X₅₅₃

POINT MAIA II

X₅₅₃ = X₅₁-of-intouch-triangle

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

X₅₅₃ (POINT MAIA II) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 7}, {1, 376}

• Related Centers: X₂ (CENTROID), X₇ (GERGONNE POINT), X₁ (INCENTER), X₃₇₆ (CENTROID OF THE ANTIPEDAL TRIANGLE OF X)

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

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

Search Value (6, 9, 13): \(0.636741730375\)

▸ Linear Combinations1 combination

• X₅₅₃ = \(5\) X₇ + X₄₁₅₇₂, \(4\) X₉₄₂ - X₉₅₀, \(2\) X₉₄₂ + X₄₂₉₂, \(5\) X₉₄₂ - \(2\) X₁₂₄₃₃, X₉₄₂ + \(2\) X₂₄₄₇₀, X₉₅₀ + \(2\) X₄₂₉₂, \(5\) X₉₅₀ - \(8\) X₁₂₄₃₃, X₉₅₀ + \(8\) X₂₄₄₇₀, \(5\) X₄₂₉₂ + \(4\) X₁₂₄₃₃, X₄₂₉₂ - \(4\) X₂₄₄₇₀, \(2\) X₁₀₁₂₂ + X₁₀₁₂₃, X₁₂₄₃₃ + \(5\) X₂₄₄₇₀, X₆₅ + \(2\) X₄₂₉₈, \(2\) X₆₅ + X₁₀₁₀₆, \(5\) X₆₅ + X₁₀₉₄₄, \(4\) X₄₂₉₈ - X₁₀₁₀₆, \(10\) X₄₂₉₈ - X₁₀₉₄₄, \(5\) X₅₄₃₄ - X₁₀₉₄₄, \(5\) X₁₀₁₀₆ - \(2\) X₁₀₉₄₄, X₇₂ - \(4\) X₁₂₄₃₆, \(2\) X₂₁₀ - \(3\) X₄₆₉₁₆, \(3\) X₃₅₄ - X₃₀₅₈, X₃₀₅₈ + \(3\) X₁₁₂₄₆, X₅₀₈₃ + \(2\) X₂₄₄₆₅, \(2\) X₃₆₄₉ + X₄₁₅₅₁, X₁₇₇₀ + \(5\) X₁₈₃₉₈, X₃₀₅₇ - \(4\) X₁₂₅₇₇, \(4\) X₃₈₁₂ - X₁₂₅₂₇, X₃₉₆₂ - \(4\) X₁₂₄₄₇, \(4\) X₅₀₄₅ - X₁₀₆₂₄, \(5\) X₅₄₃₉ - \(2\) X₁₂₅₇₂, X₆₂₈₄ - \(4\) X₆₇₄₄, \(2\) X₆₇₃₈ + X₇₃₅₄, \(3\) X₁₀₂₀₂ - X₂₈₄₅₉, X₁₀₅₇₂ - \(4\) X₁₇₇₀₆, \(5\) X₁₁₀₂₅ - \(2\) X₁₅₀₀₆, \(2\) X₁₂₅₇₅ - \(5\) X₁₇₆₀₉, X₁₅₅₅₆ - \(4\) X₃₇₅₄₄, X₁₈₉₉₀ + \(2\) X₃₁₇₉₄, \(2\) X₃₁₇₇₆ + X₃₇₇₃₀

X₅₅₃ lies on the cubic K637.

▸ Incident Central Lines341 lines

{1, 376}, {2, 7}, {4, 30304}, {6, 24177}, {10, 5221}, {11, 13159}, {12, 3828}, {20, 11518}, {30, 942}, {36, 11551}, {46, 10056}, {55, 5542}, {56, 551}, {65, 519}, {72, 12436}, {79, 10308}, {81, 3946}, {85, 42030}, {109, 9108}, {165, 3475}, {171, 24231}, {181, 1463}, {210, 5850}, {222, 5228}, {223, 7271}, {241, 4955}, {244, 41011}, {269, 45126}, {278, 39980}, {279, 39948}, {320, 3687}, {333, 24199}, {345, 17298}, {354, 516}, {377, 24391}, {381, 1210}, {388, 3339}, {390, 44841}, {428, 1876}, {481, 13389}, {482, 13388}, {497, 4312}, {498, 38068}, {515, 5902}, {524, 24471}, {528, 5083}, {534, 1407}, {535, 18838}, {536, 4032}, {540, 35650}, {542, 24235}, {545, 35652}, {547, 34753}, {549, 6147}, {552, 1412}, {554, 37772}, {559, 3639}, {664, 41823}, {752, 42053}, {903, 6648}, {938, 3543}, {940, 3663}, {946, 1709}, {962, 9841}, {999, 3656}, {1014, 40592}, {1046, 24178}, {1056, 2093}, {1081, 37773}, {1082, 3638}, {1086, 3163}, {1125, 3647}, {1155, 4995}, {1214, 1418}, {1266, 1999}, {1357, 2796}, {1358, 1366}, {1373, 1659}, {1374, 13390}, {1420, 38314}, {1427, 39974}, {1432, 43263}, {1443, 17011}, {1454, 10197}, {1465, 26740}, {1466, 11517}, {1467, 11111}, {1469, 4685}, {1617, 4428}, {1697, 11037}, {1738, 32913}, {1750, 36996}, {1770, 18398}, {1776, 3337}, {1788, 5290}, {1836, 4860}, {1837, 34648}, {1892, 5064}, {2099, 4315}, {2308, 4989}, {2325, 32933}, {2646, 12563}, {2982, 34578}, {2999, 4644}, {3008, 4641}, {3017, 23537}, {3057, 12577}, {3086, 38021}, {3146, 37723}, {3175, 17132}, {3210, 3879}, {3212, 29617}, {3241, 3340}, {3243, 17784}, {3303, 5493}, {3304, 4301}, {3333, 4295}, {3336, 3584}, {3361, 3485}, {3476, 18421}, {3486, 34628}, {3487, 3524}, {3488, 11001}, {3534, 4304}, {3545, 9612}, {3578, 3686}, {3586, 15682}, {3601, 10304}, {3654, 31397}, {3655, 4311}, {3660, 28534}, {3664, 3666}, {3665, 43054}, {3670, 5717}, {3676, 4785}, {3677, 4307}, {3729, 18141}, {3740, 5852}, {3742, 17768}, {3745, 4353}, {3746, 5557}, {3752, 17365}, {3782, 4887}, {3812, 12527}, {3817, 17728}, {3830, 5722}, {3834, 44416}, {3839, 9581}, {3873, 5853}, {3912, 32007}, {3947, 24914}, {3951, 37462}, {3962, 12447}, {3977, 18139}, {4021, 37595}, {4034, 41915}, {4035, 17740}, {4052, 40151}, {4059, 10521}, {4077, 31148}, {4102, 4431}, {4190, 11520}, {4257, 26728}, {4293, 11529}, {4310, 5269}, {4314, 34638}, {4317, 5882}, {4321, 34607}, {4327, 8270}, {4334, 42042}, {4349, 17599}, {4383, 24175}, {4413, 21060}, {4416, 19804}, {4419, 17022}, {4421, 37541}, {4512, 38053}, {4640, 25557}, {4656, 17276}, {4659, 34255}, {4664, 7201}, {4666, 44447}, {4667, 5256}, {4669, 5252}, {4684, 32932}, {4745, 40663}, {4847, 5880}, {4888, 5712}, {4891, 28530}, {4896, 17595}, {4967, 37653}, {4980, 6358}, {4982, 45222}, {5045, 10624}, {5049, 28174}, {5054, 11374}, {5071, 5714}, {5121, 33096}, {5122, 5719}, {5223, 26040}, {5270, 6901}, {5272, 24695}, {5284, 10032}, {5393, 21171}, {5425, 21578}, {5439, 12572}, {5563, 6906}, {5603, 14646}, {5698, 10582}, {5703, 15692}, {5743, 17345}, {5758, 37526}, {5759, 10857}, {5762, 11227}, {5843, 10157}, {5847, 24165}, {5919, 28228}, {6175, 6734}, {6284, 6744}, {6604, 17294}, {6703, 17235}, {6705, 26877}, {6738, 7354}, {6897, 11362}, {6904, 11523}, {7004, 40960}, {7056, 42309}, {7146, 29574}, {7175, 46922}, {7176, 29584}, {7195, 16833}, {7228, 44417}, {7247, 19797}, {7248, 10473}, {7321, 14829}, {7580, 43177}, {7702, 41565}, {7964, 43151}, {8025, 17190}, {8581, 41539}, {8703, 24929}, {8808, 10400}, {9026, 22278}, {9580, 10580}, {9613, 34627}, {9778, 10389}, {9955, 11544}, {10164, 17718}, {10165, 21165}, {10178, 38454}, {10202, 28459}, {10396, 11023}, {10569, 17642}, {10572, 17706}, {10578, 30340}, {10588, 19876}, {10592, 38083}, {10895, 38076}, {11025, 15006}, {11113, 37566}, {11375, 19883}, {11552, 30384}, {11679, 42697}, {11684, 24564}, {12351, 21636}, {12512, 37080}, {12560, 47357}, {12575, 17609}, {12625, 37435}, {12701, 21625}, {12709, 31165}, {13151, 44255}, {13624, 16137}, {14450, 41012}, {15171, 28202}, {15326, 44840}, {15556, 37544}, {15888, 43174}, {15935, 19710}, {15936, 18650}, {15956, 18655}, {16009, 16116}, {16466, 24171}, {16579, 18607}, {16834, 17079}, {16888, 17382}, {17050, 18206}, {17067, 26723}, {17074, 22464}, {17092, 28606}, {17094, 45669}, {17205, 40153}, {17364, 17490}, {17378, 41777}, {17549, 37583}, {17561, 31424}, {17757, 44848}, {18193, 26098}, {18201, 24239}, {18389, 28452}, {18625, 33150}, {18990, 28204}, {19541, 41561}, {19708, 30282}, {19723, 24796}, {19819, 36595}, {20070, 37556}, {20292, 26015}, {20367, 22097}, {21255, 32777}, {23681, 37642}, {24208, 43682}, {24210, 32857}, {24216, 33106}, {24328, 37269}, {24474, 28458}, {24588, 41140}, {24692, 29655}, {27790, 43283}, {28208, 31776}, {28301, 42044}, {28557, 32915}, {29594, 30617}, {31145, 37709}, {31995, 37655}, {34641, 41687}, {34647, 40726}, {34937, 37522}, {35242, 41870}, {36913, 40617}, {37584, 44284}, {37756, 41629}, {38055, 41166}, {39126, 42029}, {39542, 44675}, {39704, 44733}, {39782, 41150}, {41003, 41311}, {41801, 42045}, {43573, 43855}, {45638, 45639}

▸ Functional Relations19 relations

• X₅₅₃ = midpoint of X_i and X_j for these {i,j}: {65, 5434}, {354, 11246}, {11112, 24473}, {24474, 28458}

• X₅₅₃ = reflection of X_i in X_j for these {i,j}: {5434, 4298}, {10106, 5434}, {10624, 15170}, {15170, 5045}, {40998, 3742}

• X₅₅₃ = isogonal conjugate of X₃₃₆₃₅

• X₅₅₃ = isotomic conjugate of X₄₁₀₂

• X₅₅₃ = complement of X₁₇₇₈₁

• X₅₅₃ = X₁₀₃₀₈-complementary conjugate of X₁₄₁

• X₅₅₃ = X_i-Ceva conjugate of X_j for these (i,j): {7, 3649}, {4552, 3669}, {4573, 3676}, {38340, 514}

• X₅₅₃ = X_i-cross conjugate of X_j for these (i,j): {1100, 1125}, {34502, 7}

• X₅₅₃ = X_i-Dao conjugate of X_j for these (i,j): {2, 4102}, {3, 33635}, {8, 1213}, {9, 3647}, {9, 32635}, {223, 1255}, {478, 1126}, {522, 35076}, {553, 17781}, {1100, 4420}, {1125, 2321}, {1214, 6539}, {1268, 3160}, {3120, 3700}, {3634, 4060}, {3649, 24048}, {4560, 16726}, {4608, 40615}, {4886, 41809}, {6540, 10001}, {31010, 40622}, {32018, 40593}

• X₅₅₃ = cevapoint of X_i and X_j for these (i,j): {1100, 32636}, {1125, 4856}

• X₅₅₃ = crosspoint of X_i and X_j for these (i,j): {7, 1434}, {554, 1081}

• X₅₅₃ = crosssum of X_i and X_j for these (i,j): {55, 1334}, {1250, 10638}

• X₅₅₃ = trilinear pole of line {4977, 5298}

• X₅₅₃ = X₅₁-of-intouch-triangle

• X₅₅₃ = excentral-to-intouch similarity image of X₂

• X₅₅₃ = X_i-isoconjugate of X_j for these (i,j): {1, 33635}, {6, 32635}, {8, 28615}, {9, 1126}, {31, 4102}, {33, 1796}, {41, 1268}, {55, 1255}, {210, 1171}, {650, 8701}, {663, 37212}, {1334, 40438}, {2150, 6538}, {2175, 32018}, {2194, 6539}, {3063, 6540}, {3709, 4596}, {4041, 4629}

• X₅₅₃ = barycentric product X_i*X_j for these {i,j}: {7, 1125}, {12, 30593}, {56, 1269}, {57, 4359}, {65, 16709}, {75, 32636}, {85, 1100}, {86, 3649}, {190, 30724}, {226, 8025}, {269, 3702}, {273, 3916}, {278, 4001}, {279, 3686}, {307, 31900}, {331, 22054}, {348, 1839}, {552, 8013}, {651, 4978}, {658, 4976}, {664, 4977}, {903, 5298}, {934, 4985}, {1014, 4647}, {1088, 3683}, {1213, 1434}, {1230, 1412}, {1414, 30591}, {2308, 6063}, {2355, 7182}, {3261, 36075}, {3676, 4427}, {4115, 17096}, {4554, 4979}, {4573, 4988}, {4625, 4983}, {4626, 4990}, {4697, 7249}, {4856, 27818}, {4870, 39704}, {4973, 18815}, {4974, 7233}, {6358, 30581}, {17190, 43682}, {24002, 35342}

• X₅₅₃ = barycentric quotient X_i/X_j for these {i,j}: {1, 32635}, {2, 4102}, {6, 33635}, {7, 1268}, {12, 6538}, {56, 1126}, {57, 1255}, {85, 32018}, {109, 8701}, {222, 1796}, {226, 6539}, {604, 28615}, {651, 37212}, {664, 6540}, {1014, 40438}, {1100, 9}, {1125, 8}, {1213, 2321}, {1230, 30713}, {1269, 3596}, {1412, 1171}, {1414, 4596}, {1434, 32014}, {1839, 281}, {1962, 210}, {2308, 55}, {2355, 33}, {3578, 42033}, {3647, 4420}, {3649, 10}, {3676, 4608}, {3683, 200}, {3686, 346}, {3702, 341}, {3775, 3790}, {3911, 31011}, {3916, 78}, {3958, 3694}, {4001, 345}, {4046, 4082}, {4115, 30730}, {4359, 312}, {4410, 4494}, {4427, 3699}, {4565, 4629}, {4573, 4632}, {4647, 3701}, {4654, 43260}, {4697, 7081}, {4856, 3161}, {4870, 3679}, {4966, 3717}, {4969, 2325}, {4970, 27538}, {4973, 4511}, {4974, 3685}, {4975, 4723}, {4976, 3239}, {4977, 522}, {4978, 4391}, {4979, 650}, {4983, 4041}, {4984, 1639}, {4985, 4397}, {4988, 3700}, {4989, 390}, {4990, 4163}, {4992, 4147}, {5298, 519}, {6533, 3702}, {7178, 31010}, {7181, 31013}, {8013, 6057}, {8025, 333}, {8040, 4046}, {16709, 314}, {20970, 1334}, {22054, 219}, {22080, 2318}, {23201, 212}, {30581, 2185}, {30591, 4086}, {30592, 14430}, {30593, 261}, {30724, 514}, {30729, 6558}, {31900, 29}, {32636, 1}, {34502, 3634}, {35327, 3939}, {35342, 644}, {36075, 101}, {41014, 3710}, {41547, 34772}, {41820, 4886}

• X₅₅₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 7, 4654}, {2, 63, 5325}, {2, 2094, 3928}, {2, 4654, 226}, {2, 28610, 3929}, {7, 57, 226}, {7, 226, 3982}, {7, 4031, 3911}, {7, 21454, 57}, {57, 226, 3911}, {57, 4114, 3982}, {57, 4654, 2}, {57, 5219, 5435}, {57, 21454, 4031}, {65, 4298, 10106}, {142, 5325, 2}, {226, 4031, 57}, {226, 4114, 7}, {329, 5437, 5316}, {388, 3339, 4848}, {908, 27003, 6692}, {942, 4292, 950}, {942, 13369, 10122}, {942, 24470, 4292}, {1443, 17011, 47057}, {1836, 4860, 11019}, {3218, 5249, 5745}, {3218, 26842, 5249}, {3296, 6361, 1}, {3306, 5905, 3452}, {3333, 4295, 12053}, {3336, 13407, 6684}, {3339, 4355, 388}, {3649, 5298, 4870}, {3649, 32636, 1125}, {3649, 41549, 41546}, {3782, 37520, 39595}, {3911, 3982, 226}, {3916, 32636, 41547}, {3928, 6173, 2}, {4001, 4359, 3686}, {4031, 4114, 226}, {4114, 21454, 3911}, {4190, 11520, 12437}, {4312, 10980, 497}, {4641, 40688, 3008}, {4870, 5298, 1125}, {4870, 32636, 5298}, {4887, 39595, 3782}, {5221, 10404, 10}, {5228, 10481, 43035}, {6147, 37582, 13411}, {9776, 9965, 9}, {9778, 11038, 10389}, {10136, 10481, 30623}, {11019, 30424, 1836}, {17276, 37674, 4656}, {17483, 27003, 908}, {18201, 33097, 24239}