X₁₄₄

ANTICOMPLEMENT OF X₇

X₁₄₄ is the anticomplement of X₇.

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

X₁₄₄ (ANTICOMPLEMENT OF X₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 7}

• Related Centers: X₂ (CENTROID), X₇ (GERGONNE POINT)

Trilinears: \((\csc A)(\tan B/2 + \tan C/2 - \tan A/2) : :\)

Barycentrics: \(\tan B/2 + \tan C/2 - \tan A/2 : :\)

Search Value (6, 9, 13): \(6.399784419454\)

▸ Alternate Coordinates3 forms

• Barycentrics: \(1/(a - b - c) + 1/(a - b + c) + 1/(a + b - c) : :\)

• Barycentrics: \(3a^{2} - 2a(b + c) - (b - c)^{2} : :\)

• Barycentrics: \(3 a^{2} - 2 a b - b^{2} - 2 a c + 2 b c - c^{2} : :\)

▸ Linear Combinations1 combination

• X₁₄₄ = \(3\) X₂ - \(4\) X₉, \(9\) X₂ - \(8\) X₁₄₂, \(5\) X₂ - \(4\) X₆₁₇₃, \(15\) X₂ - \(16\) X₆₆₆₆, \(9\) X₂ - \(10\) X₁₈₂₃₀, \(21\) X₂ - \(20\) X₂₀₁₉₅, \(2\) X₃ - \(3\) X₂₁₁₆₈, \(3\) X₄ - \(2\) X₃₁₆₇₁, \(3\) X₇ - \(4\) X₁₄₂, X₇ - \(3\) X₆₁₇₂, \(5\) X₇ - \(6\) X₆₁₇₃, \(5\) X₇ - \(8\) X₆₆₆₆, \(3\) X₇ - \(5\) X₁₈₂₃₀, \(7\) X₇ - \(10\) X₂₀₁₉₅, \(3\) X₉ - \(2\) X₁₄₂, \(2\) X₉ - \(3\) X₆₁₇₂, \(5\) X₉ - \(3\) X₆₁₇₃, \(5\) X₉ - \(4\) X₆₆₆₆, \(6\) X₉ - \(5\) X₁₈₂₃₀, \(4\) X₉ - X₂₀₀₅₉, \(7\) X₉ - \(5\) X₂₀₁₉₅, \(2\) X₇₅ - \(3\) X₂₇₄₈₄, \(4\) X₁₄₂ - \(9\) X₆₁₇₂, \(10\) X₁₄₂ - \(9\) X₆₁₇₃, \(5\) X₁₄₂ - \(6\) X₆₆₆₆, \(4\) X₁₄₂ - \(5\) X₁₈₂₃₀, \(8\) X₁₄₂ - \(3\) X₂₀₀₅₉, \(14\) X₁₄₂ - \(15\) X₂₀₁₉₅, X₁₄₅ - \(4\) X₅₆₉₈, \(3\) X₂₁₀ - \(2\) X₁₅₅₈₇, \(3\) X₂₁₀ - X₃₁₃₉₁

X₁₄₄ lies on the Feuerbach circumhyperbola of the anticomplementary triangle, the Mandart hyperbola, the cubics K200, K202, K308, K710, K1044, K1084.

▸ Incident Central Lines360 lines

{2, 7}, {3, 5843}, {4, 2894}, {6, 3672}, {8, 516}, {10, 4312}, {12, 18231}, {20, 72}, {21, 954}, {37, 3945}, {40, 5815}, {42, 4335}, {44, 4000}, {45, 4648}, {65, 27288}, {69, 190}, {71, 27544}, {75, 391}, {77, 2324}, {78, 3522}, {85, 25001}, {86, 4747}, {92, 6994}, {100, 480}, {145, 192}, {149, 1156}, {153, 1145}, {165, 21060}, {175, 30557}, {176, 30556}, {191, 3085}, {194, 20036}, {200, 2951}, {210, 3474}, {213, 4352}, {218, 17691}, {219, 347}, {220, 279}, {238, 4310}, {239, 4452}, {241, 26669}, {269, 25930}, {281, 7282}, {306, 25734}, {320, 344}, {321, 14552}, {345, 33066}, {348, 10004}, {376, 3940}, {405, 11036}, {443, 15650}, {452, 3868}, {524, 17262}, {528, 12531}, {536, 5839}, {545, 4361}, {573, 21362}, {597, 17323}, {599, 17340}, {631, 31657}, {658, 17113}, {666, 30228}, {673, 4373}, {758, 18412}, {912, 6987}, {920, 15518}, {938, 10398}, {942, 5129}, {950, 20008}, {956, 6912}, {960, 3600}, {962, 6766}, {966, 4363}, {984, 4307}, {1001, 2975}, {1086, 16885}, {1119, 26003}, {1212, 24554}, {1213, 4470}, {1260, 7411}, {1266, 4402}, {1278, 20248}, {1418, 25067}, {1419, 3160}, {1434, 24557}, {1441, 4047}, {1633, 12329}, {1654, 2475}, {1721, 28043}, {1742, 2340}, {1743, 3663}, {1757, 24248}, {1761, 14543}, {1766, 7291}, {1788, 8165}, {1901, 31043}, {1959, 4704}, {1992, 4360}, {1999, 10889}, {2095, 6939}, {2245, 27039}, {2267, 18162}, {2287, 5781}, {2293, 24708}, {2321, 32099}, {2325, 17296}, {2345, 4643}, {2478, 5729}, {2664, 25570}, {2801, 6224}, {2895, 2897}, {3008, 3973}, {3059, 3681}, {3086, 6763}, {3091, 5805}, {3161, 3912}, {3174, 3935}, {3241, 30331}, {3243, 3623}, {3247, 4667}, {3339, 18250}, {3419, 3543}, {3434, 16112}, {3475, 3683}, {3476, 31165}, {3487, 17558}, {3523, 3916}, {3589, 17255}, {3616, 5542}, {3618, 4389}, {3619, 17273}, {3620, 4741}, {3621, 5853}, {3629, 17318}, {3630, 17309}, {3631, 17269}, {3640, 30334}, {3641, 30333}, {3648, 5696}, {3650, 5687}, {3664, 3731}, {3671, 5234}, {3686, 4659}, {3687, 10443}, {3715, 11246}, {3739, 7222}, {3758, 17258}, {3812, 28646}, {3826, 11681}, {3832, 5735}, {3839, 18482}, {3870, 4326}, {3873, 5572}, {3876, 6904}, {3946, 16670}, {3949, 24683}, {3952, 4019}, {3958, 8680}, {4001, 34255}, {4021, 16667}, {4073, 4712}, {4098, 29602}, {4110, 25278}, {4182, 20534}, {4190, 5784}, {4192, 22149}, {4292, 5785}, {4293, 5692}, {4294, 5904}, {4301, 24644}, {4308, 15829}, {4313, 11523}, {4321, 19861}, {4329, 5227}, {4343, 17018}, {4344, 7174}, {4353, 16469}, {4359, 20921}, {4370, 17267}, {4371, 4686}, {4384, 31995}, {4422, 7232}, {4430, 7671}, {4462, 6008}, {4468, 6006}, {4473, 16593}, {4512, 10578}, {4552, 20082}, {4640, 5281}, {4645, 27549}, {4652, 15717}, {4661, 7674}, {4670, 28640}, {4675, 16814}, {4678, 24393}, {4683, 33163}, {4699, 24633}, {4715, 4851}, {4748, 17303}, {4758, 28626}, {4788, 20016}, {4795, 28639}, {4847, 9812}, {4855, 21734}, {4859, 4887}, {4882, 5493}, {4888, 29571}, {4896, 25072}, {4902, 31183}, {4912, 17348}, {4915, 28228}, {5044, 17580}, {5175, 17578}, {5176, 17488}, {5177, 11662}, {5231, 9779}, {5274, 24477}, {5290, 18249}, {5302, 28629}, {5440, 10304}, {5587, 5775}, {5703, 31424}, {5708, 17559}, {5709, 5811}, {5714, 5791}, {5738, 31049}, {5739, 32933}, {5758, 7330}, {5766, 7675}, {5809, 12649}, {5832, 6871}, {5833, 9612}, {5857, 20060}, {5880, 15481}, {6007, 20012}, {6067, 11680}, {6144, 17388}, {6147, 16845}, {6244, 10307}, {6350, 32849}, {6360, 20211}, {6542, 20080}, {6600, 7676}, {6604, 32024}, {6605, 10509}, {6762, 9785}, {6764, 10624}, {6885, 31835}, {6908, 26921}, {6926, 24467}, {6995, 7717}, {7155, 17794}, {7172, 32937}, {7175, 9310}, {7201, 17451}, {7227, 17251}, {7238, 17265}, {7262, 33144}, {7277, 16777}, {7321, 17335}, {7613, 32857}, {7670, 16019}, {7672, 13601}, {7677, 24558}, {8055, 30567}, {8163, 12513}, {9533, 31627}, {9780, 30424}, {9797, 12575}, {10005, 32850}, {10177, 11025}, {10392, 24391}, {10442, 11679}, {10446, 21061}, {10580, 30330}, {10590, 17057}, {11008, 17377}, {11037, 31435}, {11160, 17373}, {12125, 12632}, {12514, 15298}, {12560, 19860}, {12630, 20014}, {12670, 14872}, {14555, 32939}, {14986, 15299}, {15254, 30340}, {15374, 28071}, {15492, 17278}, {15680, 20013}, {15913, 31527}, {16020, 24231}, {16284, 21872}, {16435, 23089}, {16552, 17753}, {16566, 28795}, {16669, 17301}, {16675, 17392}, {16713, 17139}, {17002, 26245}, {17007, 26032}, {17045, 24441}, {17051, 26105}, {17116, 17331}, {17118, 17330}, {17120, 17247}, {17132, 17151}, {17137, 27523}, {17147, 20043}, {17170, 17742}, {17183, 18206}, {17234, 31333}, {17243, 28333}, {17253, 17369}, {17261, 17316}, {17264, 17361}, {17272, 17355}, {17279, 17345}, {17281, 17344}, {17285, 21356}, {17288, 17339}, {17289, 17329}, {17298, 25101}, {17300, 29621}, {17327, 26039}, {17343, 21286}, {17375, 29583}, {17496, 20296}, {17582, 24470}, {17615, 17668}, {17776, 32859}, {18161, 21801}, {18600, 27644}, {18661, 24435}, {19742, 19789}, {19993, 20068}, {19998, 22312}, {20009, 20077}, {20018, 25264}, {20020, 20064}, {20101, 31087}, {20905, 30854}, {20930, 28974}, {20992, 21320}, {21039, 24341}, {21075, 27525}, {21084, 28124}, {21219, 30662}, {22003, 24048}, {24597, 33151}, {25000, 27541}, {26034, 32938}, {26768, 27136}, {26871, 32863}, {27268, 27475}, {27543, 28420}, {28628, 28645}, {30225, 32028}, {32007, 32100}, {33099, 33137}

▸ Functional Relations26 relations

• X₁₄₄ = reflection of X_i in X_j for these {i,j}: {2, 6172}, {4, 5779}, {7, 9}, {8, 5223}, {20, 5759}, {100, 6068}, {145, 390}, {149, 1156}, {390, 5698}, {962, 11372}, {2550, 5220}, {3868, 5728}, {4312, 10}, {4430, 7671}, {4440, 673}, {4452, 5838}, {5880, 15481}, {5905, 8545}, {8581, 960}, {9965, 12848}, {17314, 17262}, {20014, 12630}, {20059, 7}, {20533, 190}, {25722, 3059}, {30628, 14100}, {31391, 15587}

• X₁₄₄ = isogonal conjugate of X₁₁₀₅₁

• X₁₄₄ = isotomic conjugate of X₁₀₄₀₅

• X₁₄₄ = complement of X₂₀₀₅₉

• X₁₄₄ = anticomplement of X₇

• X₁₄₄ = anticomplementary conjugate of X₃₄₃₄

• X₁₄₄ = anticomplementary-isogonal conjugate of X₃₄₃₄

• X₁₄₄ = polar conjugate of the isogonal of X₂₂₁₁₇

• X₁₄₄ = perspector of anticomplementary triangle and its intouch triangle (inner Conway triangle)

• X₁₄₄ = perspector of anticomplementary triangle and the extouch triangle of ABC

• X₁₄₄ = perspector of extouch triangle and inner Conway triangle (Gemini triangle 30)

• X₁₄₄ = X₇-of-anticomplementary triangle

• X₁₄₄ = polar conjugate of isogonal conjugate of X₂₂₁₁₇

• X₁₄₄ = X_i-beth conjugate of X_j for these (i,j): (190, 144), (645, 346)

• X₁₄₄ = Conway-triangle-to-inner-Conway-triangle similarity image of X₇

• X₁₄₄ = X_i-Ceva conjugate of X_j for these (i,j): {8, 2}, {516, 20533}, {3729, 192}, {4416, 1654}, {4480, 17487}, {5223, 27484}, {31627, 3160}

• X₁₄₄ = X_i-cross conjugate of X_j for these (i,j): {165, 3160}, {3160, 2}, {21060, 16284}, {21872, 165}

• X₁₄₄ = X_i-isoconjugate of X_j for these (i,j): {1, 11051}, {6, 3062}, {31, 10405}, {56, 19605}

• X₁₄₄ = cevapoint of X_i and X_j for these (i,j): {7, 15913}, {9, 2951}, {57, 7955}, {220, 6244}, {3207, 22117}, {21060, 21872}

• X₁₄₄ = crosspoint of X_i and X_j for these (i,j): {190, 1275}, {16284, 31627}

• X₁₄₄ = crosssum of X₆₄₉ and X₁₄₉₃₆

• X₁₄₄ = crossdifference of every pair of points on line {663, 20980}

• X₁₄₄ = barycentric product X_i*X_j for these {i,j}: {1, 16284}, {8, 3160}, {9, 31627}, {75, 165}, {76, 3207}, {86, 21060}, {190, 7658}, {264, 22117}, {274, 21872}, {312, 1419}, {341, 17106}, {346, 9533}, {1275, 13609}

• X₁₄₄ = barycentric quotient X_i/X_j for these {i,j}: {1, 3062}, {2, 10405}, {6, 11051}, {9, 19605}, {165, 1}, {1419, 57}, {3160, 7}, {3207, 6}, {7658, 514}, {9533, 279}, {13609, 1146}, {15856, 30330}, {16284, 75}, {17106, 269}, {21060, 10}, {21872, 37}, {22117, 3}, {23058, 24856}, {31627, 85}

• X₁₄₄ = X_i-anticomplementary conjugate of X_j for these (i,j): {1, 3434}, {2, 21285}, {6, 7}, {8, 6327}, {9, 69}, {21, 17135}, {25, 12649}, {29, 20242}, {31, 145}, {32, 3210}, {33, 4}, {37, 2893}, {41, 2}, {42, 2475}, {43, 20350}, {48, 347}, {55, 8}, {57, 6604}, {58, 3873}, {60, 17140}, {71, 2897}, {75, 21280}, {78, 1370}, {81, 20244}, {100, 21302}, {101, 693}, {109, 3900}, {163, 4467}, {192, 20559}, {198, 5932}, {200, 3436}, {210, 1330}, {212, 20}, {213, 17778}, {219, 4329}, {220, 329}, {228, 3152}, {251, 20247}, {281, 21270}, {282, 21279}, {283, 20243}, {284, 75}, {294, 20347}, {312, 315}, {314, 17138}, {318, 11442}, {333, 17137}, {346, 21286}, {522, 21293}, {604, 4452}, {607, 5905}, {643, 512}, {644, 20295}, {645, 17217}, {646, 21304}, {650, 150}, {662, 4374}, {663, 149}, {692, 522}, {765, 3888}, {904, 29840}, {911, 9436}, {923, 4442}, {983, 25304}, {1110, 100}, {1126, 20292}, {1172, 17220}, {1174, 85}, {1212, 2890}, {1252, 21272}, {1253, 144}, {1320, 21282}, {1333, 3875}, {1334, 2895}, {1395, 11851}, {1397, 17480}, {1412, 17158}, {1415, 4025}, {1812, 18659}, {1857, 5906}, {1973, 30699}, {2053, 10453}, {2149, 664}, {2150, 4360}, {2175, 192}, {2176, 20537}, {2185, 17143}, {2188, 280}, {2191, 6601}, {2192, 962}, {2193, 17134}, {2194, 1}, {2195, 518}, {2200, 18667}, {2204, 3187}, {2212, 193}, {2251, 30577}, {2258, 388}, {2259, 1441}, {2287, 20245}, {2289, 6527}, {2299, 3868}, {2311, 30941}, {2316, 320}, {2319, 21281}, {2320, 21283}, {2321, 21287}, {2328, 3869}, {2329, 30660}, {2332, 92}, {2338, 4872}, {2340, 20344}, {2341, 17139}, {2342, 517}, {2344, 4441}, {2360, 20221}, {2361, 6224}, {3063, 4440}, {3207, 31527}, {3596, 21275}, {3683, 2891}, {3684, 20345}, {3685, 20554}, {3688, 21289}, {3689, 21290}, {3693, 20552}, {3699, 21301}, {3700, 21294}, {3709, 21221}, {3711, 21291}, {3939, 513}, {4041, 3448}, {4166, 20346}, {4182, 20555}, {4548, 21215}, {4612, 17159}, {4636, 17166}, {4845, 5057}, {4876, 20553}, {5546, 7192}, {5547, 17491}, {5548, 21297}, {6065, 3952}, {6602, 30695}, {7037, 9799}, {7054, 21273}, {7069, 2888}, {7070, 6225}, {7071, 5942}, {7072, 11415}, {7074, 6223}, {7075, 32548}, {7077, 4645}, {7084, 17784}, {7110, 21276}, {7115, 4566}, {7118, 9965}, {7156, 14361}, {7367, 189}, {8611, 13219}, {8750, 521}, {8851, 20352}, {9439, 497}, {9447, 194}, {9448, 17486}, {9456, 1266}, {10482, 3681}, {11051, 32003}, {13455, 637}, {14547, 2894}, {14827, 3177}, {14942, 20556}, {15374, 9801}, {18265, 17759}, {18889, 527}, {20967, 5484}, {21059, 7674}, {23990, 4552}, {24019, 23683}, {28615, 3879}, {32635, 20290}, {32652, 8058}, {32665, 4453}, {32674, 17896}, {32677, 22464}, {32739, 17496}, {33299, 1369}

• X₁₄₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 9965, 21454}, {2, 20059, 7}, {2, 20078, 9965}, {2, 20214, 5905}, {6, 3672, 17014}, {6, 4419, 3672}, {6, 17334, 4419}, {7, 9, 2}, {7, 6172, 9}, {7, 18230, 142}, {7, 29007, 8232}, {8, 3729, 4461}, {8, 4488, 3729}, {8, 30625, 30695}, {8, 30695, 10405}, {9, 142, 18230}, {9, 6173, 6666}, {20, 72, 20007}, {37, 3945, 29624}, {37, 4644, 3945}, {44, 17276, 4000}, {45, 17365, 4648}, {57, 18228, 2}, {63, 329, 2}, {63, 908, 5744}, {63, 17781, 329}, {69, 190, 346}, {69, 346, 29616}, {142, 18230, 2}, {190, 17347, 69}, {192, 193, 145}, {192, 20072, 193}, {193, 20073, 192}, {193, 20111, 20110}, {210, 31391, 15587}, {226, 3929, 5273}, {226, 5273, 2}, {307, 27382, 2}, {320, 344, 4869}, {320, 17336, 344}, {329, 5744, 908}, {329, 20348, 20245}, {391, 4454, 75}, {480, 11495, 100}, {672, 1423, 27624}, {672, 30946, 2}, {894, 17257, 2}, {894, 17333, 17257}, {908, 5744, 2}, {984, 24695, 4307}, {1001, 11038, 3622}, {1743, 3663, 5222}, {2287, 8822, 14953}, {2345, 4643, 5232}, {2550, 5220, 5686}, {2550, 5686, 3617}, {3161, 21296, 3912}, {3177, 3869, 20535}, {3177, 20111, 145}, {3218, 31018, 2}, {3219, 5905, 2}, {3243, 8236, 3623}, {3305, 9776, 2}, {3452, 3928, 5435}, {3452, 5435, 2}, {3487, 31445, 17558}, {3662, 26685, 2}, {3664, 3731, 5308}, {3681, 25722, 3059}, {3686, 4659, 32087}, {3715, 11246, 26040}, {3729, 4416, 8}, {3729, 4480, 4488}, {3758, 17258, 17321}, {3869, 30616, 20111}, {3911, 5328, 2}, {3911, 31142, 5328}, {3912, 25728, 3161}, {3973, 4862, 3008}, {4000, 17276, 4346}, {4357, 5749, 2}, {4363, 17332, 966}, {4416, 4480, 3729}, {4416, 4488, 4461}, {4640, 25568, 5281}, {4643, 17351, 2345}, {4652, 27383, 15717}, {4661, 20075, 20015}, {4704, 20090, 29585}, {4741, 17280, 3620}, {5226, 5745, 2}, {5296, 10436, 2}, {5698, 10394, 6872}, {5745, 28609, 5226}, {5805, 5817, 3091}, {6646, 17350, 2}, {12246, 31793, 20}, {12526, 12527, 8}, {16552, 17753, 27304}, {17120, 17247, 26626}, {17183, 18206, 26818}, {17261, 17364, 17316}, {17272, 17355, 29611}, {17273, 17354, 3619}, {17288, 17339, 29579}, {17298, 25101, 29627}, {18228, 28610, 57}, {20072, 20073, 145}, {21151, 31658, 3523}, {24477, 24703, 5274}, {24909, 25679, 2}, {24952, 25461, 2}, {26059, 26125, 2}, {26065, 27184, 2}, {27058, 27170, 2}, {27282, 27334, 2}, {27509, 28739, 2}, {29621, 32093, 17300}, {31547, 31548, 8}