X(74) = X(74) ISOGONAL CONJUGATE OF EULER INFINITY POINT

Trilinears

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

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

\(a/[2a4 - (b2 - c2)2 - a2(b2 + c2)] : :\)

Barycentrics

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

\(a^2/(3 SBSC - S2) : :\)

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 X(74). Let A’ be the point of intersection of the orthic axis and line BC, and define B’ and C’ cyclically. Let OA be the circumcenter of AB’C’, and define Let OB and OC cyclically; then the lines AOA, BOB, COC concur in X(74). (Randy Hutson, August 26, 2014)

Let A’B’C’ be the anticomplementary triangle. Let L be the line through 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 X(74). (Randy Hutson, August 26, 2014)

Let A’B’C’ be the X(3)-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 X(265), and A’’B’’C’’ is perspective to ABC at X(74), 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 X(74) 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 X(74). (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 X(74). (Randy Hutson, January 15, 2019)

The tangents at A, B, C to the Neuberg cubic K001 concur in X(74)

Let Ea be the ellipse through X(74) having foci B and C, and define Eb and Ec cyclically. The 6 minor vertices of these three ellipses lie on a pair of lines. Figure. –> Figure. (Dan Reznik, December 13, 2021)

X(74) 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, and these 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}

X(74) = 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}

X(74) = 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}

X(74) = isogonal conjugate of X(30)

X(74) = isotomic conjugate of X(3260)

X(74) = complement of X(146)

X(74) = anticomplement of X(113)

X(74) = X(1)-Ceva conjugate of X(17149)

X(74) = cevapoint of X(i) and X(j) for these (i,j): (15,16), (50,184)

X(74) = crosssum of X(i) and X(j) for these (i,j): (3,399), (616),617)

X(74) = X(i)-cross conjugate of X(j) for these (i,j): (186,54), (526,110)

X(74) = circumcircle-antipode of X(110)

X(74) = polar-circle-inverse of X(133)

X(74) = 2nd-Droz-Farney-circle-inverse of X(17854)

X(74) = Schoutte-circle-inverse of X(2715)

X(74) = 2nd-Brocard-circle-inverse of X(38520)

X(74) = inverse-in-O(15,16) of X(2715), where O(15,16) is the circle having segment X(15)X(16) as diameter

X(74) = trilinear pole of line X(6)X(647)

X(74) = &Psi;(X(6),:ref:X(647) <X(647)>)

X(74) = antipode of X(1199) in Moses-Jerabek conic

X(74) = reflection of X(477) in the Euler line

X(74) = reflection of X(842) in the Brocard axis

X(74) = reflection of X(2687) in the line X(1)X(3)

X(74) = reflection of X(1296) in the line X(3)X(351)

X(74) = {X(3),:ref:X(399) <X(399)>}-harmonic conjugate of X(1511)

X(74) = X(128)-of-excentral-triangle

X(74) = X(137)-of-hexyl-triangle

X(74) = X(1296)-of-circumsymmedial

X(74) = trilinear pole wrt circumorthic triangle of van Aubel line

X(74) = X(1577)-isoconjugate of X(2420)

X(74) = orthocentroidal-to-ABC similarity image of X(4)

X(74) = 4th-Brocard-to-circumsymmedial similarity image of X(4)

X(74) = perspector of ABC and the reflection of the Kosnita triangle in X(3)

X(74) = orthocenter of X(3)X(67)X(879)

X(74) = intersection of tangents at X(3) and X(4) to Napoleon-Feuerbach cubic, K005

X(74) = X(1317)-of-tangential-triangle is ABC is acute

X(74) = 2nd-Parry-to-ABC similarity image of X(110)

X(74) = X(80)-of-Trinh-triangle if ABC is acute

X(74) = Trinh-isogonal conjugate of X(2071)

X(74) = trilinear product of PU(86)

X(74) = perspector of ABC and the (degenerate) side-triangle of the (equilateral) circumcevian triangles of X(15) and X(16)

X(74) = homothetic center of X(15)- and X(16)-Ehrmann triangles; see X(25)

X(74) = perspector of ABC and X(15)-Ehrmann triangle

X(74) = perspector of ABC and X(16)-Ehrmann triangle

X(74) = 3rd-Parry-to-circumsymmedial similarity image of X(23)

X(74) = perspector of ABC and unary cofactor triangle of orthocentroidal triangle

X(74) = endo-homothetic center of X(4)-altimedial and X(4)-anti-altimedial triangles

X(74) = Thomson isogonal conjugate of X(523)

X(74) = Lucas isogonal conjugate of X(523)

X(74) = X(100)-of-circumorthic-triangle if ABC is acute

X(74) = perspector of ABC and 2nd anti-Parry triangle

X(74) = X(110)-of-2nd-anti-Parry-triangle

X(74) = X(9138)-of-1st-anti-Parry-triangle

X(74) = excentral-to-ABC functional image of X(5541)

X(74) = orthic-to-ABC functional image of X(128)

X(74) = trilinear pole wrt, Thomson triangle, of line X(3)X(5646)

X(74) = trilinear pole, wrt Lucas triangle, of line X(4)X(15066)

X(74) = BSS(a&srarr;a^2) of X(1156)

X(74) = areal center of pedal triangles of PU(4)

X(74) = areal center of pedal triangles of PU(5)

X(74) = areal center of pedal triangles of PU(11)

X(74) = antipode of X(32111) in Walsmith rectangular hyperbola

X(74) = orthocenter of X(6)X(110)X(3569)

X(74) = X(5541)-of-orthic-triangle if ABC is acute

X(74) = perspector of circumconic centered at X(36896)

X(74) = trilinear pole of line X(6)X(647)

X(74) = crossdifference of every pair of points on line {1636, 1637}

X(74) = psi-transform of X(14685)

X(74) = Collings transform of X(i) for these i: {125, 2088, 3134, 39174, 39987}

X(74) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {10419, 8}, {36053, 146}, {40388, 5905}, {40423, 6327}

X(74) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 36896}, {34178, 10}

X(74) = 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}

X(74) = 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}

X(74) = 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}

X(74) = 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}

X(74) = 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}

X(74) = 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}

X(74) = 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}

X(74) = 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}