X₄₂₉

EULER X₅₈-5th-SUBSTITUTION POINT

X₄₂₉ = isogonal conjugate of X₁₇₉₈

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

X₄₂₉ (EULER X₅₈-5th-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {12, 37}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₂ ({X), X₃₇ (CROSSPOINT OF INCENTER AND CENTROID)

Trilinears: \((v + w)\sec A : (w + u)\sec B : (u + v)\sec C\), where \(u : v : w\) = X₅₈; e.g., \(u = u(a,b,c) = a/(b + c)\)

Barycentrics: \((v + w)\tan A : (w + u)\tan B : (u + v)\tan C\)

Search Value (6, 9, 13): \(44.999584520517\)

▸ Alternate Coordinates1 form

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

X₄₂₉ lies on these on the cubic K253.

▸ Incident Central Lines97 lines

{1, 5130}, {2, 3}, {6, 20029}, {10, 1824}, {11, 1104}, {12, 37}, {19, 1213}, {33, 1834}, {34, 5155}, {56, 23304}, {65, 26955}, {92, 3948}, {108, 961}, {115, 20621}, {119, 136}, {120, 5139}, {125, 1902}, {135, 42423}, {208, 1892}, {226, 1426}, {264, 40828}, {281, 27040}, {318, 4451}, {343, 10441}, {495, 1068}, {607, 2238}, {956, 11401}, {958, 26377}, {960, 1211}, {1125, 40985}, {1329, 32777}, {1425, 51365}, {1452, 2245}, {1478, 27802}, {1698, 2960}, {1717, 1722}, {1785, 5530}, {1825, 40663}, {1827, 10395}, {1828, 3454}, {1835, 3649}, {1852, 52427}, {1856, 10958}, {1861, 1900}, {1862, 12019}, {1869, 3925}, {1870, 37737}, {1876, 18635}, {1878, 5087}, {1887, 8287}, {1890, 15254}, {1891, 11363}, {1896, 6530}, {1899, 5706}, {1905, 10974}, {2052, 3597}, {2354, 52087}, {2899, 11681}, {2901, 21072}, {2905, 2907}, {2970, 21664}, {2971, 34337}, {3193, 3564}, {3580, 41723}, {3695, 3701}, {4292, 26933}, {5089, 16589}, {5174, 16824}, {5307, 25466}, {5521, 31845}, {5730, 11396}, {5799, 13567}, {6198, 12135}, {6335, 17981}, {7079, 38930}, {7102, 7952}, {7283, 19839}, {7719, 46196}, {8192, 36844}, {9942, 12136}, {10896, 17111}, {11392, 17408}, {12259, 24474}, {15488, 21243}, {15888, 23710}, {15946, 32431}, {16221, 42422}, {16471, 44086}, {16747, 44146}, {20653, 21033}, {21016, 21029}, {21671, 40967}, {22345, 51414}, {24006, 31946}, {34332, 34338}, {40952, 44547}, {51870, 51879}

▸ Functional Relations25 relations

• X₄₂₉ = midpoint of X_i and X_j for these {i,j}: {4, 7414}, {108, 39990}

• X₄₂₉ = reflection of X_i in X_j for these {i,j}: {1, 51720}, {37361, 5}

• X₄₂₉ = isogonal conjugate of X₁₇₉₈

• X₄₂₉ = complement of X₁₆₀₄₉

• X₄₂₉ = nine-point circle-inverse of X₃₇₉₈₂

• X₄₂₉ = orthocentroidal-circle-inverse of X₄₁₈₅

• X₄₂₉ = polar-circle-inverse of X₁₃₂₅

• X₄₂₉ = polar conjugate of X₁₄₅₃₄

• X₄₂₉ = complement of the isogonal conjugate of X₄₃₇₀₃

• X₄₂₉ = isotomic conjugate of the isogonal conjugate of X₄₄₀₉₂

• X₄₂₉ = polar conjugate of the isotomic conjugate of X₁₂₁₁

• X₄₂₉ = polar conjugate of the isogonal conjugate of X₂₀₉₂

• X₄₂₉ = orthic-isogonal conjugate of X₁₈₂₉

• X₄₂₉ = X_i-complementary conjugate of X_j for these (i,j): {42, 478}, {3435, 1125}, {8048, 3741}, {34277, 21246}, {39167, 34851}, {40097, 8062}, {40454, 49598}, {42467, 3739}, {43703, 10}, {43742, 34831}, {46640, 4369}

• X₄₂₉ = X_i-Ceva conjugate of X_j for these (i,j): {4, 1829}, {108, 523}, {6335, 2501}, {46878, 2292}

• X₄₂₉ = X₂₀₉₂-cross conjugate of X₁₂₁₁

• X₄₂₉ = X_i-isoconjugate of X_j for these (i,j): {1, 1798}, {3, 2363}, {48, 14534}, {58, 1791}, {63, 1169}, {81, 2359}, {163, 15420}, {283, 961}, {1220, 1437}, {1790, 2298}, {4575, 4581}, {7254, 36147}, {9247, 40827}, {23189, 36098}

• X₄₂₉ = X_i-Dao conjugate of X_j for these (i,j): {3, 960}, {3, 1798}, {10, 1791}, {69, 3666}, {115, 15420}, {136, 4581}, {429, 16049}, {905, 3125}, {1169, 3162}, {1211, 1444}, {1249, 14534}, {1790, 52087}, {1812, 2092}, {2359, 40586}, {2363, 36103}, {7254, 39015}, {23189, 38992}

• X₄₂₉ = cevapoint of X₂₀₉₂ and X₄₄₀₉₂

• X₄₂₉ = crosspoint of X_i and X_j for these (i,j): {4, 41013}, {264, 40149}

• X₄₂₉ = crosssum of X_i and X_j for these (i,j): {3, 1437}, {6, 52143}, {184, 2193}

• X₄₂₉ = crossdifference of every pair of points on line {647, 23189}

• X₄₂₉ = barycentric product X_i*X_j for these {i,j}: {4, 1211}, {10, 1848}, {19, 18697}, {25, 1228}, {27, 20653}, {33, 45196}, {76, 44092}, {92, 2292}, {225, 3687}, {226, 46878}, {264, 2092}, {273, 21033}, {278, 3704}, {281, 41003}, {286, 21810}, {313, 2354}, {321, 1829}, {331, 40966}, {349, 40976}, {427, 27067}, {960, 40149}, {1824, 20911}, {1826, 4357}, {1897, 21124}, {1969, 3725}, {2052, 22076}, {3666, 41013}, {3882, 24006}, {6331, 42661}, {6335, 50330}, {7140, 16705}, {7141, 40153}, {41609, 43675}

• X₄₂₉ = barycentric quotient X_i/X_j for these {i,j}: {4, 14534}, {6, 1798}, {19, 2363}, {25, 1169}, {37, 1791}, {42, 2359}, {264, 40827}, {523, 15420}, {960, 1812}, {1193, 1790}, {1211, 69}, {1228, 305}, {1824, 2298}, {1826, 1220}, {1829, 81}, {1848, 86}, {1880, 961}, {2092, 3}, {2269, 283}, {2292, 63}, {2300, 1437}, {2354, 58}, {2501, 4581}, {3004, 15419}, {3666, 1444}, {3687, 332}, {3704, 345}, {3725, 48}, {3882, 4592}, {3965, 1792}, {4357, 17206}, {6371, 7254}, {7140, 14624}, {18697, 304}, {20653, 306}, {20967, 2193}, {21033, 78}, {21124, 4025}, {21810, 72}, {22076, 394}, {22345, 18604}, {27067, 1799}, {40149, 31643}, {40966, 219}, {40976, 284}, {41003, 348}, {41013, 30710}, {41609, 40571}, {41611, 41610}, {42661, 647}, {44092, 6}, {45196, 7182}, {45218, 23086}, {46878, 333}, {50330, 905}, {52326, 23189}

• X₄₂₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 4, 4185}, {2, 2475, 17518}, {2, 7438, 468}, {4, 5, 37368}, {4, 24, 7511}, {4, 25, 37398}, {4, 29, 1884}, {4, 403, 15763}, {4, 406, 25}, {4, 427, 1883}, {4, 451, 28}, {4, 461, 28076}, {4, 475, 4214}, {4, 860, 407}, {4, 3088, 37391}, {4, 3089, 37387}, {4, 3144, 27}, {4, 3542, 7497}, {4, 4194, 4186}, {4, 4213, 29}, {4, 4219, 1885}, {4, 4222, 428}, {4, 5136, 37226}, {4, 5142, 427}, {4, 6353, 4198}, {4, 7412, 3575}, {4, 7559, 23047}, {4, 8889, 4200}, {4, 14018, 1889}, {4, 17555, 1894}, {5, 405, 37315}, {5, 47510, 37056}, {10, 39579, 1824}, {21, 4231, 20832}, {21, 37983, 37360}, {25, 37346, 37982}, {28, 451, 468}, {225, 1826, 1867}, {235, 27687, 37982}, {403, 1594, 45168}, {406, 860, 4205}, {407, 430, 4}, {427, 1904, 4}, {430, 4205, 37398}, {431, 442, 37982}, {440, 4205, 37225}, {442, 30446, 30444}, {442, 30447, 21530}, {442, 51557, 5}, {469, 17555, 4}, {851, 27685, 18641}, {1312, 1313, 37982}, {1848, 46878, 1829}, {1904, 5142, 1883}, {3136, 27687, 442}, {4198, 6353, 17523}, {4205, 37346, 442}, {4214, 5094, 475}, {5000, 5001, 37360}, {5064, 17516, 4}, {7563, 37420, 4}, {13442, 33305, 3145}, {21530, 30444, 442}, {25984, 26019, 14011}, {26120, 27531, 7413}, {27553, 37225, 440}, {29993, 46525, 26550}, {30446, 30447, 442}

Notes

As a point on the Euler line, X₄₂₉ has Shinagawa coefficients \((\$a\$F,\$a\$(E + F) + 2abc)\).