X₇₄

ISOGONAL CONJUGATE OF EULER INFINITY POINT

The tangents at A, B, C to the Neuberg cubic K001 concur in X₇₄

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

X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {20, 68}, {24, 64}

• Related Centers: X₂₀ (DE LONGCHAMPS POINT), X₆₈ (PRASOLOV POINT), X₂₄ (PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE), X₆₄ (ISOGONAL CONJUGATE OF X₂₀)

Trilinears: \(1/(\cos A - 2 \cos B \cos C) : 1/(\cos B - 2 \cos C \cos A) : 1/(\cos C - 2 \cos A \cos B)\)

Barycentrics: \(a/(\cos A - 2 \cos B \cos C) : b/(\cos B - 2 \cos C \cos A) : c/(\cos C - 2 \cos A \cos B)\)

Search Value (6, 9, 13): \(12.657162814291\)

▸ Alternate Coordinates3 forms

• Trilinears: \(1/(3 \cos A - 2 \sin B \sin C) : 1/(3 \cos B - 2 \sin C \sin A) : 1/(3 \cos C - 2 \sin A \sin B)\)

• Trilinears: \(a/[2a^{4} - (b^{2} - c^{2})^{2} - a^{2}(b^{2} + c^{2})] : :\)

• Barycentrics: \(a^{2}/(3 S_{B}S_{C} - S^{2}) : :\)

▸ Linear Combinations2 combinations

• X₇₄ = \((r^{2} + 2rR + s^{2})\) X₁ - \(R(6r + 9R)\) X₂ + \((r^{2} + 12rR + 18R^{2} - 3s^{2})\) X₃ (Peter Moses, April 2, 2013)

• X₇₄ = \(3\) X₂ - \(4\) X₆₆₉₉, \(9\) X₂ - \(8\) X₁₂₉₀₀, \(3\) X₃ - X₃₉₉, \(3\) X₃ - \(2\) X₁₅₁₁, \(5\) X₃ - \(2\) X₅₆₀₉, \(5\) X₃ - X₁₂₃₀₈, \(4\) X₃ - X₁₄₀₉₄, \(10\) X₃ - \(7\) X₁₅₀₂₀, \(2\) X₃ - \(5\) X₁₅₀₂₁, \(22\) X₃ - \(19\) X₁₅₀₂₃, \(8\) X₃ - \(5\) X₁₅₀₃₄, \(4\) X₃ - \(3\) X₁₅₀₃₅, \(8\) X₃ - \(7\) X₁₅₀₃₆, \(13\) X₃ - \(7\) X₁₅₀₃₉, \(7\) X₃ - \(5\) X₁₅₀₄₀, X₃ - \(3\) X₁₅₀₄₁, \(15\) X₃ - \(13\) X₁₅₀₄₂, \(6\) X₃ - \(5\) X₁₅₀₅₁, \(2\) X₃ + X₁₅₀₅₄, \(2\) X₃ - \(3\) X₁₅₀₅₅, \(5\) X₃ - \(3\) X₃₂₆₀₉, \(5\) X₃ + \(4\) X₃₈₆₂₆, \(13\) X₃ - \(4\) X₃₈₆₃₂, \(5\) X₃ - \(9\) X₃₈₆₃₃, \(13\) X₃ - \(9\) X₃₈₆₃₈, \(3\) X₄ - \(4\) X₇₆₈₇, X₄ + \(2\) X₁₀₉₉₀, \(3\) X₄ - \(2\) X₁₃₂₀₂, \(2\) X₄ - \(3\) X₁₄₆₄₄, \(3\) X₄ - \(5\) X₁₅₀₈₁, X₄ - \(4\) X₂₀₄₁₇, \(4\) X₅ - \(7\) X₁₅₀₅₇, \(4\) X₅ - \(5\) X₁₅₀₅₉, \(2\) X₅ - \(3\) X₁₅₀₆₁, \(3\) X₅ - \(4\) X₄₀₆₈₅, X₆ - \(3\) X₅₆₂₁, \(2\) X₆ - \(3\) X₅₆₂₂, X₂₀ + \(2\) X₁₆₀₀₃, \(3\) X₅₁ - \(2\) X₁₁₈₀₇, \(3\) X₅₄ - \(2\) X₂₉₁₄, \(5\) X₅₄ - \(4\) X₃₂₂₂₆, \(3\) X₆₄ + X₁₇₈₁₂, \(3\) X₁₁₀ - \(2\) X₃₉₉, \(3\) X₁₁₀ - \(4\) X₁₅₁₁, \(5\) X₁₁₀ - \(4\) X₅₆₀₉, X₁₁₀ + \(2\) X₁₀₆₂₀, X₁₁₀ - \(4\) X₁₂₀₄₁, \(5\) X₁₁₀ - \(2\) X₁₂₃₀₈, \(5\) X₁₁₀ - \(7\) X₁₅₀₂₀, X₁₁₀ - \(5\) X₁₅₀₂₁, \(11\) X₁₁₀ - \(19\) X₁₅₀₂₃, \(4\) X₁₁₀ - \(5\) X₁₅₀₃₄, \(2\) X₁₁₀ - \(3\) X₁₅₀₃₅, \(4\) X₁₁₀ - \(7\) X₁₅₀₃₆, \(13\) X₁₁₀ - \(14\) X₁₅₀₃₉, \(7\) X₁₁₀ - \(10\) X₁₅₀₄₀, X₁₁₀ - \(6\) X₁₅₀₄₁, \(15\) X₁₁₀ - \(26\) X₁₅₀₄₂, \(3\) X₁₁₀ - \(5\) X₁₅₀₅₁

X₇₄ lies on the circumcircle, Walsmith rectangular hyperbola, Jerabek hyperbola, Moses-Jerabek conic, 2nd Evans circle, the cubics K001, K039, K073, K114, K130, K187, K223, K255, K279, K374, K446, K447, K448, K488, K489, K496, K499, K505, K513, K523, K524, K536, K564, K595, K596, K597, K614, K639, K668, K669, K695, K698, K724, K802, K803, K811, K816, K818, K819, K854, K905, K923, K929, K930, K1106, K1107, K1166, K1169, K1170, K1172, the curves Q001, Q030, Q125, Q138.

▸ Incident Central Lines674 lines

{1, 3464}, {2, 113}, {3, 110}, {4, 107}, {5, 3521}, {6, 112}, {10, 12368}, {11, 10767}, {12, 12372}, {13, 5618}, {14, 5619}, {15, 5668}, {16, 5669}, {20, 68}, {21, 34800}, {22, 2931}, {23, 9060}, {24, 64}, {25, 3426}, {26, 12279}, {30, 265}, {32, 9984}, {35, 73}, {36, 7343}, {40, 6011}, {49, 10226}, {50, 11079}, {51, 11807}, {52, 11806}, {54, 185}, {55, 3028}, {56, 3024}, {65, 108}, {66, 1289}, {67, 935}, {69, 99}, {70, 1288}, {71, 101}, {72, 100}, {94, 39375}, {97, 19193}, {98, 690}, {102, 2773}, {103, 2774}, {104, 7429}, {105, 2775}, {106, 2776}, {111, 2433}, {140, 14643}, {141, 14982}, {154, 35472}, {165, 2948}, {182, 9970}, {184, 3043}, {186, 1304}, {187, 248}, {195, 22949}, {287, 4235}, {290, 16077}, {323, 2071}, {325, 2855}, {352, 35188}, {371, 19111}, {372, 19110}, {381, 1539}, {382, 10113}, {386, 34453}, {389, 1173}, {394, 17838}, {402, 12369}, {403, 1514}, {468, 10293}, {477, 523}, {484, 2222}, {493, 12377}, {494, 12378}, {511, 691}, {512, 842}, {513, 2687}, {514, 2688}, {515, 2689}, {516, 2690}, {517, 1290}, {518, 2691}, {519, 2692}, {520, 2693}, {521, 2694}, {522, 2695}, {524, 2696}, {525, 2697}, {526, 9142}, {546, 15025}, {548, 20189}, {549, 5655}, {550, 930}, {573, 3031}, {574, 15920}, {578, 13472}, {631, 5972}, {675, 7433}, {689, 7470}, {695, 35476}, {759, 14127}, {789, 33805}, {805, 35002}, {827, 1176}, {841, 8675}, {858, 16167}, {901, 35000}, {907, 34817}, {915, 3657}, {924, 15453}, {927, 5195}, {931, 6876}, {934, 1439}, {952, 36158}, {1064, 29038}, {1099, 9405}, {1112, 1593}, {1113, 2575}, {1114, 2574}, {1138, 5670}, {1147, 3047}, {1151, 11462}, {1152, 11463}, {1154, 1291}, {1157, 3484}, {1177, 10423}, {1181, 8567}, {1192, 10594}, {1199, 13382}, {1243, 4219}, {1245, 32691}, {1250, 36073}, {1263, 5671}, {1276, 5672}, {1277, 5673}, {1286, 18124}, {1287, 18125}, {1292, 2836}, {1293, 2842}, {1294, 9033}, {1295, 2850}, {1296, 1350}, {1297, 2435}, {1300, 15328}, {1303, 32438}, {1305, 20291}, {1309, 38955}, {1311, 7441}, {1337, 5674}, {1338, 5675}, {1352, 41737}, {1370, 12319}, {1385, 5606}, {1428, 32290}, {1464, 36064}, {1498, 3532}, {1499, 2770}, {1510, 14979}, {1533, 32223}, {1553, 22104}, {1587, 19002}, {1588, 19001}, {1594, 6696}, {1597, 3531}, {1620, 35479}, {1650, 10745}, {1656, 34128}, {1657, 11999}, {1764, 38482}, {1853, 18434}, {1870, 19505}, {1885, 10816}, {1899, 35481}, {1903, 40117}, {1942, 2713}, {1987, 3331}, {1989, 11070}, {1993, 19456}, {1994, 13482}, {1995, 11472}, {2070, 11559}, {2080, 32694}, {2088, 11060}, {2132, 39376}, {2133, 5676}, {2330, 32289}, {2372, 22037}, {2373, 32122}, {2374, 32121}, {2393, 5505}, {2646, 11670}, {2706, 22089}, {2709, 18860}, {2720, 5172}, {2752, 3309}, {2758, 3667}, {2764, 34109}, {2766, 6001}, {2790, 15111}, {2794, 11005}, {2883, 10018}, {2930, 31884}, {2940, 16143}, {2979, 12273}, {2981, 14816}, {3003, 32681}, {3060, 12236}, {3065, 5677}, {3068, 8994}, {3069, 13969}, {3081, 38246}, {3090, 6723}, {3091, 20397}, {3100, 12888}, {3101, 12661}, {3134, 39985}, {3146, 12295}, {3147, 6225}, {3153, 19479}, {3165, 10646}, {3166, 10645}, {3184, 15526}, {3231, 9091}, {3258, 32417}, {3260, 40423}, {3428, 22586}, {3440, 5678}, {3441, 5679}, {3455, 40080}, {3466, 5680}, {3479, 5681}, {3480, 5682}, {3481, 5683}, {3482, 5684}, {3483, 5685}, {3515, 13093}, {3516, 7592}, {3518, 11381}, {3522, 14683}, {3523, 16534}, {3524, 5642}, {3528, 24981}, {3529, 15077}, {3533, 38792}, {3541, 14542}, {3542, 12250}, {3563, 35364}, {3564, 32244}, {3565, 6391}, {3566, 40118}, {3574, 35482}, {3576, 11720}, {3616, 11723}, {3627, 11801}, {3628, 15029}, {3827, 10100}, {3832, 38725}, {3843, 20396}, {3851, 15088}, {4220, 9058}, {4231, 9107}, {4296, 19469}, {4299, 18968}, {4302, 12896}, {4549, 16063}, {4558, 15919}, {5012, 12228}, {5024, 9475}, {5070, 15046}, {5085, 6593}, {5094, 7699}, {5095, 14912}, {5158, 15816}, {5169, 7706}, {5181, 10519}, {5189, 36853}, {5318, 11139}, {5321, 11138}, {5480, 22336}, {5486, 6776}, {5562, 22978}, {5584, 11460}, {5597, 12365}, {5598, 12366}, {5603, 11735}, {5640, 31861}, {5656, 35486}, {5840, 10778}, {5866, 10425}, {5870, 10815}, {5871, 10814}, {5878, 7505}, {5889, 12084}, {5893, 35487}, {5894, 18560}, {5895, 11704}, {5900, 35489}, {5916, 39424}, {5917, 39425}, {5921, 32275}, {5925, 35490}, {5961, 15469}, {6003, 12030}, {6010, 30269}, {6037, 11676}, {6055, 9144}, {6091, 35191}, {6102, 15002}, {6103, 6794}, {6143, 25563}, {6145, 6240}, {6146, 35491}, {6151, 14817}, {6197, 10119}, {6200, 6413}, {6221, 6415}, {6233, 8722}, {6236, 43273}, {6284, 12904}, {6321, 14734}, {6325, 32228}, {6353, 35512}, {6396, 6414}, {6398, 6416}, {6403, 39382}, {6515, 18932}, {6570, 10979}, {6584, 22765}, {6636, 14855}, {6644, 10546}, {6698, 10516}, {6759, 11270}, {6799, 32339}, {6811, 13654}, {6813, 13774}, {6998, 9057}, {7059, 8459}, {7060, 8449}, {7164, 8432}, {7165, 8485}, {7280, 9638}, {7325, 8476}, {7326, 8468}, {7327, 8503}, {7328, 8527}, {7329, 8504}, {7354, 12903}, {7413, 9056}, {7417, 9084}, {7423, 9061}, {7434, 9083}, {7471, 38700}, {7488, 8718}, {7492, 8717}, {7503, 25711}, {7506, 11439}, {7512, 22109}, {7514, 20791}, {7526, 10574}, {7527, 9730}, {7547, 40686}, {7550, 16836}, {7556, 40291}, {7574, 19402}, {7575, 32124}, {7576, 15321}, {7577, 23329}, {7712, 10298}, {7732, 11824}, {7733, 11825}, {7953, 41435}, {8059, 37583}, {8172, 8174}, {8173, 8175}, {8431, 8440}, {8433, 8435}, {8434, 8436}, {8437, 8441}, {8438, 8442}, {8439, 8443}, {8444, 8464}, {8445, 8465}, {8446, 8466}, {8447, 8467}, {8448, 8462}, {8450, 8470}, {8451, 8469}, {8452, 8458}, {8453, 8471}, {8454, 8472}, {8455, 8473}, {8456, 8474}, {8457, 8475}, {8460, 8478}, {8461, 8477}, {8463, 8479}, {8480, 8505}, {8481, 8508}, {8482, 8509}, {8483, 8506}, {8484, 8507}, {8486, 8510}, {8487, 8511}, {8488, 8512}, {8489, 8515}, {8490, 8516}, {8491, 8513}, {8492, 8514}, {8493, 8517}, {8494, 8518}, {8495, 8519}, {8496, 8520}, {8497, 8521}, {8498, 8522}, {8499, 8525}, {8500, 8526}, {8501, 8523}, {8502, 8524}, {8528, 8530}, {8529, 8532}, {8531, 8533}, {8537, 13248}, {8550, 35492}, {8705, 32229}, {8744, 32687}, {8998, 9540}, {9069, 14605}, {9070, 41455}, {9100, 9759}, {9129, 38698}, {9143, 10304}, {9160, 14060}, {9181, 38702}, {9202, 14538}, {9203, 14539}, {9218, 38611}, {9544, 35493}, {9545, 35494}, {9703, 35495}, {9704, 35496}, {9705, 12038}, {9706, 35498}, {9716, 35499}, {9729, 16223}, {9777, 35501}, {9781, 9786}, {9818, 9826}, {9833, 35503}, {9938, 11412}, {10102, 30230}, {10111, 18917}, {10263, 13358}, {10282, 17506}, {10303, 38795}, {10310, 13204}, {10311, 41414}, {10409, 14369}, {10410, 14368}, {10421, 12380}, {10540, 15646}, {10610, 18364}, {10619, 18368}, {10625, 12226}, {10632, 10681}, {10633, 10682}, {10638, 36072}, {10663, 11420}, {10664, 11421}, {10698, 31525}, {10880, 13287}, {10881, 13288}, {11003, 39242}, {11004, 13352}, {11012, 39633}, {11017, 22462}, {11061, 25406}, {11081, 39380}, {11086, 39381}, {11179, 41720}, {11202, 20421}, {11248, 13217}, {11249, 13218}, {11250, 12092}, {11403, 15465}, {11410, 12165}, {11411, 30552}, {11414, 12310}, {11416, 12596}, {11417, 12891}, {11418, 12892}, {11423, 11425}, {11458, 11477}, {11465, 11479}, {11466, 11480}, {11467, 11481}, {11550, 18559}, {11562, 14118}, {11589, 15404}, {11634, 38873}, {11693, 15705}, {11694, 17504}, {11699, 13624}, {11804, 15800}, {11822, 13208}, {11823, 13209}, {11826, 13213}, {11827, 13214}, {11828, 13215}, {11829, 13216}, {12017, 26206}, {12042, 18332}, {12074, 12584}, {12082, 33534}, {12085, 38260}, {12100, 13392}, {12113, 13494}, {12162, 15052}, {12225, 15133}, {12254, 13418}, {12261, 12699}, {12262, 41722}, {12278, 32140}, {12282, 12301}, {12285, 12303}, {12286, 12304}, {12287, 12305}, {12288, 12306}, {12291, 12307}, {12315, 15750}, {12505, 32311}, {12898, 34773}, {13017, 13021}, {13018, 13022}, {13145, 26711}, {13391, 32608}, {13397, 28787}, {13403, 32325}, {13414, 41519}, {13415, 41518}, {13434, 13630}, {13452, 26883}, {13474, 34484}, {13568, 15559}, {13595, 16194}, {13603, 32062}, {13621, 32137}, {13665, 13915}, {13785, 13979}, {13868, 15626}, {13935, 13990}, {14110, 30238}, {14374, 14710}, {14375, 14709}, {14457, 18912}, {14458, 41443}, {14480, 14934}, {14490, 31860}, {14536, 41522}, {14540, 39636}, {14541, 39637}, {14561, 32271}, {14639, 15359}, {14651, 16278}, {14685, 14703}, {14809, 16169}, {14830, 20404}, {14833, 19905}, {14853, 15118}, {15030, 35904}, {15043, 16222}, {15058, 17928}, {15078, 18451}, {15085, 37486}, {15089, 15801}, {15131, 23328}, {15232, 26704}, {15320, 26705}, {15322, 41456}, {15329, 39987}, {15342, 34473}, {15459, 41204}, {15478, 40047}, {15545, 38741}, {15578, 19151}, {15579, 37473}, {15664, 32692}, {15749, 33703}, {16000, 18381}, {16164, 21161}, {16186, 34210}, {16340, 20957}, {16620, 16621}, {16658, 37458}, {16868, 22802}, {16936, 35446}, {18317, 20123}, {18324, 20773}, {18358, 26156}, {18363, 34563}, {18439, 37814}, {18445, 25487}, {18551, 21308}, {18916, 18947}, {19051, 42215}, {19052, 42216}, {19121, 19138}, {19168, 19172}, {19361, 32329}, {19376, 26283}, {19406, 19482}, {19407, 19483}, {19424, 19507}, {19425, 19508}, {19454, 19484}, {19455, 19485}, {20186, 40119}, {22115, 34152}, {22329, 38894}, {22535, 22549}, {23061, 37477}, {23240, 35442}, {25328, 29181}, {25641, 36172}, {25738, 34350}, {26861, 33923}, {26914, 26927}, {26915, 26935}, {26916, 26936}, {28788, 30268}, {29299, 37620}, {29317, 32273}, {30250, 34935}, {30257, 41454}, {30270, 39639}, {31074, 34796}, {31133, 40909}, {31384, 40097}, {31724, 34798}, {32235, 35268}, {32237, 37953}, {32251, 39588}, {32274, 36990}, {32349, 37970}, {32581, 42299}, {32618, 40894}, {32619, 40895}, {32620, 40916}, {33962, 35447}, {34007, 43577}, {34207, 39417}, {34298, 34310}, {34435, 38850}, {34437, 38851}, {34440, 38852}, {34568, 41433}, {34594, 37403}, {35265, 37952}, {35373, 40390}, {35465, 39373}, {35834, 42267}, {35835, 42266}, {36034, 36069}, {36071, 36131}, {36193, 38609}, {37426, 43356}, {37475, 41670}, {37948, 43572}, {38263, 39562}, {38323, 41171}, {38936, 39372}, {39808, 39812}, {39837, 39841}

▸ Functional Relations79 relations

• X₇₄ = midpoint of X_i and X_j for these {i,j}: {1, 9904}, {3, 10620}, {4, 12244}, {20, 3448}, {40, 33535}, {64, 10117}, {110, 15054}, {125, 10990}, {265, 20127}, {476, 14508}, {1350, 16010}, {1657, 12902}, {2935, 17835}, {5889, 13201}, {6241, 12281}, {6776, 32247}, {7725, 7726}, {9862, 18331}, {10264, 14677}, {11412, 12284}, {12163, 12302}, {12270, 15100}, {12283, 32249}, {12317, 12383}, {13491, 15101}, {15545, 38741}, {16003, 16111}, {32608, 35452}

• X₇₄ = reflection of X_i in X_j for these {i,j}: {1, 11709}, {3, 12041}, {4, 125}, {20, 16111}, {23, 32110}, {52, 11806}, {110, 3}, {113, 6699}, {125, 20417}, {146, 113}, {185, 17855}, {186, 21663}, {265, 10264}, {323, 10564}, {382, 10113}, {399, 1511}, {477, 36164}, {895, 11579}, {974, 15151}, {1112, 16270}, {1199, 34468}, {1498, 15647}, {1533, 32223}, {1539, 20304}, {1553, 22104}, {1986, 974}, {2930, 33851}, {2935, 11598}, {3146, 12295}, {3448, 16003}, {3627, 11801}, {5504, 12901}, {5622, 5621}, {5627, 40630}, {5655, 549}, {5921, 32275}, {6241, 17854}, {6321, 15535}, {7722, 185}, {7728, 5}, {7731, 1986}, {7978, 1}, {9138, 19902}, {9140, 20126}, {9144, 6055}, {9934, 13289}, {9970, 182}, {10113, 20379}, {10263, 13358}, {10540, 15646}, {10698, 31525}, {10706, 2}, {10721, 4}, {10733, 265}, {10752, 6}, {10767, 11}, {11005, 15357}, {11562, 40647}, {11579, 32305}, {11699, 13624}, {12111, 7723}, {12112, 1495}, {12121, 550}, {12244, 10990}, {12290, 12292}, {12292, 15738}, {12295, 36253}, {12308, 5609}, {12368, 10}, {12369, 402}, {12381, 10065}, {12382, 10081}, {12383, 16163}, {12505, 32311}, {12584, 14810}, {12699, 12261}, {12778, 3579}, {12825, 12358}, {12898, 34773}, {13202, 7687}, {13417, 389}, {14094, 110}, {14157, 186}, {14480, 14934}, {14683, 30714}, {14833, 19905}, {14982, 141}, {14989, 34150}, {15035, 15055}, {15054, 10620}, {15055, 15041}, {15063, 5972}, {15107, 3581}, {15131, 23328}, {15329, 39987}, {15463, 32607}, {15472, 19457}, {15800, 11804}, {15801, 15089}, {16105, 11746}, {16163, 37853}, {18332, 12042}, {18781, 18780}, {19140, 5092}, {19506, 20299}, {20127, 14677}, {20957, 16340}, {22115, 34152}, {22265, 98}, {23061, 37477}, {23236, 34153}, {23315, 6696}, {30714, 38726}, {32111, 468}, {32234, 6776}, {34150, 12079}, {34153, 548}, {36172, 25641}, {36193, 38609}, {36990, 32274}, {37477, 37950}, {38520, 38641}, {38789, 34128}, {38790, 1539}, {38791, 6723}, {38898, 13630}, {39985, 3134}, {41720, 11179}, {41737, 1352}, {43572, 37948}, {43574, 2071}, {43576, 7464}

• X₇₄ = isogonal conjugate of X₃₀

• X₇₄ = isotomic conjugate of X₃₂₆₀

• X₇₄ = complement of X₁₄₆

• X₇₄ = anticomplement of X₁₁₃

• X₇₄ = X₁-Ceva conjugate of X₁₇₁₄₉

• X₇₄ = cevapoint of X_i and X_j for these (i,j): (15, 16), (50, 184)

• X₇₄ = crosssum of X_i and X_j for these (i,j): (3, 399), (616),617)

• X₇₄ = X_i-cross conjugate of X_j for these (i,j): (186, 54), (526, 110)

• X₇₄ = circumcircle-antipode of X₁₁₀

• X₇₄ = polar-circle-inverse of X₁₃₃

• X₇₄ = 2nd-Droz-Farney-circle-inverse of X₁₇₈₅₄

• X₇₄ = Schoutte-circle-inverse of X₂₇₁₅

• X₇₄ = 2nd-Brocard-circle-inverse of X₃₈₅₂₀

• X₇₄ = inverse-in-O(15,16) of X₂₇₁₅, where O(15,16) is the circle having segment X₁₅X₁₆ as diameter

• X₇₄ = trilinear pole of line X₆X₆₄₇

• X₇₄ = Ψ(X₆,X₆₄₇)

• X₇₄ = antipode of X₁₁₉₉ in Moses-Jerabek conic

• X₇₄ = reflection of X₄₇₇ in the Euler line

• X₇₄ = reflection of X₈₄₂ in the Brocard axis

• X₇₄ = reflection of X₂₆₈₇ in the line X₁X₃

• X₇₄ = reflection of X₁₂₉₆ in the line X₃X₃₅₁

• X₇₄ = {X₃,X₃₉₉}-harmonic conjugate of X₁₅₁₁

• X₇₄ = X₁₂₈-of-excentral-triangle

• X₇₄ = X₁₃₇-of-hexyl-triangle

• X₇₄ = X₁₂₉₆-of-circumsymmedial

• X₇₄ = trilinear pole wrt circumorthic triangle of van Aubel line

• X₇₄ = X₁₅₇₇-isoconjugate of X₂₄₂₀

• X₇₄ = orthocentroidal-to-ABC similarity image of X₄

• X₇₄ = 4th-Brocard-to-circumsymmedial similarity image of X₄

• X₇₄ = perspector of ABC and the reflection of the Kosnita triangle in X₃

• X₇₄ = orthocenter of X₃X₆₇X₈₇₉

• X₇₄ = intersection of tangents at X₃ and X₄ to Napoleon-Feuerbach cubic, K005

• X₇₄ = X₁₃₁₇-of-tangential-triangle is ABC is acute

• X₇₄ = 2nd-Parry-to-ABC similarity image of X₁₁₀

• X₇₄ = X₈₀-of-Trinh-triangle if ABC is acute

• X₇₄ = Trinh-isogonal conjugate of X₂₀₇₁

• X₇₄ = trilinear product of PU(86)

• X₇₄ = perspector of ABC and the (degenerate) side-triangle of the (equilateral) circumcevian triangles of X₁₅ and X₁₆

• X₇₄ = homothetic center of X₁₅- and X₁₆-Ehrmann triangles; see X₂₅

• X₇₄ = perspector of ABC and X₁₅-Ehrmann triangle

• X₇₄ = perspector of ABC and X₁₆-Ehrmann triangle

• X₇₄ = 3rd-Parry-to-circumsymmedial similarity image of X₂₃

• X₇₄ = perspector of ABC and unary cofactor triangle of orthocentroidal triangle

• X₇₄ = endo-homothetic center of X₄-altimedial and X₄-anti-altimedial triangles

• X₇₄ = Thomson isogonal conjugate of X₅₂₃

• X₇₄ = Lucas isogonal conjugate of X₅₂₃

• X₇₄ = X₁₀₀-of-circumorthic-triangle if ABC is acute

• X₇₄ = perspector of ABC and 2nd anti-Parry triangle

• X₇₄ = X₁₁₀-of-2nd-anti-Parry-triangle

• X₇₄ = X₉₁₃₈-of-1st-anti-Parry-triangle

• X₇₄ = excentral-to-ABC functional image of X₅₅₄₁

• X₇₄ = orthic-to-ABC functional image of X₁₂₈

• X₇₄ = trilinear pole wrt, Thomson triangle, of line X₃X₅₆₄₆

• X₇₄ = trilinear pole, wrt Lucas triangle, of line X₄X₁₅₀₆₆

• X₇₄ = BSS(a→a^2) of X₁₁₅₆

• X₇₄ = areal center of pedal triangles of PU(4)

• X₇₄ = areal center of pedal triangles of PU(5)

• X₇₄ = areal center of pedal triangles of PU(11)

• X₇₄ = antipode of X₃₂₁₁₁ in Walsmith rectangular hyperbola

• X₇₄ = orthocenter of X₆X₁₁₀X₃₅₆₉

• X₇₄ = X₅₅₄₁-of-orthic-triangle if ABC is acute

• X₇₄ = perspector of circumconic centered at X₃₆₈₉₆

• X₇₄ = trilinear pole of line X₆X₆₄₇

• X₇₄ = crossdifference of every pair of points on line {1636, 1637}

• X₇₄ = psi-transform of X₁₄₆₈₅

• X₇₄ = Collings transform of X_i for these i: {125, 2088, 3134, 39174, 39987}

• X₇₄ = X_i-anticomplementary conjugate of X_j for these (i,j): {10419, 8}, {36053, 146}, {40388, 5905}, {40423, 6327}

• X₇₄ = X_i-complementary conjugate of X_j for these (i,j): {31, 36896}, {34178, 10}

• X₇₄ = X_i-Ceva conjugate of X_j for these (i,j): {2, 36896}, {30, 2132}, {1304, 14380}, {1494, 14919}, {2349, 15627}, {5627, 3470}, {9139, 9717}, {10419, 14385}, {14919, 15291}, {15395, 110}, {16077, 2394}, {16080, 8749}, {34568, 647}, {40384, 6}, {40423, 2}

• X₇₄ = X_i-cross conjugate of X_j for these (i,j): {3, 10419}, {6, 40384}, {184, 11079}, {186, 54}, {526, 110}, {647, 34568}, {686, 4558}, {1464, 1}, {1495, 6}, {3003, 2}, {6000, 4}, {9142, 111}, {9409, 112}, {9717, 9139}, {11081, 2981}, {11086, 6151}, {12112, 14483}, {13289, 38534}, {13754, 14264}, {14157, 1173}, {14264, 5627}, {14380, 1304}, {16186, 523}, {18877, 14919}, {21650, 265}, {21663, 3}, {40352, 8749}

• X₇₄ = cevapoint of X_i and X_j for these (i,j): {3, 13754}, {6, 1495}, {15, 16}, {50, 184}, {55, 2245}, {512, 2088}, {523, 3134}, {654, 3270}, {3269, 9409}, {3581, 4550}, {5663, 39987}, {6000, 39174}, {11074, 40355}, {18877, 40352}, {42789, 42790}

• X₇₄ = crosspoint of X_i and X_j for these (i,j): {1, 7164}, {3, 8431}, {4, 1138}, {13, 8462}, {14, 8452}, {15, 8445}, {16, 8455}, {30, 2133}, {399, 8486}, {484, 7327}, {616, 8535}, {617, 8536}, {1157, 8487}, {1263, 8439}, {1337, 8489}, {1338, 8490}, {1494, 16080}, {2132, 8534}, {3065, 3466}, {3440, 3441}, {3464, 8488}, {3465, 7328}, {3479, 8491}, {3480, 8492}, {3481, 8494}, {5667, 8493}, {7059, 7325}, {7060, 7326}, {7165, 7329}, {8446, 8471}, {8456, 8479}, {8495, 8529}, {8496, 8531}, {8499, 8501}, {8500, 8502}, {18878, 39295}

• X₇₄ = crosssum of X_i and X_j for these (i,j): {1, 3464}, {2, 39358}, {3, 399}, {4, 5667}, {13, 5623}, {14, 5624}, {15, 5668}, {16, 5669}, {74, 2132}, {484, 3465}, {616, 617}, {1138, 5670}, {1157, 3484}, {1263, 5671}, {1276, 5672}, {1277, 5673}, {1337, 5674}, {1338, 5675}, {1495, 3284}, {1650, 9033}, {2088, 21731}, {2133, 5676}, {3065, 5677}, {3081, 3163}, {3440, 5678}, {3441, 5679}, {3466, 5680}, {3479, 5681}, {3480, 5682}, {3481, 5683}, {3482, 5684}, {3483, 5685}, {7059, 8459}, {7060, 8449}, {7164, 8432}, {7165, 8485}, {7325, 8476}, {7326, 8468}, {7327, 8503}, {7328, 8527}, {7329, 8504}, {8172, 8174}, {8173, 8175}, {8431, 8440}, {8433, 8435}, {8434, 8436}, {8437, 8441}, {8438, 8442}, {8439, 8443}, {8444, 8464}, {8445, 8465}, {8446, 8466}, {8447, 8467}, {8448, 8462}, {8450, 8470}, {8451, 8469}, {8452, 8458}, {8453, 8471}, {8454, 8472}, {8455, 8473}, {8456, 8474}, {8457, 8475}, {8460, 8478}, {8461, 8477}, {8463, 8479}, {8480, 8505}, {8481, 8508}, {8482, 8509}, {8483, 8506}, {8484, 8507}, {8486, 8510}, {8487, 8511}, {8488, 8512}, {8489, 8515}, {8490, 8516}, {8491, 8513}, {8492, 8514}, {8493, 8517}, {8494, 8518}, {8495, 8519}, {8496, 8520}, {8497, 8521}, {8498, 8522}, {8499, 8525}, {8500, 8526}, {8501, 8523}, {8502, 8524}, {8528, 8530}, {8529, 8532}, {8531, 8533}, {12113, 12369}

• X₇₄ = X_i-isoconjugate of X_j for these (i,j): {1, 30}, {2, 2173}, {3, 1784}, {6, 14206}, {9, 6357}, {19, 11064}, {31, 3260}, {37, 18653}, {57, 7359}, {63, 1990}, {74, 1099}, {75, 1495}, {76, 9406}, {92, 3284}, {100, 11125}, {110, 36035}, {113, 36053}, {162, 9033}, {163, 41079}, {190, 14399}, {240, 35912}, {265, 35201}, {304, 14581}, {402, 9390}, {561, 9407}, {647, 24001}, {648, 2631}, {649, 42716}, {651, 14400}, {653, 14395}, {656, 4240}, {661, 2407}, {662, 1637}, {759, 6739}, {799, 14398}, {811, 9409}, {823, 1636}, {896, 9214}, {897, 5642}, {1494, 42074}, {1511, 2166}, {1568, 2190}, {1577, 2420}, {1650, 24000}, {1725, 15454}, {1749, 3471}, {1895, 11589}, {1959, 35906}, {2153, 41887}, {2154, 41888}, {2159, 36789}, {2349, 3163}, {3431, 18486}, {5620, 16164}, {5664, 32678}, {6149, 14254}, {8772, 36891}, {9408, 33805}, {11251, 36062}, {14208, 23347}, {16163, 36119}, {24019, 41077}, {32679, 41392}, {34334, 35200}, {36037, 42750}

• X₇₄ = barycentric product X_i*X_j for these {i,j}: {1, 2349}, {3, 16080}, {4, 14919}, {6, 1494}, {7, 15627}, {15, 36308}, {16, 36311}, {30, 40384}, {31, 33805}, {63, 36119}, {69, 8749}, {75, 2159}, {76, 40352}, {92, 35200}, {94, 14385}, {98, 35910}, {99, 2433}, {110, 2394}, {111, 36890}, {112, 34767}, {249, 12079}, {253, 15291}, {264, 18877}, {287, 35908}, {305, 40354}, {323, 5627}, {340, 11079}, {470, 39377}, {471, 39378}, {520, 15459}, {524, 9139}, {525, 1304}, {526, 39290}, {647, 16077}, {648, 14380}, {671, 9717}, {850, 32640}, {1073, 10152}, {1495, 31621}, {1577, 36034}, {2867, 15292}, {2966, 32112}, {2986, 14264}, {2987, 36875}, {3003, 40423}, {3260, 40353}, {3265, 32695}, {3267, 32715}, {3269, 42308}, {3470, 13582}, {3580, 10419}, {4558, 18808}, {7799, 40355}, {9033, 34568}, {14208, 36131}, {15412, 36831}, {20123, 40391}, {22455, 37638}, {30474, 32681}, {40050, 40351}

• X₇₄ = barycentric quotient X_i/X_j for these {i,j}: {1, 14206}, {2, 3260}, {3, 11064}, {6, 30}, {15, 41887}, {16, 41888}, {19, 1784}, {25, 1990}, {30, 36789}, {31, 2173}, {32, 1495}, {50, 1511}, {55, 7359}, {56, 6357}, {58, 18653}, {100, 42716}, {110, 2407}, {111, 9214}, {112, 4240}, {162, 24001}, {184, 3284}, {186, 14920}, {187, 5642}, {216, 1568}, {248, 35912}, {323, 6148}, {512, 1637}, {520, 41077}, {523, 41079}, {526, 5664}, {560, 9406}, {574, 13857}, {647, 9033}, {649, 11125}, {661, 36035}, {663, 14400}, {667, 14399}, {669, 14398}, {810, 2631}, {1304, 648}, {1384, 35266}, {1494, 76}, {1495, 3163}, {1501, 9407}, {1576, 2420}, {1946, 14395}, {1974, 14581}, {1976, 35906}, {1989, 14254}, {1990, 34334}, {2088, 3258}, {2159, 1}, {2173, 1099}, {2245, 6739}, {2349, 75}, {2394, 850}, {2420, 3233}, {2433, 523}, {2987, 36891}, {3003, 113}, {3049, 9409}, {3163, 23097}, {3269, 1650}, {3284, 16163}, {3310, 42750}, {3457, 36299}, {3458, 36298}, {3470, 37779}, {5063, 10564}, {5158, 1531}, {5627, 94}, {8739, 6110}, {8740, 6111}, {8749, 4}, {9139, 671}, {9406, 42074}, {9407, 9408}, {9408, 3081}, {9409, 14401}, {9412, 34582}, {9717, 524}, {10152, 15466}, {10317, 16165}, {10419, 2986}, {11060, 14583}, {11063, 10272}, {11074, 14993}, {11079, 265}, {12079, 338}, {14264, 3580}, {14380, 525}, {14385, 323}, {14560, 41392}, {14579, 3471}, {14581, 16240}, {14642, 11589}, {14910, 15454}, {14919, 69}, {15166, 14499}, {15167, 14500}, {15291, 20}, {15395, 39295}, {15451, 14391}, {15459, 6528}, {15627, 8}, {16077, 6331}, {16080, 264}, {18320, 38610}, {18808, 14618}, {18877, 3}, {19622, 16164}, {21906, 2682}, {22455, 43530}, {32112, 2799}, {32640, 110}, {32681, 1302}, {32695, 107}, {32715, 112}, {33805, 561}, {34397, 39176}, {34417, 18487}, {34568, 16077}, {34767, 3267}, {34952, 14397}, {35200, 63}, {35908, 297}, {35910, 325}, {36034, 662}, {36064, 38340}, {36119, 92}, {36131, 162}, {36308, 300}, {36311, 301}, {36430, 18484}, {36831, 14570}, {36890, 3266}, {36896, 146}, {39201, 1636}, {39290, 35139}, {39377, 40709}, {39378, 40710}, {39380, 10217}, {39381, 10218}, {40135, 13202}, {40351, 1974}, {40352, 6}, {40354, 25}, {40355, 1989}, {40384, 1494}, {40385, 39263}, {40388, 1300}, {40423, 40832}, {41336, 20772}, {42658, 14345}, {42671, 6793}, {43083, 18557}

• {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 146, 113}, {3, 110, 15035}, {3, 399, 1511}, {3, 1511, 15051}, {3, 5609, 15020}, {3, 6241, 1614}, {3, 11456, 11464}, {3, 12041, 15055}, {3, 12174, 9707}, {3, 12308, 32609}, {3, 14094, 15034}, {3, 14264, 14385}, {3, 15035, 15036}, {3, 15041, 12041}, {3, 15054, 14094}, {3, 32138, 11440}, {3, 32139, 11449}, {3, 33533, 41462}, {3, 38497, 38555}, {4, 125, 14644}, {4, 15081, 7687}, {4, 26937, 26917}, {5, 15061, 15059}, {6, 9412, 9408}, {20, 11457, 12289}, {24, 64, 12290}, {35, 19470, 10088}, {36, 7727, 10091}, {110, 15020, 32609}, {110, 15021, 15055}, {110, 15035, 15034}, {110, 15051, 1511}, {110, 15055, 3}, {113, 146, 10706}, {113, 6699, 2}, {125, 7687, 15081}, {125, 12244, 10721}, {125, 13202, 7687}, {184, 11204, 35473}, {185, 3520, 54}, {185, 11430, 15032}, {185, 13293, 15463}, {185, 32607, 13198}, {186, 12112, 1495}, {265, 10264, 9140}, {265, 20126, 10264}, {323, 2071, 10564}, {323, 10564, 43574}, {376, 12317, 12383}, {376, 12383, 16163}, {378, 1986, 15472}, {378, 5890, 15033}, {378, 10605, 5890}, {378, 17835, 7731}, {381, 38790, 1539}, {382, 38724, 10113}, {399, 1511, 110}, {399, 15042, 32609}, {548, 34153, 38723}, {549, 10272, 38794}, {616, 617, 6148}, {974, 1986, 5890}, {974, 2935, 15472}, {974, 11598, 378}, {974, 19457, 5622}, {1181, 8567, 35477}, {1204, 3357, 4}, {1495, 12112, 14157}, {1498, 32534, 26882}, {1511, 12358, 15066}, {1511, 15051, 15035}, {1539, 20304, 381}, {1995, 11472, 16261}, {2914, 15032, 12227}, {2930, 31884, 33851}, {2935, 5621, 19457}, {2935, 10605, 1986}, {2935, 10606, 11598}, {2935, 15151, 5622}, {2935, 19457, 378}, {3003, 40353, 36896}, {3357, 39174, 38937}, {3520, 7722, 15463}, {3520, 15032, 11430}, {3627, 15027, 15044}, {4550, 37470, 2}, {5092, 19140, 15462}, {5609, 32609, 110}, {5621, 17835, 974}, {5622, 10752, 6}, {5627, 14989, 34150}, {5642, 6053, 20125}, {5655, 38794, 10272}, {5890, 7731, 1986}, {5972, 38727, 631}, {6200, 12375, 10819}, {6241, 11464, 11456}, {6241, 11468, 3}, {6396, 12376, 10820}, {6723, 36518, 3090}, {6723, 38791, 36518}, {7488, 10575, 8718}, {7687, 13202, 4}, {7687, 15081, 14644}, {7722, 32607, 54}, {7725, 19059, 10752}, {7726, 19060, 10752}, {7728, 15061, 5}, {7731, 19457, 15033}, {8749, 18877, 15291}, {9140, 10733, 265}, {9408, 9412, 112}, {9717, 14264, 39239}, {9717, 39239, 3470}, {9786, 35502, 9781}, {9904, 11709, 7978}, {10065, 10081, 1}, {10113, 20379, 38724}, {10264, 20127, 10733}, {10605, 10606, 378}, {10605, 11598, 15472}, {10605, 19457, 974}, {10606, 17835, 2935}, {10620, 11454, 43578}, {10620, 12041, 110}, {10620, 13171, 17854}, {10620, 15021, 15035}, {10620, 15041, 3}, {10620, 15055, 14094}, {10620, 38633, 5609}, {10721, 14644, 4}, {10990, 20417, 4}, {11250, 34783, 34148}, {11413, 12163, 11412}, {11430, 15032, 54}, {11454, 15072, 3}, {11456, 11464, 1614}, {11598, 15151, 19457}, {12041, 15041, 15021}, {12041, 15054, 15035}, {12041, 38626, 32609}, {12079, 34150, 5627}, {12121, 38788, 550}, {12219, 12901, 43574}, {12244, 20417, 14644}, {12308, 32609, 5609}, {12308, 38633, 3}, {12358, 12412, 110}, {12358, 12825, 11459}, {12371, 12374, 10767}, {12381, 12382, 7978}, {13198, 15463, 54}, {13293, 17855, 13198}, {13293, 32607, 3520}, {13491, 32210, 3}, {13621, 33541, 32137}, {13630, 14130, 13434}, {14094, 15035, 110}, {14094, 15055, 15036}, {14264, 14385, 3470}, {14264, 15468, 110}, {14385, 39239, 9717}, {14480, 38701, 14934}, {14643, 38728, 140}, {14677, 20126, 10733}, {15021, 15054, 3}, {15021, 15055, 12041}, {15034, 15036, 15035}, {15041, 32609, 38633}, {15041, 34469, 17854}, {15054, 15055, 110}, {15057, 15059, 15061}, {15063, 38727, 5972}, {15072, 15100, 12270}, {15080, 15100, 399}, {16163, 37853, 376}, {17835, 19457, 1986}, {18933, 37643, 15081}, {19059, 19060, 6}, {20126, 20127, 265}, {20427, 26937, 4}, {22462, 33539, 11017}, {23236, 38723, 34153}, {31954, 31955, 5890}, {32616, 32617, 6241}, {34150, 40630, 12079}, {36518, 38729, 6723}, {38626, 38633, 110}, {38632, 38638, 110}, {38641, 38653, 110}, {38650, 38661, 110}, {38729, 38791, 3090}, {38937, 39174, 38933}

Notes

Let T be the triangle formed by reflecting the orthic axis in the sidelines of ABC; then T is perspective to ABC, and the perspector is X₇₄. Let A' be the point of intersection of the orthic axis and line BC, and define B' and C' cyclically. Let O_A be the circumcenter of AB'C', and define Let O_B and O_C cyclically; then the lines AO_A, BO_B, CO_C concur in X₇₄. (Randy Hutson, August 26, 2014)

Let K_aK_bK_c be the anticomplementary triangle. Let L be the line through K_a parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X₇₄. (Randy Hutson, August 26, 2014)

Let A'B'C' be the X₃-Fuhrmann triangle. Let A″ be the reflection of A in line B'C', and define B″ and C″ cyclically. Then A″B″C″ is inversely similar to ABC, with similitude center X₂₆₅, and A″B″C″ is perspective to ABC at X₇₄, which is also the orthocenter of A″B″C″. (Randy Hutson, August 26, 2014)

In Hyacinthos 8129 (10/4/03), Floor van Lamoen noted that if X₇₄ is denoted by J, then each of the points A, B, C,J is J of the other three, in analogy with the well known property of orthocentric systems (that is, each of the points A, B, C,H is the orthocenter of the other three).

Let A'B'C' be the orthocentroidal triangle and A″B″C″ the anti-orthocentroidal triangle. Let A* be the reflection of A″ in B'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₇₄. (Randy Hutson, December 10, 2016)

Let A'B'C' be the anti-orthocentroidal triangle. Let A″ be the reflection of A in line B'C', and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₇₄. (Randy Hutson, January 15, 2019)

Let E_a be the ellipse through X₇₄ having foci B and C, and define E_b and E_c cyclically. The 6 minor vertices of these three ellipses lie on a pair of lines. Figure. (Dan Reznik, December 13, 2021)