(8\) X16239, \(2\) X2 + \(3\) X17504, \(23\) X2 + \(5\) X17538, \(37\) X2 - \(5\) X17578, \(25\) X2 + X17800, \(7\) X2 + \(5\) X19708, \(11\) X2 - \(5\) X19709, \(8\) X2 + X19710, \(2\) X2 + \(7\) X19711, \(19\) X2 + \(13\) X21734, \(17\) X2 + \(11\) X21735, \(10\) X2 - \(3\) X23046, \(10\) X2 - X33699, \(29\) X2 - X33703, \(7\) X2 + \(4\) X33923, \(3\) X2 + \(2\) X34200, \(13\) X2 - \(8\) X35018, \(37\) X2 - \(55\) X35381, \(123\) X2 - \(65\) X35382, \(143\) X2 - \(5\) X35384, \(39\) X2 - X35400, \(69\) X2 - \(11\) X35401, \(87\) X2 - \(13\) X35402, \(27\) X2 - \(5\) X35403, \(12\) X2 - X35404, \(289\) X2 - \(11\) X35405, \(347\) X2 - \(13\) X35406, \(139\) X2 - \(5\) X35407, \(69\) X2 - \(2\) X35408, \(71\) X2 - \(3\) X35409, \(327\) X2 + \(11\) X35410, \(381\) X2 + \(13\) X35411, \(333\) X2 + \(13\) X35412, \(165\) X2 + \(8\) X35413, \(171\) X2 + \(5\) X35414, \(521\) X2 - \(55\) X35416, \(321\) X2 - \(88\) X35417, \(79\) X2 + \(33\) X35418, \(799\) X2 - \(65\) X35419, \(73\) X2 + \(26\) X35421, \(39\) X2 - \(5\) X35434, \(345\) X2 + \(13\) X35435, \(8\) X2 - \(3\) X38071, \(17\) X2 - \(3\) X38335, \(17\) X2 - \(5\) X41099, \(19\) X2 - \(7\) X41106, \(23\) X2 + \(8\) X41981, \(11\) X2 + \(6\) X41982, X2 + \(6\) X41983, \(11\) X2 - \(12\) X41984, \(13\) X2 - \(12\) X41985, \(47\) X2 - \(24\) X41986, \(23\) X2 - \(6\) X41987, \(39\) X2 - \(8\) X41988, \(43\) X2 - \(20\) X41989, \(34\) X2 - \(13\) X41990, \(34\) X2 - \(11\) X41991, \(22\) X2 - \(23\) X41992, \(13\) X2 + \(4\) X44245, X2 + \(8\) X44580, \(4\) X2 + \(7\) X44682, \(12\) X2 + X44903, \(25\) X2 - \(14\) X44904, \(19\) X2 - \(12\) X45757, \(17\) X2 - \(24\) X45758, \(4\) X2 + \(3\) X45759, \(13\) X2 - \(20\) X45760, \(15\) X2 + \(28\) X45761, \(57\) X2 - \(14\) X45762, \(11\) X2 - \(13\) X46219, \(13\) X2 + \(8\) X46332, \(25\) X2 + \(3\) X46333, \(8\) X2 + \(5\) X46853(and many more)

▸ Incident Central Lines13 lines

{2, 3}, {15, 395}, {16, 396}, {35, 496}, {36, 495}, {141, 542}, {182, 524}, {230, 574}, {302, 617}, {303, 616}, {511, 597}, {517, 551}, {543, 1153}

X549 lies on the cubic K1224.

▸ Incident Central Lines927 lines

{1, 3653}, {2, 3}, {6, 21843}, {7, 38065}, {8, 38066}, {9, 38067}, {10, 13624}, {11, 5010}, {12, 7280}, {13, 10646}, {14, 10645}, {15, 395}, {16, 396}, {17, 5351}, {18, 5352}, {32, 9300}, {35, 496}, {36, 495}, {39, 5306}, {40, 3656}, {46, 37737}, {49, 43572}, {51, 13363}, {52, 10627}, {54, 34483}, {55, 10072}, {56, 10056}, {57, 5719}, {61, 16773}, {62, 16772}, {69, 12017}, {74, 5655}, {76, 32516}, {83, 42788}, {95, 1494}, {98, 8724}, {99, 11632}, {103, 38774}, {104, 38762}, {109, 38786}, {110, 11694}, {113, 14677}, {114, 9167}, {115, 3054}, {119, 44847}, {125, 34153}, {126, 10163}, {128, 38429}, {141, 542}, {142, 38080}, {143, 10625}, {148, 17006}, {156, 10984}, {165, 5886}, {182, 524}, {183, 6390}, {184, 40111}, {185, 11591}, {187, 3815}, {216, 3163}, {226, 5122}, {230, 574}, {252, 25042}, {265, 15051}, {302, 617}, {303, 616}, {316, 37647}, {323, 15087}, {325, 7771}, {343, 37513}, {355, 7987}, {371, 13966}, {372, 8981}, {373, 13364}, {386, 5453}, {389, 5447}, {394, 44683}, {397, 5237}, {398, 5238}, {399, 13392}, {476, 11749}, {484, 5444}, {485, 6410}, {486, 6409}, {487, 32810}, {488, 32811}, {498, 5204}, {499, 5217}, {500, 3216}, {511, 597}, {515, 3828}, {516, 11230}, {517, 551}, {519, 1385}, {523, 9175}, {525, 45321}, {527, 31657}, {528, 1484}, {529, 31659}, {530, 6771}, {531, 6774}, {532, 33458}, {533, 33459}, {535, 23961}, {538, 13334}, {539, 10610}, {541, 5972}, {543, 1153}, {544, 6712}, {553, 6147}, {566, 14836}, {567, 43574}, {568, 2979}, {569, 43652}, {572, 17330}, {575, 8584}, {582, 37522}, {590, 6396}, {591, 43119}, {598, 11669}, {599, 3564}, {615, 6200}, {626, 34510}, {627, 33612}, {628, 33613}, {641, 13821}, {642, 13701}, {671, 21166}, {754, 13335}, {804, 9126}, {912, 11227}, {930, 12026}, {946, 19883}, {952, 3576}, {960, 40296}, {970, 5482}, {971, 38113}, {993, 3035}, {999, 5218}, {1001, 35238}, {1040, 37729}, {1071, 31835}, {1078, 3933}, {1092, 13336}, {1125, 3579}, {1141, 6592}, {1147, 37515}, {1151, 5420}, {1152, 5418}, {1154, 3917}, {1155, 4870}, {1157, 14141}, {1209, 13561}, {1216, 6102}, {1263, 34837}, {1296, 38806}, {1327, 42261}, {1328, 42260}, {1329, 5267}, {1340, 39023}, {1341, 39022}, {1350, 18583}, {1352, 13692}, {1384, 7736}, {1387, 5119}, {1482, 38314}, {1499, 11183}, {1503, 11178}, {1506, 14537}, {1539, 12900}, {1587, 6450}, {1588, 6449}, {1614, 43607}, {1621, 35000}, {1698, 18357}, {1699, 28178}, {1737, 37600}, {1991, 43118}, {1992, 5050}, {1994, 15037}, {2079, 9722}, {2080, 12150}, {2482, 2782}, {2548, 5023}, {2549, 37637}, {2771, 10176}, {2780, 11176}, {2794, 22247}, {2797, 45682}, {2808, 38772}, {2818, 38784}, {2826, 45314}, {3017, 4256}, {3053, 31401}, {3055, 5475}, {3060, 13340}, {3068, 6398}, {3069, 6221}, {3070, 41952}, {3071, 41951}, {3086, 10385}, {3095, 33706}, {3098, 3589}, {3104, 36364}, {3105, 36365}, {3241, 5657}, {3295, 7288}, {3304, 31452}, {3306, 37584}, {3311, 13935}, {3312, 9540}, {3314, 34623}, {3329, 9301}, {3336, 16139}, {3357, 16252}, {3359, 11729}, {3398, 41624}, {3411, 42504}, {3412, 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{6715, 38620}, {6716, 38621}, {6717, 38622}, {6718, 38600}, {6719, 38623}, {6720, 9530}, {6721, 22505}, {6722, 22515}, {6723, 10113}, {6776, 21356}, {6781, 7603}, {7171, 7308}, {7294, 7741}, {7354, 10592}, {7581, 13903}, {7582, 13961}, {7610, 7618}, {7688, 34618}, {7689, 9820}, {7691, 8254}, {7697, 11149}, {7746, 11648}, {7748, 18362}, {7750, 7769}, {7752, 11057}, {7755, 9607}, {7756, 39563}, {7757, 11171}, {7761, 44377}, {7762, 10357}, {7763, 7767}, {7772, 31457}, {7775, 47101}, {7776, 32829}, {7777, 34734}, {7783, 19570}, {7786, 9821}, {7789, 7815}, {7792, 35002}, {7793, 7837}, {7801, 10104}, {7804, 15491}, {7818, 10256}, {7835, 34624}, {7840, 21445}, {7844, 44381}, {7846, 35248}, {7853, 31274}, {7857, 7884}, {7930, 9873}, {7931, 9862}, {7967, 31145}, {7988, 28182}, {7991, 31425}, {7999, 10574}, {8143, 27784}, {8182, 11184}, {8227, 16192}, {8273, 11499}, {8550, 20190}, {8552, 44818}, {8591, 14651}, {8596, 38635}, {8667, 34511}, {8717, 15448}, {8718, 43614}, {8722, 10796}, {8725, 31268}, {8726, 37700}, {8760, 44567}, {9127, 10166}, {9140, 15035}, {9143, 32609}, {9159, 15111}, {9172, 33962}, {9177, 46998}, {9517, 45319}, {9521, 45318}, {9543, 43387}, {9655, 10588}, {9668, 10589}, {9681, 43794}, {9698, 35007}, {9703, 11003}, {9709, 30478}, {9734, 44401}, {9746, 28915}, {9778, 28216}, {9780, 18525}, {9826, 37511}, {9829, 31961}, {9833, 40686}, {9880, 14971}, {9940, 24475}, {9955, 19862}, {10022, 29069}, {10031, 19914}, {10039, 37605}, {10053, 12351}, {10069, 12350}, {10095, 45186}, {10171, 28150}, {10172, 28164}, {10175, 28160}, {10197, 26286}, {10199, 26285}, {10202, 24473}, {10222, 31447}, {10247, 34631}, {10269, 11194}, {10523, 14792}, {10541, 15533}, {10564, 32225}, {10575, 45959}, {10576, 42259}, {10577, 42258}, {10601, 37483}, {10606, 32620}, {10653, 11481}, {10654, 11480}, {10706, 14643}, {10707, 34474}, {10708, 38690}, {10709, 38691}, {10710, 38692}, {10711, 38693}, {10712, 38694}, {10713, 38695}, {10714, 23239}, {10715, 38696}, {10716, 38697}, {10717, 14666}, {10718, 38699}, {10902, 10943}, {10942, 34606}, {10979, 42459}, {10990, 38795}, {10991, 38751}, {10992, 38740}, {11006, 18332}, {11064, 33533}, {11123, 16220}, {11160, 14912}, {11180, 25406}, {11204, 15311}, {11207, 45620}, {11208, 45621}, {11235, 26492}, {11236, 26487}, {11239, 16203}, {11240, 16202}, {11249, 40726}, {11362, 15178}, {11381, 45958}, {11412, 37481}, {11430, 13567}, {11438, 23292}, {11444, 31834}, {11459, 20791}, {11485, 37641}, {11486, 37640}, {11488, 42115}, {11489, 42116}, {11614, 39601}, {11623, 36521}, {11645, 24206}, {11693, 16003}, {11801, 12121}, {11804, 12893}, {12024, 44665}, {12045, 13570}, {12152, 45623}, {12153, 45624}, {12161, 37514}, {12162, 14128}, {12228, 19129}, {12243, 13188}, {12245, 37624}, {12295, 15088}, {12307, 22051}, {12322, 13798}, {12323, 13678}, {12324, 14530}, {12355, 13172}, {12506, 47074}, {12512, 22793}, {12816, 42431}, {12817, 42432}, {13108, 32522}, {13157, 14379}, {13226, 18446}, {13289, 32600}, {13292, 37476}, {13329, 17392}, {13338, 30537}, {13352, 41588}, {13353, 34148}, {13367, 31804}, {13372, 14072}, {13399, 44110}, {13416, 14708}, {13419, 17712}, {13434, 13482}, {13857, 32110}, {13886, 43511}, {13939, 43512}, {13971, 31439}, {14061, 38730}, {14131, 34466}, {14216, 17821}, {14385, 40630}, {14417, 44202}, {14561, 31884}, {14639, 38731}, {14641, 44870}, {14644, 38723}, {14711, 32523}, {14712, 17005}, {14855, 15030}, {14915, 15082}, {14988, 31165}, {15012, 15606}, {15018, 15038}, {15020, 23236}, {15033, 15360}, {15034, 15057}, {15042, 15081}, {15056, 18439}, {15066, 18445}, {15068, 17811}, {15072, 18435}, {15092, 39809}, {15305, 33879}, {15345, 16768}, {15462, 34319}, {15484, 15655}, {15489, 37536}, {15740, 44763}, {15801, 43600}, {15805, 37498}, {15806, 43601}, {16092, 46634}, {16163, 20304}, {16266, 36752}, {16317, 20481}, {16336, 35729}, {16654, 44082}, {16760, 38611}, {16808, 42145}, {16809, 42144}, {16964, 41122}, {16965, 41121}, {16966, 36969}, {16967, 36970}, {16984, 43453}, {17004, 47286}, {17008, 31859}, {17074, 23071}, {17398, 37508}, {17414, 32232}, {17595, 39544}, {17702, 34128}, {17757, 34698}, {17825, 44413}, {18016, 35720}, {18243, 22936}, {18285, 31378}, {18319, 22104}, {18388, 35254}, {18390, 47296}, {18391, 37606}, {18400, 23332}, {18483, 19878}, {18492, 19872}, {18546, 34504}, {18581, 42122}, {18582, 42123}, {18913, 19347}, {18914, 19357}, {18925, 26944}, {20112, 32479}, {20194, 38010}, {20299, 34782}, {20376, 23358}, {20379, 30714}, {20398, 36523}, {20575, 30269}, {20576, 30270}, {20583, 39561}, {21160, 31158}, {21164, 31142}, {21165, 31164}, {21309, 37665}, {21975, 35888}, {22102, 38617}, {22112, 32269}, {22236, 42149}, {22238, 42152}, {22253, 37667}, {22268, 35887}, {22660, 43839}, {22677, 41146}, {22799, 38759}, {22837, 32157}, {22937, 33668}, {23041, 31166}, {23267, 43382}, {23273, 43383}, {23300, 35228}, {23320, 23333}, {24239, 37589}, {24299, 33595}, {24390, 34745}, {24813, 41138}, {24827, 40480}, {24953, 34746}, {25524, 35239}, {25561, 29012}, {25565, 29317}, {26398, 45696}, {26422, 45697}, {26498, 45699}, {26507, 45698}, {28236, 38176}, {29181, 38136}, {29331, 41140}, {29959, 30532}, {30209, 44560}, {30308, 34595}, {30392, 34747}, {30435, 31400}, {31235, 38761}, {31253, 33697}, {31267, 34778}, {31379, 38609}, {31508, 37704}, {31662, 34641}, {31670, 38072}, {31671, 38073}, {31672, 38075}, {31673, 38076}, {31729, 32156}, {31744, 31762}, {31788, 31838}, {31945, 47050}, {32006, 32839}, {32063, 35260}, {32358, 35602}, {32419, 43120}, {32421, 43121}, {32767, 41362}, {32808, 45508}, {32809, 45509}, {32817, 32869}, {32904, 35721}, {33521, 38775}, {33537, 33540}, {33549, 46025}, {33604, 42926}, {33605, 42927}, {33899, 37837}, {34209, 47084}, {34224, 43608}, {34229, 46951}, {34312, 38700}, {34339, 44663}, {34486, 34699}, {34584, 36518}, {34841, 38608}, {35268, 35283}, {35731, 42563}, {35770, 43888}, {35771, 43887}, {35820, 42582}, {35821, 42583}, {36362, 41021}, {36363, 41020}, {36383, 36767}, {36634, 37699}, {36770, 41042}, {36836, 40694}, {36843, 40693}, {37472, 43651}, {37478, 37649}, {37506, 43653}, {37510, 46922}, {37525, 37728}, {37633, 45923}, {37698, 42043}, {37809, 42849}, {38031, 47357}, {38139, 38318}, {38141, 38319}, {38716, 38796}, {39242, 44569}, {39524, 42295}, {40329, 42329}, {41112, 42151}, {41113, 42150}, {41133, 43461}, {41313, 46475}, {41961, 41968}, {41962, 41967}, {41973, 42503}, {41974, 42502}, {41979, 42648}, {41980, 42647}, {42085, 42143}, {42086, 42146}, {42090, 42095}, {42091, 42098}, {42093, 42585}, {42094, 42584}, {42096, 42111}, {42097, 42114}, {42099, 42107}, {42100, 42110}, {42101, 42914}, {42102, 42915}, {42108, 42918}, {42109, 42919}, {42112, 42475}, {42113, 42474}, {42119, 42129}, {42120, 42132}, {42125, 42688}, {42128, 42689}, {42130, 42139}, {42131, 42142}, {42157, 42163}, {42158, 42166}, {42159, 42591}, {42162, 42590}, {42164, 42434}, {42165, 42433}, {42263, 42274}, {42264, 42277}, {42266, 42270}, {42267, 42273}, {42275, 42601}, {42276, 42600}, {42417, 42525}, {42418, 42524}, {42429, 42594}, {42430, 42595}, {42506, 42793}, {42507, 42794}, {42520, 42635}, {42521, 42636}, {42532, 42977}, {42533, 42976}, {42629, 42693}, {42630, 42692}, {42682, 42906}, {42683, 42907}, {42777, 43199}, {42778, 43200}, {42783, 43628}, {42784, 43629}, {42797, 43773}, {42798, 43774}, {42815, 42968}, {42816, 42969}, {42817, 43463}, {42818, 43464}, {42890, 43547}, {42891, 43546}, {42892, 43232}, {42893, 43233}, {42928, 43548}, {42929, 43549}, {42950, 42984}, {42951, 42985}, {42952, 43485}, {42953, 43486}, {42956, 43005}, {42957, 43004}, {42996, 43372}, {42997, 43373}, {43006, 43020}, {43007, 43021}, {43014, 43308}, {43015, 43309}, {43105, 43645}, {43106, 43646}, {43242, 43554}, {43243, 43555}, {43248, 43310}, {43249, 43311}, {43394, 43573}, {43489, 43499}, {43490, 43500}, {43575, 43836}, {43604, 44516}, {43619, 44541}, {43620, 44526}, {44204, 44564}, {44555, 45969}, {44813, 44820}, {45331, 46127}, {45410, 45420}, {45411, 45421}, {46980, 46987}, {46981, 46986}, {46983, 46990}, {46984, 46989}

▸ Functional Relations34 relations

  • X549 = midpoint of Xi and Xj for these (i,j): {1, 3654}, {2, 3}, {4, 3534}, {5, 8703}, {20, 3830}, {40, 3656}, {74, 5655}, {98, 8724}, {99, 11632}, {110, 20126}, {140, 12100}, {165, 5886}, {376, 381}, {382, 11001}, {405, 44284}, {427, 44261}, {442, 44255}, {546, 15690}, {547, 34200}, {548, 5066}, {550, 3845}, {568, 2979}, {599, 11179}, {631, 15693}, {632, 15711}, {858, 44265}, {1350, 20423}, {1352, 43273}, {1368, 44273}, {1656, 19708}, {1657, 15682}, {2072, 44280}, {2482, 6055}, {3060, 13340}, {3091, 15695}, {3095, 33706}, {3098, 5476}, {3104, 36364}, {3105, 36365}, {3146, 15685}, {3241, 34718}, {3431, 44751}, {3522, 19709}, {3523, 15701}, {3524, 5054}, {3525, 15716}, {3526, 15698}, {3530, 11812}, {3543, 15681}, {3545, 15688}, {3576, 26446}, {3627, 19710}, {3628, 15759}, {3655, 3679}, {3819, 16836}, {3839, 15689}, {3843, 15697}, {3860, 44245}, {3917, 9730}, {4421, 45700}, {4669, 5882}, {4677, 37727}, {4930, 34744}, {5050, 10519}, {5055, 10304}, {5071, 14093}, {5188, 44422}, {5473, 25154}, {5474, 25164}, {5569, 7622}, {5648, 11579}, {5652, 21733}, {5656, 35450}, {5657, 10246}, {5731, 5790}, {5890, 23039}, {6054, 14830}, {6175, 28460}, {6194, 32447}, {6321, 12117}, {6863, 19704}, {6865, 19706}, {6958, 19705}, {6996, 19703}, {7502, 44287}, {7610, 7618}, {7775, 47101}, {7998, 40280}, {8182, 11184}, {8550, 22165}, {8598, 15980}, {8667, 34511}, {9126, 16235}, {9880, 38738}, {10031, 19914}, {10109, 33923}, {10124, 14891}, {10156, 33575}, {10164, 10165}, {10182, 10193}, {10192, 23328}, {10297, 47031}, {10564, 32225}, {10625, 21969}, {10717, 14666}, {11006, 18332}, {11050, 20128}, {11112, 28459}, {11113, 28458}, {11123, 16220}, {11171, 22712}, {11194, 45701}, {11202, 23329}, {11231, 17502}, {11250, 44278}, {11295, 44461}, {11296, 44465}, {11539, 17504}, {11585, 44268}, {11623, 36521}, {12101, 12103}, {12108, 44580}, {12243, 13188}, {12355, 13172}, {12506, 47074}, {13083, 13084}, {13626, 35232}, {13627, 35231}, {13632, 13634}, {13633, 13635}, {13857, 32110}, {14070, 44441}, {14417, 44202}, {14561, 31884}, {14639, 38731}, {14643, 15055}, {14644, 38723}, {14855, 15030}, {14869, 19711}, {14893, 15691}, {15035, 15061}, {15072, 18435}, {15122, 18579}, {15360, 37477}, {15561, 34473}, {15640, 17800}, {15644, 21849}, {15678, 47032}, {15683, 15684}, {15686, 15687}, {15692, 15694}, {15696, 41099}, {15699, 45759}, {15700, 15702}, {15704, 33699}, {15706, 15709}, {15707, 15708}, {15712, 15713}, {15715, 15723}, {15718, 15721}, {15719, 15720}, {15760, 44285}, {15764, 15765}, {15819, 21163}, {16092, 46634}, {17525, 37401}, {18281, 18324}, {18546, 34504}, {18570, 44262}, {19548, 19707}, {21153, 38122}, {21154, 38760}, {21166, 38224}, {28452, 37428}, {31145, 34748}, {31694, 44250}, {34152, 44282}, {34726, 44442}, {35018, 46332}, {35404, 44903}, {35955, 37348}, {36362, 41021}, {36363, 41020}, {36439, 36457}, {37506, 43653}, {37922, 44450}, {37950, 44266}, {38692, 38764}, {38693, 38752}, {38697, 38776}, {38716, 38796}, {38727, 38793}, {38737, 38748}, {41490, 41491}, {41982, 47478}, {44210, 44218}, {44882, 47354}, {46264, 47353}, {46980, 46987}, {46981, 46986}, {46983, 46990}, {46984, 46989}, {47088, 47089}, {47097, 47333}

  • X549 = reflection of Xi in Xj for these (i,j): {2, 140}, {3, 12100}, {4, 5066}, {5, 2}, {20, 15690}, {51, 13363}, {110, 11694}, {140, 11812}, {376, 34200}, {381, 547}, {382, 12101}, {428, 23410}, {546, 10109}, {547, 10124}, {548, 15759}, {550, 8703}, {597, 10168}, {632, 15713}, {3524, 41983}, {3530, 44580}, {3534, 548}, {3543, 14893}, {3545, 47599}, {3627, 3845}, {3628, 11540}, {3656, 5901}, {3830, 546}, {3839, 47478}, {3845, 5}, {3853, 3860}, {3860, 35018}, {5055, 47598}, {5066, 3628}, {5476, 3589}, {5655, 10272}, {5946, 5892}, {7575, 18579}, {8584, 575}, {8703, 3}, {9771, 7619}, {9909, 33591}, {10109, 16239}, {10154, 34351}, {10182, 46265}, {10192, 10182}, {10263, 21849}, {10283, 38028}, {11001, 12103}, {11178, 20582}, {11539, 5054}, {11563, 44282}, {11812, 12108}, {12040, 7622}, {12100, 3530}, {12101, 3850}, {13468, 34506}, {13490, 10127}, {14269, 14892}, {14869, 15701}, {14892, 41985}, {14893, 11737}, {15060, 10170}, {15067, 3819}, {15597, 1153}, {15681, 15691}, {15682, 3853}, {15686, 376}, {15687, 381}, {15689, 41982}, {15690, 33923}, {15699, 11539}, {15704, 3534}, {15711, 15712}, {15712, 15693}, {15713, 631}, {15714, 15692}, {16160, 44257}, {16509, 15597}, {17504, 3524}, {17525, 12104}, {19710, 550}, {19711, 3523}, {20423, 18583}, {21849, 5462}, {21850, 5476}, {21969, 143}, {22165, 40107}, {23039, 44324}, {23046, 5055}, {23328, 10193}, {25154, 20252}, {25164, 20253}, {31649, 15673}, {31650, 28465}, {33699, 4}, {34200, 14891}, {34331, 34478}, {34351, 34477}, {34510, 40344}, {35404, 15687}, {36523, 20398}, {37967, 37904}, {38028, 10165}, {38034, 11230}, {38042, 11231}, {38071, 15699}, {38111, 38122}, {38112, 26446}, {38136, 38317}, {38137, 38171}, {38138, 38042}, {38139, 38318}, {38140, 10172}, {38141, 38319}, {38229, 34127}, {39884, 47354}, {41099, 12812}, {41990, 5067}, {44204, 44564}, {44213, 15330}, {44245, 46332}, {44257, 6675}, {44262, 6676}, {44265, 18571}, {44266, 468}, {44267, 47332}, {44268, 43615}, {44270, 16238}, {44275, 6677}, {44278, 10020}, {44280, 37968}, {44282, 44452}, {44422, 11272}, {44682, 19711}, {44903, 15686}, {45759, 17504}, {46853, 15711}, {47313, 12105}, {47353, 18358}, {47354, 24206}, {47478, 41984}, {47598, 14890}

  • X549 = isogonal conjugate of X14483

  • X549 = complement of X381

  • X549 = anticomplement of X547

  • X549 = complement of the isogonal conjugate of X3431

  • X549 = isotomic conjugate of the isogonal conjugate of X44109

  • X549 = isogonal conjugate of the isotomic conjugate of X44148

  • X549 = isotomic conjugate of the polar conjugate of X6749

  • X549 = Thomson isogonal conjugate of X5888

  • X549 = psi-transform of X9143

  • X549 = Xi-complementary conjugate of Xj for these (i,j): {3431, 10}, {43530, 20305}

  • X549 = X18317-Ceva conjugate of X30

  • X549 = X44109-cross conjugate of X6749

  • X549 = X1-isoconjugate of X14483

  • X549 = Xi-Dao conjugate of Xj for these (i,j): {3, 14483}, {381, 549}

  • X549 = crosssum of X6 and X34417

  • X549 = circumcircle-inverse of X37924

  • X549 = orthocentroidal circle inverse of X5055

  • X549 = orthoptic circle of the Steiner inellipe inverse of X10989

  • X549 = first Droz-Farney circle inverse of X30745

  • X549 = ninepoint circle of medial triangle inverse of X15122

  • X549 = center of the Vu pedal-centroidal circle of X20

  • X549 = X3-of-X2-Brocard-triangle

  • X549 = intersection of tangents to Evans conic at X13 and X14

  • X549 = inverse-in-circle-O(PU(5)) of X140

  • X549 = center of inverse-in-{circumcircle, nine-point circle}-inverter of de Longchamps line

  • X549 = trisector nearest X3 of segment X3X5

  • X549 = X115-of-McCay-triangle

  • X549 = Euler line intercept, other than X5, of conic {X5,X13,X14,X15,X16}}

  • X549 = QA-P39 (Midpoint of QA-P12 and QA-P20) of quadrangle ABCX2 (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/encyclopedia-of-quadri-figures/quadrangle-objects/artikelen-qa/204-qa-p39.html)

  • X549 = barycentric product Xi*Xj for these {i,j}: {6, 44148}, {69, 6749}, {76, 44109}

  • X549 = barycentric quotient Xi/Xj for these {i,j}: {6, 14483}, {6749, 4}, {44109, 6}, {44148, 76}

  • X549 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {2, 4, 5055}, {2, 5, 15699}, {2, 20, 3545}, {2, 140, 11539}, {2, 376, 381}, {2, 381, 547}, {2, 382, 47478}, {2, 548, 23046}, {2, 550, 38071}, {2, 631, 5054}, {2, 1656, 47599}, {2, 1657, 14892}, {2, 3522, 3839}, {2, 3523, 3524}, {2, 3524, 3}, {2, 3526, 47598}, {2, 3528, 14269}, {2, 3530, 17504}, {2, 3534, 5066}, {2, 3543, 5071}, {2, 3545, 1656}, {2, 3830, 10109}, {2, 3839, 3090}, {2, 3843, 45757}, …, {3, 4, 548}, {3, 5, 550}, {3, 20, 33923}, {3, 24, 7525}, {3, 140, 5}, {3, 376, 34200}, {3, 381, 376}, {3, 382, 3522}, {3, 547, 15686}, {3, 550, 46853}, {3, 631, 140}, {3, 632, 3627}, {3, 1006, 7508}, {3, 1656, 20}, {3, 1657, 3528}, {3, 2070, 6636}, {3, 2072, 44249}, {3, 3090, 12103}, {3, 3091, 44245}, {3, 3523, 3530}, {3, 3524, 12100}, {3, 3525, 546}, {3, 3526, 4}, {3, 3530, 15712}, {3, 3533, 3853}, {3, 3534, 10304}, {3, 3545, 15690}, {3, 3548, 12362}, {3, 3549, 31829}, {3, 3628, 15704}, {3, 3830, 15688}, {3, 3832, 41981}, {3, 3851, 15696}, {3, 5020, 35243}, {3, 5054, 2}, {3, 5055, 3534}, {3, 5070, 1657}, {3, 5071, 15691}, {3, 5079, 17538}, {3, 6640, 12605}, {3, 6644, 7502}, {3, 6863, 31775}, {3, 6883, 6914}, {3, 6958, 31789}, {3, 6989, 37281}, {3, 7393, 7526}, {3, 7395, 12084}, {3, 7483, 37356}, {3, 7484, 7514}, {3, 7489, 6909}, … , {4, 5, 3857}, {4, 376, 15683}, {4, 548, 15704}, {4, 631, 10303}, {4, 3524, 15698}, {4, 3526, 3628}, {4, 3628, 5}, {4, 5054, 11540}, {4, 5055, 5066}, {4, 5066, 23046}, {4, 5072, 3856}, {4, 7486, 5072}, {4, 10303, 3526}, {4, 10304, 3534}, {4, 15683, 15684}, {4, 15698, 10304}, {4, 15706, 15759}, {4, 15709, 2}, {4, 15717, 3}, {4, 23046, 3845}, {4, 35472, 37931}, {4, 37453, 37942}, {4, 37942, 44957}, {4, 45761, 15714}, {4, 46333, 15640}, {5, 140, 632}, {5, 376, 35404}, {5, 550, 3627}, {5, 1368, 37938}, {5, 3524, 15711}, {5, 3530, 44682}, {5, 3627, 3858}, {5, 3845, 38071}, {5, 11539, 2}, {5, 12100, 45759}, {5, 14869, 140}, {5, 15686, 15687}, {5, 15687, 381}, {5, 15704, 4}, {5, 15712, 3}, {5, 15713, 11539}, {5, 15714, 15686}, {5, 17504, 8703}, {5, 19711, 17504}, {5, 23046, 5066}, {5, 23335, 33332}, {5, 31651, 44286}, {5, 33699, 23046}, {5, 34200, 44903}, {5, 34477, 16532}, {5, 41991, 3851}, {5, 43893, 46030}, {5, 44222, 5499}, {5, 44682, 46853}, {5, 45759, 19710}, {6, 42912, 42633}, {6, 42913, 42634}, {10, 13624, 34773}, {10, 34773, 37705}, {13, 10646, 42943}, {13, 42943, 42118}, {13, 42955, 43544}, {14, 10645, 42942}, {14, 42942, 42117}, {14, 42954, 43545}, {15, 16242, 395}, {16, 16241, 396}, {17, 5351, 42148}, {18, 5352, 42147}, {20, 1656, 546}, {20, 3524, 15716}, {20, 3525, 1656}, {20, 3545, 3830}, {20, 10299, 3}, {20, 15022, 4}, {20, 15688, 15690}, {20, 15705, 19708}, {20, 16239, 5}, {20, 19708, 15688}, {21, 13747, 17527}, {22, 12106, 37936}, {24, 1595, 7715}, {24, 14787, 23410}, {25, 35486, 37935}, {35, 496, 10386}, {35, 3582, 3058}, {35, 5433, 496}, {36, 3584, 5434}, {36, 5432, 495}, {40, 25055, 3656}, {55, 10072, 15170}, {61, 16963, 43229}, {62, 16962, 43228}, {74, 38794, 10272}, {98, 41134, 8724}, {140, 382, 41992}, {140, 546, 16239}, {140, 547, 10124}, {140, 548, 3628}, {140, 631, 14869}, {140, 3523, 15712}, {140, 3524, 8703}, {140, 3530, 3}, {140, 3628, 3526}, {140, 3850, 46219}, {140, 5054, 15713}, {140, 5066, 47598}, {140, 6676, 44452}, {140, 8703, 15699}, {140, 10124, 15694}, {140, 10212, 14118}, {140, 11276, 37251}, {140, 11540, 15709}, {140, 11737, 15723}, {140, 11812, 5054}, {140, 12101, 41984}, {140, 12108, 631}, {140, 14891, 381}, {140, 15331, 7399}, {140, 15690, 47599}, {140, 15692, 15687}, {140, 15693, 17504}, {140, 15698, 23046}, {140, 15700, 15686}, {140, 15706, 33699}, {140, 15707, 19711}, {140, 15711, 38071}, {140, 15712, 550}, {140, 15717, 15704}, {140, 15718, 15714}, {140, 15759, 5055}, {140, 16197, 16238}, {140, 16239, 3525}, {140, 16976, 18570}, {140, 17504, 3845}, {140, 19711, 45759}, {140, 33923, 1656}, {140, 34004, 10125}, {140, 34200, 547}, {140, 35018, 3533}, {140, 37298, 31650}, {140, 37968, 46029}, {140, 41983, 12100}, {140, 44245, 5070}, {140, 44580, 3524}, {140, 44682, 3627}, {140, 45761, 15684}, {140, 47598, 11540}, {186, 427, 37458}, {186, 15246, 3}, {186, 37118, 427}, {187, 3815, 18907}, {230, 574, 15048}, {371, 13966, 19116}, {372, 8981, 19117}, {376, 547, 15687}, {376, 631, 15702}, {376, 3091, 35400}, {376, 3523, 15718}, {376, 3524, 15692}, {376, 3543, 15681}, {376, 5054, 10124}, {376, 5071, 3543}, {376, 10124, 5}, {376, 14891, 15714}, {376, 15078, 47333}, {376, 15681, 15691}, {376, 15683, 3534}, {376, 15686, 550}, {376, 15687, 44903}, {376, 15692, 3}, {376, 15694, 547}, {376, 15700, 14891}, {376, 15702, 2}, {376, 15703, 14893}, {376, 15718, 12100}, {376, 15721, 15694}, {376, 15723, 11737}, {376, 33246, 37461}, {376, 34200, 8703}, {376, 35404, 19710}, {376, 44214, 44213}, {378, 468, 1596}, {381, 547, 5}, {381, 1657, 35434}, {381, 3524, 14891}, {381, 3534, 15684}, {381, 3543, 14893}, {381, 5054, 15694}, {381, 5071, 11737}, {381, 8703, 44903}, {381, 10201, 47334}, {381, 12100, 15714}, {381, 14093, 15681}, {381, 14891, 8703}, {381, 15681, 3543}, {381, 15682, 41988}, {381, 15684, 4}, {381, 15686, 35404}, {381, 15687, 3845}, {381, 15692, 34200}, {381, 15693, 15700}, {381, 15694, 2}, {381, 15700, 3}, {381, 15701, 15721}, {381, 15702, 10124}, {381, 15703, 5071}, {381, 15714, 550}, {381, 15718, 15692}, {381, 15721, 140}, {381, 15723, 15703}, {381, 31152, 31181}, {381, 34006, 37901}, {381, 34200, 15686}, {381, 35434, 14269}, {381, 44211, 44212}, {381, 44214, 44211}, {381, 44903, 3627}, {382, 3090, 3850}, {382, 3522, 12103}, {382, 3839, 12101}, {382, 15689, 11001}, {382, 19709, 3839}, {382, 46219, 3090}, {389, 5447, 6101}, {395, 16242, 42121}, {396, 16241, 42124}, {404, 7483, 8728}, {404, 37291, 7483}, {442, 4188, 17563}, {465, 466, 26906}, {474, 6910, 6675}, {484, 5444, 15950}, {498, 5204, 18990}, {499, 5217, 15171}, {546, 1656, 5}, {546, 3530, 10299}, {546, 3628, 15022}, {546, 10109, 3545}, {546, 15705, 8703}, {546, 16239, 1656}, {546, 33923, 20}, {546, 47599, 10109}, {547, 3530, 15692}, {547, 8703, 35404}, {547, 10124, 2}, {547, 11737, 5071}, {547, 11812, 15702}, {547, 12100, 34200}, {547, 14891, 376}, {547, 14893, 11737}, {547, 15330, 44211}, {547, 15683, 23046}, {547, 15686, 3845}, {547, 15691, 14893}, {547, 15692, 8703}, {547, 15700, 15714}, {547, 15702, 11539}, {547, 15714, 44903}, {547, 15718, 17504}, {547, 15759, 15683}, {547, 35404, 38071}, {547, 41981, 35401}, {547, 44213, 44212}, {547, 44580, 15718}, {547, 45761, 10304}, {547, 46332, 35400}, {548, 3526, 5}, {548, 3530, 15717}, {548, 3628, 4}, {548, 3856, 17800}, {548, 5055, 33699}, {548, 10304, 8703}, {548, 11540, 5055}, {548, 11812, 15709}, {548, 14890, 2}, {548, 15684, 15686}, {548, 15704, 550}, {548, 15759, 10304}, {548, 47598, 5066}, {550, 632, 5}, {550, 15699, 3845}, {550, 15711, 45759}, {550, 16532, 10154}, {550, 35404, 44903}, {550, 44682, 3}, {550, 44903, 15686}, {550, 45759, 8703}, {590, 6396, 42216}, {590, 41946, 35822}, {597, 10168, 38110}, {599, 5085, 11179}, {615, 6200, 42215}, {615, 41945, 35823}, {618, 619, 141}, {618, 5092, 47611}, {619, 5092, 47610}, {631, 3523, 3}, {631, 3524, 2}, {631, 3530, 5}, {631, 3534, 14890}, {631, 5054, 11812}, {631, 7485, 10257}, {631, 10299, 3525}, {631, 12100, 11539}, {631, 15692, 15694}, {631, 15698, 15709}, {631, 15700, 10124}, {631, 15702, 15721}, {631, 15706, 11540}, {631, 15707, 12100}, {631, 15708, 15701}, {631, 15712, 632}, {631, 15717, 3526}, {631, 15718, 547}, {631, 15719, 3524}, {631, 15720, 12108}, {631, 15722, 41983}, {631, 19711, 15699}, {631, 41983, 8703}, {631, 44580, 17504}, {632, 3845, 15699}, {632, 3857, 3628}, {632, 8703, 38071}, {632, 15699, 2}, {632, 15712, 46853}, {632, 17504, 19710}, {632, 35404, 547}, {632, 44682, 550}, {632, 45759, 3845}, {632, 46853, 3858}, {858, 10298, 44239}, {858, 37347, 39504}, {993, 3035, 3820}, {1006, 17549, 28443}, {1078, 7799, 37671}, {1092, 13336, 32046}, {1113, 1114, 37924}, {1125, 3579, 22791}, {1141, 6592, 14073}, and many more

Notes

As a point on the Euler line, X549 has Shinagawa coefficients \((5,-3)\).

Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X3 wrt Pa. Define Lb, Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. X141 of triangle A'B'C' is X549. X549 is also the centroid of the six points of tangency of lines from X3 to Pa, Pb, and Pc. (Randy Hutson, July 20, 2016)

In the construction of the Dao 6-point circle (see X5569), X549 is the centroid of AbAcBcBaCaCb. (Randy Hutson, July 20, 2016)