| PART 1: | Introduction and Centers X(1) - X(1000) | PART 2: | Centers X(1001) - X(3000) | PART 3: | Centers X(3001) - X(5000) |
| PART 4: | Centers X(5001) - X(7000) | PART 5: | Centers X(7001) - X(10000) | PART 6: | Centers X(10001) - X(12000) |
| PART 7: | Centers X(12001) - X(14000) | PART 8: | Centers X(14001) - X(16000) | PART 9: | Centers X(16001) - X(18000) |
| PART 10: | Centers X(18001) - X(20000) | PART 11: | Centers X(20001) - X(22000) | PART 12: | Centers X(22001) - X(24000) |
| PART 13: | Centers X(24001) - X(26000) | PART 14: | Centers X(26001) - X(28000) | PART 15: | Centers X(28001) - X(30000) |
| PART 16: | Centers X(30001) - X(32000) | PART 17: | Centers X(32001) - X(34000) | PART 18: | Centers X(34001) - X(36000) |
| PART 19: | Centers X(36001) - X(38000) | PART 20: | Centers X(38001) - X(40000) | PART 21: | Centers X(40001) - X(42000) |
| PART 22: | Centers X(42001) - X(44000) | PART 23: | Centers X(44001) - X(46000) | PART 24: | Centers X(46001) - X(48000) |
| PART 25: | Centers X(48001) - X(50000) | PART 26: | Centers X(50001) - X(52000) | PART 27: | Centers X(52001) - X(54000) |
| PART 28: | Centers X(54001) - X(56000) | PART 29: | Centers X(56001) - X(58000) | PART 30: | Centers X(58001) - X(60000) |
| PART 31: | Centers X(60001) - X(62000) | PART 32: | Centers X(62001) - X(64000) | PART 33: | Centers X(64001) - X(66000) |
| PART 34: | Centers X(66001) - X(68000) | PART 35: | Centers X(68001) - X(70000) | PART 36: | Centers X(70001) - X(72000) |
X(3001) lies on these lines: {3, 6}, {67, 36214}, {86, 22150}, {141, 20819}, {157, 20806}, {160, 12220}, {206, 33801}, {231, 6036}, {237, 9019}, {311, 34845}, {316, 36187}, {323, 9142}, {325, 523}, {338, 21531}, {524, 20975}, {868, 34827}, {1503, 44716}, {1576, 22151}, {1594, 44145}, {1634, 2393}, {1990, 2967}, {2450, 44388}, {3964, 34777}, {3981, 34811}, {4383, 21493}, {4558, 11416}, {5094, 7778}, {6033, 7574}, {6593, 9407}, {6660, 18374}, {7495, 7792}, {7774, 16063}, {7819, 42442}, {8705, 9155}, {9766, 31152}, {9969, 35222}, {9971, 11328}, {13409, 26905}, {13857, 16186}, {14570, 14957}, {15270, 28710}, {16776, 37338}, {17710, 20775}, {18181, 37796}, {19151, 43722}, {21284, 44089}, {23333, 39113}, {34834, 38998}, {35296, 39231}, {37444, 41761}, {38987, 44380}, {40073, 40361}
X(3001) = midpoint of X(14570) and X(14957)
X(3001) = reflection of X(i) in X(j) for these {i,j}: {237, 34990}, {338, 21531}, {1634, 36212}, {5201, 3003}
X(3001) = isotomic conjugate of X(2367)
X(3001) = isotomic conjugate of the isogonal conjugate of X(2387)
X(3001) = X(31)-isoconjugate of X(2367)
X(3001) = crosssum of X(6) and X(21177)
X(3001) = crossdifference of every pair of points on line {32, 523}
X(3001) = barycentric product X(76)*X(2387)
X(3001) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2367}, {2387, 6}
X(3001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {216, 3313, 8266}, {237, 22087, 34990}, {325, 45279, 45921}, {325, 45921, 45918}, {570, 11574, 41328}, {3014, 45918, 45921}, {9737, 11511, 5063}, {20819, 23635, 141}, {22151, 37183, 1576}, {45279, 45921, 3014}
X(3002) lies on these lines: 3,6 36,1951 241,514 978,1047 1758,1945
X(3002) = crosssum of X(6) and X(851)
X(3002) = complement of isotomic conjugate of X(37142)lies on the cubics K277, K489, K1170, K1171, and these lines: on lines {2, 3260}, {3, 6}, {30, 6128}, {31, 8606}, {37, 7359}, {44, 13006}, {53, 235}, {54, 43753}, {74, 32681}, {111, 9060}, {112, 32710}, {114, 45921}, {115, 11799}, {157, 1974}, {160, 6467}, {186, 8749}, {206, 40947}, {230, 231}, {237, 2393}, {248, 1177}, {393, 847}, {395, 40696}, {396, 40695}, {419, 1632}, {524, 34990}, {607, 2178}, {608, 2164}, {615, 8963}, {648, 44375}, {672, 22059}, {895, 35298}, {1084, 23967}, {1100, 2649}, {1194, 5306}, {1196, 34811}, {1213, 16699}, {1249, 3147}, {1495, 9142}, {1503, 44437}, {1513, 45279}, {1545, 5321}, {1546, 5318}, {1576, 44102}, {1634, 8681}, {1885, 6748}, {1942, 1987}, {1989, 14993}, {2079, 37933}, {2174, 2197}, {2252, 8776}, {2269, 22058}, {2351, 44077}, {2386, 21177}, {2790, 44227}, {3133, 12235}, {3148, 19136}, {3163, 18334}, {3164, 41760}, {3199, 13881}, {3580, 18609}, {3815, 30739}, {4383, 21494}, {4558, 35296}, {5181, 11672}, {5191, 9407}, {5305, 16618}, {5412, 44193}, {5413, 44192}, {5486, 43718}, {5702, 46453}, {6509, 26958}, {6663, 13383}, {7113, 7117}, {7202, 43034}, {7297, 14936}, {7493, 7735}, {7669, 18374}, {7736, 15302}, {7745, 34664}, {8623, 18371}, {8962, 32787}, {9027, 9155}, {9145, 32127}, {9220, 39565}, {9300, 43957}, {9475, 38987}, {9723, 40318}, {9822, 35222}, {9969, 23635}, {10313, 37929}, {11188, 37465}, {11328, 29959}, {13567, 46832}, {14060, 37183}, {14569, 15508}, {14575, 41593}, {14581, 35372}, {14984, 44221}, {15073, 37114}, {15355, 39576}, {16237, 44138}, {16238, 45847}, {17907, 41770}, {18122, 34827}, {18362, 36430}, {18877, 21663}, {19118, 44200}, {20775, 32366}, {21639, 39231}, {23357, 44174}, {23980, 35090}, {34104, 39170}, {34394, 36296}, {34395, 36297}, {35370, 44895}, {35486, 39575}, {36982, 41373}, {37778, 41678}, {39176, 44281}, {40350, 44533}, {40423, 46230}, {41758, 45172}
X(3003) = midpoint of X(i) and X(j) for these {i,j}: {237, 20975}, {3001, 5201}
X(3003) = reflection of X(36212) in X(34990)
X(3003) = isogonal conjugate of X(2986)
X(3003) = isotomic conjugate of X(40832)
X(3003) = complement of X(3260)
X(3003) = Brocard-circle-inverse of X(5063)
X(3003) = Moses-circle-inverse of X(15544)
X(3003) = Moses-radical-circle-inverse of X(16319)
X(3003) = Schoutte-circle-inverse of X(10564)
X(3003) = Moses-Parry-circle-inverse of X(3018)
X(3003) = complement of the isogonal conjugate of X(40352)
X(3003) = complement of the isotomic conjugate of X(74)
X(3003) = isogonal conjugate of the isotomic conjugate of X(3580)
X(3003) = isotomic conjugate of the polar conjugate of X(44084)
X(3003) = isogonal conjugate of the polar conjugate of X(403)
X(3003) = polar conjugate of the isotomic conjugate of X(13754)
X(3003) = perspector of circumconic centered at X(113)
X(3003) = center of circumconic that is locus of trilinear poles of lines passing through X(113)
X(3003) = Brocard-axis intercept of line through X(371)-Ceva conjugate of X(372) and X(372)-Ceva conjugate of X(371)
X(3003) = center of bicevian conic of X(15) and X(16)
X(3003) = crossdifference of every pair of points on line X(3)X(523)
X(3003) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 113}, {74, 2887}, {560, 3163}, {810, 16177}, {1304, 21259}, {1494, 21235}, {2159, 141}, {2349, 626}, {2433, 21253}, {8749, 20305}, {9139, 21256}, {15627, 21244}, {18877, 18589}, {23995, 31945}, {32640, 4369}, {32715, 8062}, {33805, 40379}, {35200, 1368}, {36034, 512}, {36119, 21243}, {36131, 30476}, {40351, 16583}, {40352, 10}, {40354, 226}, {44769, 42327}
X(3003) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 113}, {186, 1495}, {403, 44084}, {476, 512}, {687, 924}, {1299, 25}, {2407, 526}, {3580, 13754}, {6344, 51}, {8749, 6}, {15329, 21731}, {18609, 1725}, {18879, 110}, {32640, 647}, {38534, 184}, {46106, 6000}
X(3003) = X(21731)-cross conjugate of X(15329)
X(3003) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2986}, {2, 36053}, {31, 40832}, {63, 1300}, {75, 14910}, {92, 5504}, {162, 15421}, {525, 36114}, {656, 687}, {661, 18878}, {662, 15328}, {1109, 18879}, {1577, 10420}, {2173, 40423}, {2349, 15454}, {6149, 40427}, {10419, 14206}, {14208, 32708}, {15470, 32680}, {24006, 43755}, {36102, 39986}, {36151, 39988}
X(3003) = crosspoint of X(i) and X(j) for these (i,j): {2, 74}, {6, 1989}, {54, 43766}, {110, 18879}, {249, 1304}, {403, 3580}
X(3003) = crosssum of X(i) and X(j) for these (i,j): {2, 323}, {4, 15262}, {6, 30}, {115, 9033}, {1650, 32320}, {3936, 32933}, {5504, 14910}
X(3003) = trilinear pole of line {686, 21731}
X(3003) = crossdifference of every pair of points on line {3, 523}
X(3003) = barycentric product X(i)*X(j) for these {i,j}: {1, 1725}, {3, 403}, {4, 13754}, {6, 3580}, {30, 14264}, {37, 18609}, {67, 12824}, {69, 44084}, {74, 113}, {92, 2315}, {99, 21731}, {112, 6334}, {131, 1299}, {155, 16172}, {184, 44138}, {186, 39170}, {265, 1986}, {399, 18781}, {523, 15329}, {526, 41512}, {647, 16237}, {648, 686}, {895, 12828}, {1177, 12827}, {1300, 34333}, {1989, 34834}, {5663, 39985}, {10419, 34104}, {10693, 12826}, {11557, 33565}, {11744, 12825}, {18879, 39021}, {38534, 46085}
X(3003) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40832}, {6, 2986}, {25, 1300}, {31, 36053}, {32, 14910}, {74, 40423}, {110, 18878}, {112, 687}, {113, 3260}, {184, 5504}, {403, 264}, {512, 15328}, {647, 15421}, {686, 525}, {1495, 15454}, {1576, 10420}, {1725, 75}, {1986, 340}, {1989, 40427}, {2315, 63}, {3580, 76}, {5663, 39988}, {6334, 3267}, {12824, 316}, {12827, 1236}, {12828, 44146}, {13754, 69}, {14264, 1494}, {14270, 15470}, {14583, 39375}, {15329, 99}, {16172, 46746}, {16237, 6331}, {18609, 274}, {18781, 40705}, {21731, 523}, {23357, 18879}, {32661, 43755}, {32676, 36114}, {34397, 38936}, {34834, 7799}, {39170, 328}, {40352, 10419}, {40354, 40388}, {41512, 35139}, {44084, 4}, {44138, 18022}
X(3003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 5063}, {6, 50, 3284}, {6, 216, 570}, {6, 566, 39}, {6, 570, 5421}, {6, 1609, 571}, {6, 8553, 577}, {6, 8573, 13345}, {6, 11063, 50}, {6, 13338, 5007}, {6, 18573, 566}, {15, 16, 10564}, {32, 5158, 6}, {39, 216, 566}, {39, 566, 570}, {50, 11063, 187}, {53, 9722, 1879}, {74, 36896, 40353}, {187, 3284, 50}, {187, 40135, 3284}, {216, 800, 6}, {230, 1990, 16310}, {230, 2493, 3291}, {230, 16303, 3018}, {230, 16306, 6103}, {230, 16308, 2493}, {230, 16310, 231}, {231, 3018, 16310}, {232, 3291, 2493}, {232, 6103, 14580}, {371, 372, 13352}, {566, 18573, 216}, {577, 46432, 6}, {1990, 11062, 232}, {1990, 16310, 3018}, {1990, 16328, 11062}, {2028, 2029, 15544}, {2493, 16308, 232}, {3284, 40135, 6}, {5063, 40320, 571}, {5201, 46127, 3001}, {7669, 18374, 42671}, {8105, 8106, 3018}, {8607, 8608, 8609}, {11081, 11086, 1495}, {13338, 41335, 6}, {15166, 15167, 6}, {16303, 16310, 1990}, {23635, 40981, 9969}, {34569, 46216, 50}, {35296, 37784, 4558}, {40135, 46216, 18365}, {41196, 41197, 14961}, {41678, 44893, 37778}, {46203, 46222, 50}
X(3004) lies on these lines: 241,514 325,523 522,2526 661,918
X(3004) = isogonal conjugate of X(32736)
X(3004) = isotomic conjugate of X(8707)
X(3004) = crossdifference of every pair of points on line X(32)X(55)
X(3004) = X(i)-Ceva conjugate of X(j) for these (i,j): (274,1086), (1014,1565), (1441,1111)
X(3004) = crosssum of X(6) and X(2483)
Let L denote the line at infinity. Then (Lemoine axis) = (isogonal conjugate of isotomic conjugate of L) and (de Longchamps line) = (isotomic conjugate of isogonal conjugate of L), and X(3005) = (Lemoine axis)∩(de Longchamps line). (Randy Hutson, February 10, 2016)
X(3005) lies on the bicevian conic of X(2) and X(512) and these lines: 2,881 187,237 325,523 661,756 689,783 826,2474 878,2623
X(3005) = reflection of X(i) in X(j) for these (i,j): (669,647), (2528,2525)
X(3005) = isogonal conjugate of X(4577)
X(3005) = isotomic conjugate of X(689)
X(3005) = X(i)-Ceva conjugate of X(j) for these (i,j): (512,688), (523,826), (827,6), (1634,39)
X(3005) = X(2531)-cross conjugate of X(688)
X(3005) = crosspoint of X(i) and X(j) for these (i,j): (6,827), (39,1634), (512,523)
X(3005) = crosssum of X(i) and X(j) for these (i,j): (2,826), (99,110), (512,1194)
X(3005) = complementary conjugate of X(7668)
X(3005) = complement of anticomplementary conjugate of X(39346)
X(3005) = crossdifference of every pair of points on line X(2)X(32)
X(3005) = perspector of circumconic centered at X(3124)
X(3005) = center of circumconic that is locus of trilinear poles of lines passing through X(3124)
X(3005) = X(2)-Ceva conjugate of X(3124)
X(3005) = radical center of {circumcircle, Brocard circle, symmedial circle}
X(3006) lies on these lines: 1,2 38,2887 321,2886 325,523 752,896
X(3006) = isotomic conjugate of X(675)
X(3006) = anticomplement of X(3011)
X(3007) lies on these lines: 1,7 5,1441 325,523 953,1305
X(3007) = isotomic conjugate of X(2370)
X(3007) = crosspoint of X(264) and X(903)
X(3007) = crosssum of X(184) and X(902)
X(3007) = anticomplement of X(8756)
X(3007) = de-Longchamps-circle-inverse of X(38941)
X(3008) lies on these lines: 1,2 44,527 57,169 101,1429 190,1266 218,226 238,516 241,514 379,1724 443,1453 536,2325 1445,1723
X(3008) = midpoint of X(i) and X(j) for these (i,j): (44,1086), (190,1266), (238,1738)
X(3008) = isotomic conjugate of X(36807)
X(3008) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,3021), (666,514)
X(3008) = cevapoint of X(1279) and X(2348)
X(3008) = X(3021)-cross conugate of X(7)
X(3008) = crosssum of X(6) and X(672)
X(3008) = complement of X(3912)
X(3008) = inverse-in-Steiner-circumellipse of X(145)
X(3008) = inverse-in-Steiner-inellipse of X(1)
X(3009) lies on these lines: 1,2 31,172 37,1964 101,2210 187,237 192,1740 213,2308 238,660 292,672 536,2234 694,2054 741,1931 872,1100 1914,2109 2111,2116 2113,2114
X(3009) = isogonal conjugate of X(3226)
X(3009) = complement of X(20352)
X(3009) = anticomplement of X(20340)
X(3009) = X(i)-Ceva conjugate of X(j) for these (i,j): (238,672), (660,649), (727,6), (1911,42)
X(3009) = crosspoint of X(i) and X(j) for these (i,j): (1,292), (6,727), (1463,1575)
X(3009) = crosssum of X(i) and X(j) for these (i,j): (1,239), (2,726)
X(3009) = crossdifference of every pair of points on line X(2)X(649)
X(3010) lies on these lines: 1,7 31,1951 187,237
X(3011) lies on these lines: 1,2 3,1072 11,1279 12,1104 25,225 31,226 100,1738 105,2006 111,2690 142,750 230,231 238,908 459,1068 851,2223 1447,1758 1612,1838 1836,3049
X(3011) = complement of X(3006)
X(3011) = crosspoint of X(2) and X(675)
X(3011) = crosssum of X(6) and X(674)
X(3011) = polar conjugate of isotomic conjugate of X(9028)
X(3011) = X(63)-isoconjugate of X(9085)
X(3011) = PU(4)-harmonic conjugate of X(7649)
X(3012) lies on these lines: 1,7 230,231
X(3013) lies on these lines: 6,13 44,513
X(3013) = X(477)-Ceva conjugate of X(55)
X(3013) = crosssum of X(1) and X(3013)
X(3014) lies on these lines: 6,13 325,523 868,2854
X(3014) = isotomic conjugate of X(35568)
X(3015) lies on these lines: 6,13 241,514
X(3016) lies on these lines: 6,13 187,237 316,323 543,2421 1843,2971 2088,2493
X(3016) = reflection of X(2088) in X(2493)
X(3016) = crossdifference of every pair of points on line X(2)X(526)
X(3017) lies on these lines: 1,2 6,13 12,1126 30,58
X(3017) = X(58)-of-orthocentroidal triangleX(3018) lies on these lines: 6,13 111,1302 112,393 230,231 1249,2165 1609,2079
X(3018) = complement of X(35520)
X(3018) = crosspoint of X(2) and X(477)
X(3018) = polar conjugate of isotomic conjugate of X(17702)
X(3018) = X(63)-isoconjugate of X(32710)
X(3019) lies on these lines: 1,7 6,13
Centers X(3020)-X(3026) are points on the incircle, as noted by Milorad Stevanovic (Hyacinthos, Dec. 7-8, 2004)
X(3020) lies on the incircle and this line: 1086,3023
X(3021) lies on the incircle and these lines: 1,1358 11,55 56,1292 354,1357 1317,2826 1361,2814 1362,2820 1364,2835 1682,3034 2775,3028 2788,3027 2795,3023 2809,3022 2836,3024
X(3021) = reflection of X(i) in X(j) for these (i,j): (8,3039), (1358,1)
X(3021) = X(7)-Ceva conjugate of X(3008)
X(3021) = crosspoint of X(7) and X(3008)
X(3021) = X(1297)-of-intouch-triangle
X(3021) = X(105)-of-Mandart-incircle-triangle
X(3021) = homothetic center of intangents triangle and reflection of extangents triangle in X(105)
X(3022) lies on the incircle and these lines: 1,1362 11,116 12,118 33,181 56,103 65,1360 101,2291 150,497 152,388 926,2170 928,1364 950,2784 1282,1697 1317,2801 1359,2823 1361,2099 1397,2192 1682,3033 2772,3028 2774,3024 2786,3023 2809,3021
X(3022) = reflection of X(i) in X(j) for these (i,j): (8,3041), (1362,1)
X(3022) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,650), (55,657)
X(3022) = crosspoint of X(i) and X(j) for these (i,j): (7,650), (55,657), (607,663)
X(3022) = crosssum of X(i) and X(j) for these (i,j): (7,658), (55,651), (100,144), (279,934), (348,664)
X(3022) = X(99)-of-intouch-triangle
X(3022) = X(101)-of-Mandart-incircle-triangle
X(3022) = trilinear product of vertices of intangents triangle
X(3022) = trilinear product of vertices of Mandart-incircle triangle
X(3022) = homothetic center of intangents triangle and reflection of extangents triangle in X(101)
X(3022) = intersection, other than vertices of intouch triangle, of incircle and Privalov conic
X(3023) lies on the incircle and these lines: 1,2782 11,115 12,114 56,98 65,1355 147,388 148,497 330,1916 690,3024 1317,2783 1359,2791 1361,2792 1500,1569 1682,3029 2023,2275 2786,3022 2795,3021
X(3023) = reflection of X(3027) in X(1)
X(3023) = X(99)-of-Mandart-incircle triangle
X(3023) = homothetic center of intangents triangle and reflection of extangents triangle in X(99)
The circumcircle of the incentral triangle intersects the incircle in 2 points, X(11) and X(3024). (Randy Hutson, August 29, 2018)
X(3024) lies on the incircle and these lines: 1,3028 11,125 12,113 33,1112 35,1511 56,74 65,1354 146,388 690,3023 1317,2771 1359,2778 1361,2779 1469,2781 1682,3031 2774,3022 2836,3021
X(3024) = reflection of X(3028) in X(1)
X(3024) = X(1029)-Ceva conjugate of X(650)
X(3024) = incentral isogonal conjugate of X(523)
X(3024) = X(930)-of-intouch-triangle
X(3024) = X(110)-of-Mandart-incircle triangle
X(3024) = homothetic center of intangents triangle and reflection of extangents triangle in X(110)
X(3025) lies on the incircle and these lines: 11,513 36,1464 56,953 517,1317 840,901 1319,1361
X(3025) = X(476)-of-intouch-triangle
X(3024) = reflection of X(11) in line X(1)X(3)
X(3026) lies on the incircle and these lines: 150,388 274,1682 1111,1357 1365,1565
Antipodal Pairs on Circles
In response to Stevanovic's findings (X(3000) to X(3006)), Peter Moses noted (Hyacinthos, 12/9/2004) a method for mapping a pair of antipodal points on one circle to an antipodal pair on another circle. The method depends on centers of similitude:
Suppose O1 and O2 are circles, that P is on O1 and that P' is the antipode of P on O1. Let U be the internal center of similitude (insimilicenter) of O1 and O2, and V the exsimilicenter. Define Q = PU ∩P' V and Q' = PV ∩P' U. Then on O2, point Q' is the antipode of Q. Moreover, the lines PP' and QQ' are parallel.
The method follows from introductory material on centers of similitude (e.g., H. S. M. Coxeter, Introduction to Geometry, 2nd edition, page 70, which cites Nathan Altshiller-Court, College Geometry, 2nd edition, page 184). If you have The Geometer's Sketchpad, you can view Antipodal Pairs.
In the following examples, suppose P = p : q : r (trilinears).
Example 1. O1 = circumcircle and O2 = incircle. In this case, insimilicenter(O1, O2) = X(55), and exsimilicenter(O1, O2) = X(56). The antipodal points on O2 given by the Moses mapping are
Q = ((b - c)2p + a(bq + cr))(b + c - a) : :
Q' = ((b + c)2p + a(bq + cr))/(b + c - a) : : .
Example 2. O1 = circumcircle and O2 = Apollonius circle, with center X(970). Here, insimilicenter(O1, O2) = X(573), and exsimilicenter(O1, O2) = X(386). A point on O2 is
Q' = (a + b)(a + c)(b + c)2p - a(bc + ca + ab + b2 + c2)(bq + cr) : : .
Example 3. O1 = circumcircle and O2 = Spieker circle. In this case, insimilicenter(O1, O2) = X(958), and exsimilicenter(O1, O2) = X(1376).
Q = (a - b - c)[((b + c)(b - c)2 + a(b2 + c2))p + (ab + ac + 2bc + a2)(bq + cr)] : :
Q' = (b3 + c3 - ab2 - ac2 - bc2 - b2c)p + (ab + ac - 2bc - a2)(bq + cr) : :
Example 4. O1 = incircle and O2 = Spieker circle. In this case, insimilicenter(O1, O2) = X(2), and exsimilicenter(O1, O2) = X(8).
Q = (bp + cq)/a : (cr + ap)/b : (ap + bq)/c = complement of P
Q' = 2abc(b + c)p - bc(a - b - c)(bq + cr) : :
Example 5. O1 = circumcircle and O2 = sine-triple-angle circle. In this case, insimilicenter(O1, O2) = X(1147), and exsimilicenter(O1, O2) = X(184). A point on O2 is
Q = a[a(a2 - b2)(a2 - c2)p - a2(b2 -a2 + c2)(bq + cr)] : :
X(3027) lies on the incircle and these lines: 1,2782 11,114 12,115 56,99 65,1356 147,497 148,388 192,1916 226,1365 350,1281 553,1357 690,3028 950,2784 1359,2798 2023,2276 2788,3021
X(3027) = reflection of X(3023) in X(1)
X(3027) = crosssum of X(55) and X(2311)
X(3027) = X(98)-of-Mandart-incircle triangle
X(3027) = homothetic center of intangents triangle and reflection of extangents triangle in X(98)
X(3028) lies on the incircle and these lines: 1,3024 11,113 12,125 34,1112 36,1464 56,110 57,2948 65,1365 146,497 690,3027 1359,2850 1367,1439 1469,2854 1870,1986 2646,2779 2772,3022 2775,3021
X(3028) = reflection of X(3024) in X(1)
X(3028) = crosssum of X(55) and X(2341)
X(3028) = X(1141)-of-intouch-triangle
X(3028) = X(74)-of-Mandart-incircle-triangle
X(3028) = homothetic center of intangents triangle and reflection of extangents triangle in X(74)
X(3029) lies on the Apollonius circle and these lines: 10,115 98,573 99,386 114,2051 181,3027 690,2782 970,2782 1682,3023 2786,3033 2787,3032 2795,3034 2796,3030
X(3029) = complement of X(38481)
Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X(3030) = X(107) of A'B'C'. (Randy Hutson, August 29, 2018))
X(3030) lies on the Apollonius circle and these lines: 10,11 43,57 106,386 373,2177 1293,2291 2796,3029 2832,3034 2842,3031
X(3030) = reflection of X(1357) in X(1054)
X(3030) = complement of X(38478)
X(3030) = inverse-in-excircles-radical-circle of X(11)
X(3030) = X(110)-of-Apollonius-triangle
X(3031) lies on the Apollonius circle and these lines: 10,125 43,2948 74,573 110,386 113,2051 690,3029 1017,2092 2774,3033 2836,3034 2842,3030
X(3031) = complement of X(38482)
Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X(3032) = X(74) of A'B'C'. (Randy Hutson, August 29, 2018)
X(3032) lies on the Apollonius circle and these lines: 10,11 100,386 104,573 119,2051 214,1015 528,3034 952,970 1695,1768 2787,3029
X(3032) = complement of X(35636)
X(3032) = inverse-in-excircles-radical-circle of X(121)
Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X(3032) = X(98) of A'B'C'. (Randy Hutson, August 29, 2018)
X(3033) lies on the Apollonius circle and these lines: 10,116 43,57 101,386 103,573 118,2051 440,2968 1490,2808 2774,3031 2786,3029
X(3033) = complement of X(38479)Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle. X(3030) = X(112) of A'B'C'. (Randy Hutson, August 29, 2018)
X(3034) lies on the Apollonius circle and these lines: 10,116 105,386 528,3032 2795,3029 2832,3030 2836,3031
X(3034) = inverse-in-excircles-radical-circle of X(116)Let P = X(100) and G=X(2); let GA be the centroid of the triangle BCP. Define GB and GC cyclically. Then G, GA, GB, GC are the vertices of a quadrilateral that is homothetic to the cyclic quadrilateral having vertices A, B, C, P. The center of homothety is X(3035). Moreover, X(3035) is the centroid of both quadrilaterals and is the Feuerbach point of the medial triangle. (Randy Hutson, 9/23/2011)
X(3035) lies on the Spieker circle and these lines: 1,1145 2,11 3,119 8,1317 9,1768 10,140 12,404 36,529 40,1537 80,1698 104,631 153,2551 230,1575 405,2932 468,1861 474,498 516,1638 549,993 620,2787 632,1484 676,2804 899,1818 908,1155 960,2800 1125,1387 1532,2077 2801,3041 2826,3039 2827,3038
X(3035) = midpoint of X(i) and X(j) for these (i,j): (1,1145), (3,119), (8,1317), (10,214), (11,100), (40,1537), (908,1155), (1532,2077)
X(3035) = reflection of X(i) in X(j) for these (i,j): (1387,1125), (3036,10)
X(3035) = isogonal conjugate of X(18771)
X(3035) = isotomic conjugate of isogonal conjugate of X(20958)
X(3035) = complement of X(11)
X(3035) = X(885)-Ceva conjugate of X(518)
X(3035) = centroid of ABCX(100)
X(3035) = polar conjugate of isogonal conjugate of X(22055)
X(3035) = crossdifference of every pair of points on line X(665)X(9259)
X(3035) = Kosnita(X(100),X(2)) point
X(3036) lies on the Spieker circle and these lines: 2,1217 8,11 9,80 10,140 104,1376 153,2550 355,1158 519,1387 960,2802 1706,1768
X(3036) = midpoint of X(i) and X(j) for these (i,j): (8,11),
(80,1145)
X(3036) = reflection of X(3035) in X(10)
X(3036) = complement of X(1317)
X(3037) lies on the Spieker circle and these lines: 2, 1356 741,958
X(3037) = complement of X(1356)
X(3038) lies on the Spieker circle and these lines: 2,1357 9,1054 106,958 121,124 960,2802 1293,1376 2810,3041 2815,3042 2827,3035 2832,3039
X(3038) = complement of X(1357)
X(3039) lies on the Spieker circle and these lines: 2,1358 8,3021 9,80 960,2809 1292,1376 2814,3042 2826,3035 2832,3038 2835,3040
X(3039) = midpoint of X(8) and X(3021)
X(3039) = complement of X(1358)
X(3039) = crosssum of X(6) and X(1357)
X(3040) lies on the Spieker circle and these lines: 2,1361 8,1364 10,2818 102,1376 109,958 117,2886 124,1329 151,2550 928,3041 956,1795 960,2800 2835,3039
X(3040) = midpoint of X(8) and X(1364)
X(3040) = reflection of X(3042) in X(10)
X(3040) = complement of X(1361)
X(3041) lies on the Spieker circle and these lines: 2,1362 8,3022 9,1282 10,2808 101,958 103,1376 118,124 150,2551 152,2550 928,3040 960,2809 2801,3035
X(3041) = midpoint of X(8) and X(3022)
X(3041) = complement of X(1362)
X(3042) lies on the Spieker circle and these lines: 2,1364 8,1361 10,2818 72,1845 102,958 109,1376 117,1329 118,124 151,2551 474,1795 960,2817 2814,3039 2815,3039
X(3042) = midpoint of X(i) and X(j) for these (i,j): (8,1361),
(72,1845)
X(3042) = reflection of X(3040) in X(10)
X(3042) = complement of X(1364)
X(3043) lies on the sine-triple-angle circle and these lines: 4,110 49,3047 54,125 74,184 186,323 215,3028 378,399 542,3044 1614,2777 1993,2931 2477,3024 2771,3045 2772,3046 2780,3048
X(3043) = reflection of X(3047) in X(49)
X(3044) lies on the sine-triple-angle circle and these lines: 49,2782 54,114 98,1147 99,184 215,3023 542,3043 543,3048 690,3047 2477,3027 2786,3046
X(3045) lies on the sine-triple-angle circle and these lines: 11,110 49,952 54,119 100,184 2771,3043 2805,3048
X(3046) lies on the sine-triple-angle circle and these lines: 49,2808 54,118 101,184 103,1147 215,3022 2772,3043 2774,3047 2786,3044 2813,3048
X(3047) lies on the sine-triple-angle circle and these lines: 2,98 49,3043 54,113 74,1147 156,265 193,1177 206,895 215,3024 690,3044 1112,1994 2477,3028 2774,3046 2854,3048
X(3047) = reflection of X(3043) in X(49)
X(3048) lies on the sine-triple-angle circle and these lines: 110,126 111,184 1147,1296 2780,3043 2805,3045 2813,3046 2854,3047
Let L be the isotomic conjugate of the polar conjugate of the Lemoine axis (i.e., line X(520)X(647)). Let M be the polar conjugate of the isotomic conjugate of the Lemoine axis (i.e., line X(512)X(1692)). Then X(3049) = L∩M. (Randy Hutson, June 7, 2019)
X(3049) lies on these lines: 6,523 112,2713 250,2715 421,2501 512,1692 520,647 669,688 924,2485 1510,2492
X(3049) = midpoint of X(i) and X(j) for these (i,j): (6,3050), (2451,3288)
X(3049) = reflection of X(2451) in X(6)
X(3049) = isogonal conjugate of X(6331)
X(3049) = X(i)-Ceva conjugate of X(j) for these (i,j): (2623,512), (2715,237)
X(3049) = cevapoint of X(647) and X(2524)
X(3049) = crosspoint of X(i) and X(j) for these (i,j): (6,1576), (512,647)
X(3049) = crosssum of X(i) and X(j) for these (i,j): (2,950), (4,2489), (99,648), (427,2501), (647,1899)
X(3049) = crossdifference of every pair of points on line X(4)X(69)
X(3049) = isogonal conjugate of isotomic conjugate of X(647)
X(3049) = polar conjugate of isotomic conjugate of X(39201)
X(3049) = perspector of hyperbola {{A,B,C,X(3),X(25)}}
X(3049) = barycentric product of PU(109)
X(3049) = barycentric product of Jerabek hyperbola intercepts of Lemoine axis
X(3050) lies on these lines: 6,523 50,647 512,1691 520,2506 669,2513
X(3050) = midpoint of X(3049) and X(3288)
X(3050) = reflection of X(6) in X(3049)
X(3050) = X(1576)-Ceva conjugate of X(6)
X(3050) = crosspoint of X(i) and X(j) for these (i,j): (83,110), (112,275)
X(3050) = crosssum of X(i) and X(j) for these (i,j): (39,523), (216,525)
X(3050) = crossdifference of every pair of points on line X(5)X(141) (the complement of the Brocard axis)
Let L be the isogonal conjugate of the isotomic conjugate of the Brocard axis (i.e., line X(32)X(184)). Let M be the isotomic conjugate of the isogonal conjugate of the Brocard axis (i.e., line X(2)X(69)). Then X(3051) = L∩M. (Randy Hutson, March 21, 2019)
X(3051) lies on these lines: 2,6 25,263 31,1911 32,184 42,1197 51,1196 99,703 110,251 213,2308 321,2235 511,1194 669,881 1078,1207 1180,2979 1627,1691
X(3051) = isogonal conjugate of X(308)
X(3051) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,39), (110,669), (694,237), (1576,2531)
X(3051) = X(2531)-cross conjugate of X(1576)
X(3051) = crosspoint of X(i) and X(j) for these (i,j): (6,32), (39,1843)
X(3051) = crosssum of X(i) and X(j) for these (i,j): (2,76), (83,1799)
Let A'B'C' be the tangential triangle of ABC, and let L be the line through X(1) parallel to BC. Let A'' = L∩B'C', and define B'' and C'' cyclically. Let A* = B'B''∩C'C'', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X*(3445), and the lines A'A*, B'B*, C'C* concur in X(3052). (Angel Montesdeoca, April 29, 2016)
X(3052) lies on these lines: 3,595 6,31 31,42 32,220 40,1104 44,200 45,612 56,1149 57,1279 101,1384 109,1407 154,2352 171,1001 221,1464 222,2078 238,1376 518,1707 614,1155 692,1397 1260,2220 1333,2256 1460,1486 1836,3011 2176,2223 2192,2342
X(3052) = isogonal conjugate of X(4373)
X(3052) = crossdifference of every pair of points on line X(514)X(4521) (the complement of the Gergonne line)
X(3052) = X(56)-Ceva conjugate of X(6)
X(3052) = crosspoint of X(i) and X(j) for these (i,j): (109,1252), (1420,1743)
X(3052) = crosssum of X(522) and X(1086)
Let Γ1 be the circumcircle, Γ2 the 2nd Lemoine circle, and Γ3 the circle {{X(371),X(372),PU(1),PU(39)}} (which has center X(32)). The three circles intersect in two points, at which let L and L' be the lines tangent to Γ1. Then X(3035) = L∩L'. (Randy Hutson, January 29, 2015)
X(3053) lies on these lines: 3,6 4,230 22,1184 24,112 25,1611 35,609 56,1914 64,248 76,1003 115,382 140,2548 154,237 218,2251 550,2549 988,1100 999,2241 1385,1572 1498,1971 2176,2223
X(3053) = isogonal conjugate of X(2996)
X(3053) = inverse-in-circumcircle of X(1692)
X(3053) = X(25)-Ceva conjugate of X(6)
X(3053) = crosspoint of X(i) and X(j) for these (i,j): (112,249), (193,459)
X(3053) = crosssum of X(115) and X(525)
X(3053) = radical center of Lucas(-cot ω) circles
X(3053) = vertex, other than X(6), of the hyperbola {{X(6),PU(1),PU(2)}}
X(3053) = intersection of diagonals of trapezoid PU(2)PU(39)
X(3053) = insimilicenter of circles with diameters X(371)X(372) and X(1151)X(1152)
X(3053) = trilinear pole wrt tangential triangle of orthic axis
X(3053) = crossdifference of every pair of points on line X(523)X(4885) (complement of orthic axis)
X(3053) = {X(1687),X(1688)}-harmonic conjugate of X(5171)
X(3053) = Schoutte-circle-inverse of X(33878)
X(3053) = insimilicenter of circle centered at X(1151) through X(372) and circle centered at X(1152) through X(371); the exsimilicenter is X(1350)
X(3053) = insimilicenter of circle centered at X(371) through X(1152) and circle centered at X(372) through X(1151); the exsimilicenter is X(1351)
See X(13) for a discussion of the Evans conic. If you have The Geometer's Sketchpad, you can view X(3054).
X(3054) and X(3055) were contributed by Peter J. C. Moses, Jan 14, 2005.
X(3054) lies on these lines: 2,6 5,187 39,632 53,468 111,930 115,549 140,574 216,2493 626,2031 1384,1656
X(3054) = midpoint of X(590) and X(615)
X(3054) = crosspoint of X(2) and X(7607)
X(3054) = crosssum of X(6) and X(576)
X(3055) lies on these lines: 2,6 5,574 32,632 115,547 140,187 570,2493 1384,2548
X(3055) = complement of X(37688)
X(3055) = crosspoint of X(2) and X(7608)
X(3055) = crosssum of X(6) and X(575)
X(3055) = center of conic {{X(5),X(13),X(14),X(15),X(16)}}
X(3056) lies on these lines: 1,256 3,613 6,31 11,141 33,1843 35,182 37,263 51,612 56,1350 69,350 87,291 144,145 210,391 573,2223 611,1351
X(3056) = reflection of X(1469) in X(1)
X(3056) = isogonal conjugate of isotomic conjugate of X(3705)
X(3056) = crosspoint of X(i) and X(j) for these (i,j): (1,2319), (284,314), (982,3061)
X(3056) = crosssum of X(i) and X(j) for these (i,j): (1,1423), (226,1402)
X(3056) = X(6)-of-Mandart-incircle-triangle
X(3056) = homothetic center of intangents triangle and reflection of extangents triangle in X(6)
Let Oa be the circle centered at the A-excenter and passing trough the A-intouch point. Define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(3057). (Randy Hutson, February 10, 2016)
X(3057) divides segment X(1)X(3) in the ratio r/(R-r). (Randy Hutson, February 10, 2016)
X(3057) is the radical center of circles {Oa}, {Ob}, {Oc} used in the construction of the Ursa-minor and Ursa-major triangles; see preamble before X(17603). (Randy Hutson, June 27, 2018)
Let A'B'C' be the excentral triangle. X(3057) is the radical center of the 2nd Droz-Farny circles of triangles A'BC, B'CA, C'AB. (Randy Hutson, June 27, 2018)
X(3057) lies on these lines: 1,3 4,1000 8,210 10,11 12,946 19,2256 21,643 33,1829 34,1902 37,1953 45,374 72,519 140,1387 144,145 190,1222 200,2136 219,2264 220,2082 227,1457 278,1888 355,1479 388,962 496,1737 595,2361 614,1616 1212,1334 1317,1364 2809,3021
X(3057) = reflection of X(i) in X(j) for these (i,j): (8,960), (65,1)
X(3057) = isogonal conjugate of X(1476)
X(3057) = complement of X(14923)
X(3057) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,1201), (190,650), (1293,513)
X(3057) = cevapoint of X(1) and X(2943)
X(3057) = crosspoint of X(1) and X(8)
X(3057) = crosssum of X(i) and X(j) for these (i,j): (56,269), (1,56), (55,1743)
X(3058) lies on these lines: 1,30 2,11 12,381 72,519 376,1250 496,549 516,553 529,2098 541,3028 542,3024 543,3023 544,3022 551,2646 595,3017 597,2330
X(3058) = inverse-in-Feuerbach-hyperbola of X(3826)
X(3058) = X(2)-of-Mandart-incircle-triangle
X(3058) = homothetic center of intangents triangle and reflection of extangents triangle in X(2)
Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is perspective to the extouch triangle at X(3059). (Randy Hutson, December 2, 2017)
X(3059) lies on the Mandart hyperbola and these lines: 7,8 9,55 37,2340 72,516 78,1001 142,354 971,2951
X(3059) = reflection of X(i) in X(j) for these (i,j): (65,2550), (390,960), (6068,14740)
X(3059) = cevapoint of X(1) and X(2942)
X(3059) = crosspoint of X(8) and X(200)
X(3059) = crosssum of X(i) and X(j) for these (i,j): (56,269), (57,1617)
X(3059) = complement of X(30628)
X(3059) = excentral-to-ABC barycentric image of X(7)
X(3059) = antipode of X(6068) in the Mandart hyperbola
X(3059) = extouch-isogonal conjugate of X(20588)
X(3060) is the second of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see X(3567).
X(3060) = centroid of the intersections, other than A, B, C, of circles {{X(4),B,C}}, {{X(4),C,A}}, {{X(4),A,B}} and lines BC, CA, AB. (Randy Hutson, March 25, 2016)
Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. X(3060) = X(2)-of-A'B'C'. (Randy Hutson, January 29, 2018)
X(3060) lies on these lines: 2,51 3,143 4,52 6,22 20,389 23,184 25,110 26,54 143,1173 156,195 371,588 372,589 394,1995 648,1629 1853,2781 1992,2393
X(3060) = reflection of X(2) in X(51)X(3061) lies on these lines: 1,6 2,257 10,262 21,2344 39,986 63,1429 78,2082 169,997 304,1921 330,335
X(3061) = isogonal conjugate of X(7132)
X(3061) = isotomic conjugate of isogonal conjugate of X(20665)
X(3061) = complement of X(3212)
X(3061) = complementary conjugate of X(20338)
X(3061) = X(3056)-cross conjugate of X(982)
X(3061) = crosssum of X(6) and X(1403)
If you have The Geometer's Sketchpad, you can view X(3062).
In the plane of a triangle ABC, let
A' = A-excircle of ABFC, and define B' and C' cyclically;
A1 = orthopole of BC wrt B'C', and define B1 and C1 cyclically;
Then X(2) = A1B1C1-to-ABC orthology center, and X(3062) = ABC-to-A1B1C1 orthology center. (Ivan Pavlov, August 21, 2022)
X(3062) lies on these lines: 1,971 7,1699 8,144 9,165 80,2093 269,2310 294,1721 515,1000 943,1490 1156,1445 1320,2801 2335,2947
X(3062) = reflection of X(i) in X(j) for these (i,j): (1768,1156), (2951,9)
X(3062) = isogonal conjugate of X(165)
X(3062) = cevapoint of X(513) and X(2310)
X(3062) = X(57)-cross conjugate of X(1)
X(3062) = crosssum of X(55) and X(1615)
X(3062) = perspector of ABC and 2nd antipedal triangle of X(1)
X(3062) = trilinear product of vertices of 6th mixtilinear triangle
X(3063) is the perspector of the triangle ABC and the tangential triangle of the conic that passes through the points A, B, C, X(55), and X(56).
X(3063) lies on these lines: 6,513 59,919 112,2714 521,650 649,854 657,663 661,832 665,1459 667,788 834,2483
X(3063) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39025), (644,55), (649,667), (692,2175), (919,2223), (1415,31), (2192, 3022), (2423,1960)
X(3063) = isogonal conjugate of X(4554)
X(3063) = cevapoint of X(798) and X(3049)
X(3063) = crosspoint of X(i) and X(j) for these (i,j): (6,692), (31,1415), (55,644), (649,663), (651,1037)
X(3063) = crosssum of X(i) and X(j) for these (i,j): (2,693), (37,2533), (190,664), (497,650), (513,2275)
X(3063) = crossdifference of every pair of points on the line X(7)X(8)
X(3063) = intersection of trilinear polars of X(55) and X(56)
X(3063) = trilinear product of PU(103)
X(3063) = perspector of hyperbola {{A,B,C,X(55),X(56)}}
X(3063) = barycentric product of Feuerbach hyperbola intercepts of Lemoine axis
X(3064) lies on radical axis of Mandart circle and excircles radical circle. (Randy Hutson, December 2, 2017)
X(3064) lies on these lines: 19,649 112,2689 225,770 230,231 243,522
X(3064) = isogonal conjugate of X(1813)
X(3064) = X(i)-Ceva conjugate of X(j) for these (i,j): (108,1856), (158,2310), (278,11), (653,4), (1783,1826), (1897,33)
X(3064) = cevapoint of X(661) and X(2501)
X(3064) = X(i)-cross conjugate of X(j) for these (i,j): (661,650), (663,522), (2170,19), (2310,158)
X(3064) = crosspoint of X(i) and X(j) for these (i,j): (4,653), (92,1897), (1172,1783)
X(3064) = crosssum of X(i) and X(j) for these (i,j): (3,652), (48,1459), 905,1214)
X(3064) = crossdifference of every pair of points on the line X(3)X(73)
X(3064) = pole wrt polar circle of trilinear polar of X(664) (line X(2)X(7))
X(3064) = polar conjugate of X(664)
X(3064) = trilinear pole of line X(2310)X(8735)
X(3064) = trilinear product X(4)*X(650)
X(3064) = trilinear product of Feuerbach hyperbola intercepts of orthic axis
X(3065) lies on the Neuberg cubic and these lines: 1,399 4,1768 8,191 9,1030 11,79 21,214 30,80 202,1251 314,1227 758,1320 1157,3465 1389,2800 2132,3466 2346,2801
X(3065) = reflection of X(i) in X(j) for these (i,j): (79,11), (484,1749)
X(3065) = isogonal conjugate of X(484)
X(3065) = cevapoint of X(i) and X(j) for these (i,j): (654,2310), (1962,2245)
X(3065) = X(36)-cross conjugate of X(1)
X(3065) = trilinear pole of line X(650)X(1100)
X(3065) = antigonal conjugate of X(79)
X(3065) = X(2914)-of-excentral-triangle
Suppose X is a point, with isogonal conjugate X- 1. It is well known that the pedal triangle of X is homothetic to the antipedal triangle of X- 1. The Lemoine homothetic center, X(1285), is the homothetic center when X = X(6), and X(3066) is the homothetic center when X = X(2). See also X(1285). (Peter Moses, Dec. 7, 2005)
In general if X = x : y : z (trilinears), then the homothetic center is given by
a(x + y cos C)(x + z cos B) : b(y + z cos A)(y + x cos C): c(z + x cos B)(z + y cos A).This is a correction for trilinears given in TCCT, p. 188.
X(3066) lies on these lines: 2,1350 3,373 6,110 25,182 51,394 107,458 125,381
X(3066) = barycentric product of vertices of submedial triangle
The Hofstadter ellipse E(0) is described at X(359). The point other than A, B, and C in which E(0) meets the circumcircle is X(3067). As with X(359) and X(360), this is a transcendental center, with "exposed angles" A, B, C in its coordinates.
X(3068): Let A'B'C' be the outer Vecten triangle. Let A" be the trilinear pole of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(3068). (Randy Hutson, July 21, 2017)
X(3068) lies on these lines: {1,1336}, {2,6}, {3,1587}, {4,371}, {5,1588}, {8,7969}, {9,5393}, {10,13893}, {11,13895}, {12,13896}, {13,13917}, {14,13916}, {17,22921}, {18,22876}, {19,1659}, {20,1151}, {24,8276}, {25,13884}, {30,6221}, {32,638}, {39,8963}, {40,13912}, {53,19410}, {54,8995}, {55,13887}, {56,18965}, {64,8991}, {68,13909}, {74,8994}, {76,8992}, {79,16148}, {80,8988}, {83,5491}, {84,8987}, {98,8980}, {99,8997}, {100,13922}, {104,13913}, {110,8998}, {112,13923}, {113,19060}, {114,19056}, {115,19109}, {119,19082}, {125,19111}, {127,19115}, {132,19094}, {140,3312}, {148,13657}, {171,606}, {184,18924}, {186,9682}, {214,19078}, {216,1590}, {238,605}, {239,1267}, {262,22720}, {265,13915}, {372,631}, {376,6200}, {381,6199}, {382,23253}, {387,2047}, {388,2067}, {393,493}, {402,13894}, {427,5410}, {468,5411}, {486,3090}, {487,5286}, {488,6423}, {489,5254}, {490,3053}, {494,13900}, {497,2066}, {498,3301}, {499,3299}, {511,22717}, {515,9583}, {516,9616}, {530,13646}, {531,13645}, {532,22918}, {533,22873}, {538,13647}, {542,13640}, {543,13642}, {546,23263}, {548,6455}, {549,6398}, {550,6449}, {571,13439}, {577,1589}, {588,2165}, {618,19074}, {619,19076}, {620,19108}, {629,19070}, {630,19072}, {632,6428}, {637,1504}, {640,7375}, {641,13771}, {642,19105}, {671,13908}, {754,13648}, {894,5391}, {920,1123}, {946,1702}, {958,19014}, {1001,18999}, {1030,21566}, {1033,15210}, {1075,8954}, {1078,18993}, {1124,3086}, {1125,13959}, {1131,3146}, {1132,3590}, {1147,19062}, {1152,3523}, {1181,6807}, {1209,19096}, {1249,3535}, {1297,13918}, {1327,13920}, {1328,13848}, {1335,3085}, {1370,11417}, {1376,19000}, {1378,19843}, {1449,5405}, {1503,7374}, {1511,19052}, {1579,7400}, {1583,8573}, {1586,3087}, {1599,1609}, {1650,19018}, {1656,6417}, {1657,6407}, {1698,13936}, {1703,6684}, {1788,2362}, {1899,18923}, {2043,5335}, {2044,5334}, {2271,21909}, {2482,19058}, {2548,5058}, {2550,5415}, {2883,19088}, {3035,19112}, {3071,3091}, {3074,3077}, {3075,3076}, {3083,3554}, {3084,3553}, {3088,3093}, {3089,3092}, {3096,19012}, {3103,12251}, {3147,10881}, {3183,22838}, {3186,8956}, {3297,14986}, {3317,10195}, {3462,8955}, {3485,16232}, {3522,6409}, {3524,6396}, {3525,5420}, {3526,6418}, {3528,9680}, {3529,6453}, {3530,6450}, {3534,6445}, {3543,6437}, {3545,6565}, {3594,10303}, {3616,7968}, {3624,13971}, {3627,6447}, {3628,6427}, {3634,13947}, {3647,19080}, {3832,23261}, {3855,23275}, {3934,19089}, {4254,16433}, {4296,9634}, {4297,9615}, {4423,13940}, {5020,19005}, {5021,21992}, {5054,6395}, {5055,18510}, {5056,6431}, {5059,6429}, {5067,10577}, {5070,6500}, {5120,16432}, {5124,21567}, {5200,10132}, {5218,5414}, {5305,11313}, {5409,6805}, {5413,6353}, {5432,19037}, {5433,18995}, {5449,19061}, {5461,19057}, {5480,7000}, {5552,19048}, {5597,13890}, {5598,13891}, {5599,19008}, {5600,19010}, {5889,12239}, {5972,19110}, {6036,19055}, {6118,19102}, {6119,19104}, {6193,8909}, {6202,8396}, {6260,19068}, {6292,19092}, {6302,9112}, {6303,9113}, {6392,6462}, {6410,15717}, {6411,10304}, {6412,15692}, {6424,7388}, {6438,15708}, {6446,15693}, {6448,12108}, {6451,8703}, {6452,12100}, {6456,15712}, {6468,15683}, {6480,11001}, {6481,15719}, {6501,13961}, {6502,7288}, {6519,15704}, {6643,10897}, {6669,19073}, {6670,19075}, {6673,19071}, {6674,19069}, {6689,19095}, {6696,19087}, {6699,19059}, {6701,19079}, {6702,19077}, {6704,19091}, {6705,19067}, {6713,19081}, {6720,19114}, {6776,6811}, {6808,10982}, {6813,14853}, {7160,13914}, {7386,11513}, {7392,10961}, {7493,11418}, {7494,11514}, {7615,13660}, {7737,9675}, {7738,11293}, {7815,13938}, {7846,19011}, {8222,19032}, {8223,19034}, {8280,8889}, {8383,19219}, {8882,16032}, {9648,15338}, {9663,15326}, {9681,22644}, {9683,12088}, {9686,13346}, {9690,15681}, {9691,17800}, {9694,11413}, {9695,12082}, {9722,15234}, {9732,21737}, {9780,13973}, {9862,13674}, {10192,17820}, {10198,13965}, {10200,13964}, {10266,13919}, {10527,19050}, {10533,11206}, {10665,11411}, {10819,12383}, {11265,14790}, {11284,13943}, {11294,12323}, {11442,11447}, {11457,11462}, {11485,18585}, {11486,15765}, {11542,18587}, {11543,18586}, {12221,13881}, {12240,15043}, {12317,12375}, {12318,12424}, {12319,12891}, {12320,12960}, {12321,12961}, {12322,12962}, {12324,12964}, {12325,12965}, {12376,20125}, {12864,19086}, {13025,13045}, {13026,13046}, {13089,19098}, {13203,13287}, {13650,13711}, {13662,13720}, {13701,22541}, {13774,15118}, {13821,19100}, {13832,13833}, {15183,19017}, {15722,17851}, {15819,19064}, {18457,18531}, {19051,20304}, {19084,22966}, {19174,19183}, {19420,19436}, {19421,19438}, {22466,22976}, {22555,22960}
X(3068) = isogonal conjugate of X(493)
X(3068) = complement of X(1270)
X(3068) = X(i)-Ceva conjugate of X(j) for these (i,j): (393,3069), (1585,4)
X(3068) = crosspoint of X(2) and X(1131)
X(3068) = crosssum of X(6) and X(1151)
X(3068) = {X(2),X(6)}-harmonic conjugate of X(3069)
X(3068) = X(2)-of-1st-tri-squares-triangle
X(3068) = X(2)-of-1st-tri-squares-central-triangle
X(3068) = orthologic center of these triangles: 1st tri-squares to outer-Vecten
X(3068) = homothetic center of ABC and 3rd tri-squares central triangle
Let A'B'C' be the inner Vecten triangle. Let A" be the trilinear pole of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(3069). (Randy Hutson, July 21, 2017)
X(3069) lies on these lines: 1,1123 2,6 3,1588 4,372 5,1587 20,1152 32,637 171,605 216,1589 238,606 393,494 489,3053 577,1590 589,2165 894,1267 920,1336 946,1703 1600,1609 3074,3076 3075,3077
X(3069) = isogonal conjugate of X(494)
X(3069) = complement of X(1271)
X(3069) = X(i)-Ceva conjugate of X(j) for these (i,j): (393,3068), (1586,4)
X(3069) = crosspoint of X(2) and X(1132)
X(3069) = crosssum of X(6) and X(1152)
X(3069) = {X(2),X(6)}-harmonic conjugate of X(3068)
X(3069) = X(2)-of-2nd-tri-squares-triangle
X(3069) = X(2)-of-2nd-tri-squares-central-triangle
X(3069) = homothetic center of ABC and 4th tri-squares central triangle
Let A'B'C' be the outer Vecten triangle and A"B"C" the inner Vecten triangle. Let A* be the isogonal conjugate, wrt A'B'C', of A", and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(3070). (Randy Hutson, July 11, 2019)
X(3070) lies on the the Evans conic and these lines: 2,490 3,485 4,6 5,372 20,1151 30,371 487,1991 642,2482 1124,1479 1335,1478 1837,2362
X(3070) = reflection of X(3071) in X(5254)
X(3070) = complement of X(490)
X(3070) = crosspoint of X(4) and X(485)
X(3070) = crosssum of X(3) and X(371)
X(3070) = {X(5),X(372)}-harmonic conjugate of X(615)
X(3070) = {X(4),X(6)}-harmonic conjugate of X(3071)
Let A'B'C' be the outer Vecten triangle and A"B"C" the inner Vecten triangle. Let A* be the isogonal conjugate, wrt A"B"C", of A', and define B* and C* cyclically. The lines A"A*, B"B*, C"C* concur in X(3071). (Randy Hutson, July 11, 2019)
X(3071) lies on the Evans conic and these lines: 2,489 3,486 4,6 5,371 11,2067 12,2066 20,1152 30,372 488,591 641,2482 1124,1478 1335,1479 1836,2362
X(3071) = reflection of X(3070) in X(5254)
X(3071) = complement of X(489)
X(3071) = crosspoint of X(4) and X(486)
X(3071) = crosssum of X(3) and X(372)
X(3071) = {X(5),X(371)}-harmonic conjugate of X(590)
X(3071) = {X(4),X(6)}-harmonic conjugate of X(3070)
X(3072) lies on these lines: 1,3 2,602 4,31 5,238 10,580 12,2361 20,601 47,1478 58,515 155,1740 181,389 225,2190 255,388 283,1010 411,1064 497,1497 578,1397 595,946 912,1046 944,1468 947,1412 990,1158 1056,1496 1254,1870 2179,2201
X(3073) lies on these lines: 1,90 2,601 3,238 4,31 5,171 11,1399 20,602 21,1064 40,1724 47,1479 55,3074 57,1777 58,946 104,1201 109,1210 255,497 388,1497 515,595 516,580 578,2175 605,1587 606,1588 631,748 651,1066 774,1870 943,2293 1058,1496 1460,1598 1467,2956
X(3074) lies on these lines: 1,6 2,255 3,1745 4,212 5,1936 10,275 12,2361 28,2183 40,1888 47,171 55,3073 73,1006 165,1777 191,1735 201,1870 226,580 283,908 388,602 603,631 1167,1785 1698,1771 1794,1838 2182,2939 3068,3077 3069,3076
X(3074) = {X(2),X(255)}-harmonic conjugate of X(3075)
X(3075) lies on these lines: 1,3 2,255 4,603 5,1935 11,1399 29,58 47,238 81,1816 109,946 158,1430 222,1745 389,1364 412,1785 750,1406 1393,1870 1699,1777 3068,3076 3069,3077
X(3075) = {X(2),X(255)}-harmonic conjugate of X(3074)
X(3075) = X(3)-gimel conjugate of X(35)
X(3076) lies on these lines: 1,606 6,255 31,1125 47,605 109,1702 212,372 371,603 601,2066 1587,1936 1588,1935 3068,3075 3069,3074
X(3077) lies on these lines: 1,605 6,255 31,1135 47,606 109,1703 212,371 372,603 602,2067 1587,1935 1588,1936 3068,3074 3069,3075
Let A'B'C' be the cevian triangle of a point X = x : y : z (trilinears), let L(A) be the line through A parallel to B'C', and define L(B) and L(C) cyclically. The lines L(A), L(B), L(C) form a triangle homothetic to A'B'C', with homothetic center
ax2(by + cz) : by2(cz + ax) : cz2(ax + by).
We denote this point by D(X) and call it the Danneels point of X. (Eric Danneels, Hyacinthos, Jan. 28, 2005). In addition to the discussion below, see the preamble to X(8012) and the Danneels points beginning at X(8012).
If X lies on the line at infinity, then D(X) = X(2) of cevian triangle of X. (Randy Hutson, Jul 23, 2015)
The formula is simpler in barycentrics: if U = u : v : w, then
D(U) = u2(v + w) : v2(w + u) : w2(u + v).
If X is line the Euler line, then D(X) is on the Euler line. A proof follows. Let a1, b1, c1 be cos A, cos B, cos C, respectively, so that the Euler line is given parametrically as x(t) : y(t) : z (t) by
x = b1c1 +
ta1
y = c1a1 +
tb1
z = a1b1 +
tc1,
where t is an arbitrary function homogenous of degree 0 in a, b, c. Then D(X) is the point U = u : v : w given by
u = ax2(by + cz)
v = by2(cz + ax)
w = cz2(ax + by).
A point P = p : q : r is on the Euler line if
a(b2 - c2)(b2 + c2 -
a2)p
+ b(c2 - a2)(c2 + a2 -
b2)q
+ c(a2 - b2)(a2 + b2
-c2)r = 0.
It is easy to check that the point U satisfies this equation. (This sort of algebraic proof is more inclusive than a geometric proof, because here, a,b,c are indeterminates or variables. They can, for example, take values that are not sidelengths of a triangle. Here, cos A is defined as (b2 + c2 - a2)/(2bc), so that no dependence on a geometric angle is necessary.)
The appearance of (i,j) in the following list means that D(X(i)) = X(j):
(1,42) (2,2) (3,418)
(4,25) 6,3051 (7,57)
(8,200) (69,394) (75,321)
(100,55) (110,84) (264,324)
(366,367) (651,222) (653,196)
(1113,25) (1114,25) (1370,455) For an extension of this list, see the preamble to X(8012).
X(3078) lies on this line: (2,3)
X(3078) = X(5)-Ceva conjugate of X(233)
X(3078) = X(288)-isoconjugate of X(2167)X(3079) lies on these lines: {2, 3}, {154, 1249}, {204, 1394}, {459, 1503}, {1495, 6524}, {1498, 2131}, {5562, 5910}
X(3079) = X(20)-Ceva conjugate of X(1249)
X(3079) = X(1073)-isoconjugate of X(2184)
As a point on the Euler line, X(3080) has Shinagawa coefficients ((E + F)3F - (E - F)FS2, -(E + F)4 + (E + F)(2E - F)S2).
X(3080) lies on these lines: {2, 3}, {682, 1196}
X(3080) = X(25)-Ceva conjugate of X(1196)
X(3080) =X(326)-isoconjugate of X(683)As a point on the Euler line, X(3081) has Shinagawa coefficients (27F2 - S2, -27(E + F)F + 9S2).
X(3081) lies on this line: (2,3)
X(3081) = reflection of X(1650) in X(1651)
X(3081) = X(30)-Ceva conjugate of X(3163)
X(3081) = centroid of cevian triangle of X(30)
X(3081) = barycentric cube of X(30)
Let (OA) be the I-excircle of triangle IBC, where I = X(1), and define (OB) and (OC) cyclically. Let A' be the touchpoint of (OA) and side BC, and define B' and C' cyclically. Let A" be the insimilicenter of (OB) and (OC), and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(3082). (Randy Hutson, July 11, 2019)
If you have The Geometer's Sketchpad, you can view X(3082).
X(3082) lies on these lines: 8,178 174,176
For a construction and relationships to other points, see the preamble just before X(37994).
X(3083) lies on these lines: 1,2 11,1591 12,1592 33,1586 34,1585 35,1600 36,1599 38,494 55,1584 56,1583 176,329 394,1124 1038,1590 1040,1589
X(3083) = isogonal conjugate of polar conjugate of X(46744)
X(3083) = isotomic conjugate of isogonal conjugate of X(605)
X(3083) = isotomic conjugate of polar conjugate of X(6212)
X(3083) = isotomic conjugate of complement of X(37881)
X(3083) = isotomic conjugate of anticomplement of X(40651)
X(3083) = {X(1),X(2)}-harmonic conjugate of X(3084)
X(3084) lies on these lines: 1,2 11,1592 12,1591 33,1585 34,1586 35,1599 36,1600 37,493 55,1583 56,1584 175,329 394,1335 1038,1589 1040,1590
X(3084) = isogonal conjugate of polar conjugate of X(46745)
X(3084) = isotomic conjugate of isogonal conjugate of X(606)
X(3084) = isotomic conjugate of polar conjugate of X(6213)
X(3084) = {X(1),X(2)}-harmonic conjugate of X(3083)
X(3085) lies on these lines: 1,2 3,388 4,12 5,497 7,46 11,1058 20,35 28,197 31,3074 33,3089 34,3088 37,158 40,226 56,631 69,611 100,377 140,999 144,191 171,255 212,3072 227,278 238,1497 346,1089 390,1479 405,2551 42,954 443,1376 496,1656 601,1935 750,1496 756,774 942,1788 944,2646 1124,3069 1335,3068 1588,2066 2241,2548
X(3085) = {X(1),X(2)}-harmonic conjugate of X(3086)
Let Ab = 0 : 2R : c, Ac = 0 : b : 2R, Bc = a : 0 : 2R, Ba = 2R : 0 : c, Ca = 2R : b : 0, Cb = a : 2R : 0 be the six points in the construction of the Paasche conic at X(37861) and X(37881). Let Ka be the point of intersection of the lines tangent to the Paasche conic at Ba and Ca, and define Kb and Kc cyclically. The triangle KaKbKc is perspective to ABC, and the perspector is X(3086); see also X(37884) and X(3086) . (Vijay Krishna, April 15, 2020)
For a construction and relationships to other points, see the preamble just before X(37994).
X(3086) lies on these lines: 1,2 3,496 4,11 5,388 7,90 12,1056 20,36 31,3075 33,3088 34,3089 35,390 55,631 57,946 69,613 171,1497 238,255 269,1256 406,1104 443,2886 495,1656 602,1936 603,3073 748,1496 944,1319 1124,3068 1335,3069 1588,2067 2242,2548
X(3086) = crosspoint of X(i) and X(j) for these (i,j): (2,1440), (1123,1336)
X(3086) = crosssum of X(1124) and X(1335)
X(3086) = {X(1),X(2)}-harmonic conjugate of X(3085)
Let A'B'C' be the anti-Euler triangle. X(3087) is the radical center of the circumcircles of triangles A'BC, B'CA, C'AB. (Randy Hutson, June 7, 2019)
X(3087) lies on these lines: 2,95 4,6 19,1877 20,216 29,1778 32,3088
X(3087) = polar conjugate of X(8797)
As a point on the Euler line, X(3088) has Shinagawa coefficients (F,E).
X(3088) lies on these lines: 2,3 32,3087 33,3086 34,3085
X(3088) = inverse-in-orthocentroidal-circle of X(3089)
X(3088) = {X(2),X(4)}-harmonic conjugate of X(3089)
x As a point on the Euler line, X(3089) has Shinagawa coefficients (F, -E).
X(3089) lies on these lines: 2,3 33,3085 34,3086 3068,3092 3069,3093
X(3089) = inverse-in-orthocentroidal-circle of X(3088)
X(3089) = anticomplement of X(3546)
X(3089) = {X(2),X(4)}-harmonic conjugate of X(3088)
X(3089) = {X(4),X(24)}-harmonic conjugate of X(20)
As a point on the Euler line, X(3090) has Shinagawa coefficients (2,1).
X(3090) lies on these lines: 2,3 11,1058 12,1056 69,576 76,1007 110,569 155,1199 182,1614 233,1249 373,389 388,499 485,3069 486,3068 497,498 575,1352 590,1588 615,1587 748,3072 750,3073 944,1125 946,1698 1493,2888
X(3090) = reflection of X(i) in X(j) for these (i,j): (3523,3526), (3528,3523)
X(3090) = isotomic conjugate of X(36948)
X(3090) = complement of X(3523)
X(3090) = anticomplement of X(3526)
X(3090) = circumcircle-inverse of X(37939)
X(3090) = nine-point-circle-inverse of anticomplement of X(37971)
X(3090) = nine-point-circle-inverse of complement of X(37945)
X(3090) = nine-point-circle-inverse of radical trace of Grebe circle and de Longchamps circle
X(3090) = polar conjugate of isogonal conjugate of X(36751)
X(3090) = inverse-in-orthocentroidal-circle of X(631)
X(3090) = homothetic center of orthocentroidal triangle and X(5)-Brocard triangle
X(3090) = homothetic center of anti-Euler triangle and cross-triangle of ABC and Euler triangle
X(3090) = homothetic center of Euler triangle and cross-triangle of ABC and anti-Euler triangle
X(3090) = homothetic center of ABC and cross-triangle of Euler and anti-Euler triangles
X(3090) = homothetic center of X(2)-altimedial and X(3)-anti-altimedial triangles
X(3090) = homothetic center of X(4)-altimedial and X(20)-anti-altimedial triangles
X(3090) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,3525), (2,4,631), (2,5,4), (2,20,140), (2,384,32970), (3,4,3529), (3,5,3091), (4,5,3545), (4,20,15682), (4,3544,3091), (76,1007,32818), (373,389,15024)
As a point on the Euler line, X(3091) has Shinagawa coefficients (1,2).
Let A'B'C' be the Euler triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3091). (Randy Hutson, December 10, 2016)
Let A'B'C' be the Euler triangle. Let A* be the centroid of AB'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3091). (Randy Hutson, December 10, 2016)
X(3091) lies on these lines: 2,3 7,1210 8,908 10,962 11,153 12,497 68,1173 110,578 114,148 115,147 116,152 118,150 119,149 124,151 125,146 145,355 155,1994 156,567 193,576 194,262 226,938 253,264 324,1093 347,1838 390,1479 485,1132 486,1131 495,1058 496,1056 516,1698 569,1614 637,1271 638,1270 1007,1975 1329,2550 1348,2543 1349,2542 1454,1776 1478,3086 1506,2549 1676,2547 1677,2546 1788,1836 1853,2883 2009,2545 2010,2544 2551,2886 3068,3071 3069,3070
X(3091) = midpoint of X(4) and X(631)
X(3091) = reflection of X(i) in X(j) for these (i,j): (3,632), (631,1656), (1656,5), (3522,631)
X(3091) = inverse-in-orthocentroidal-circle of X(20)
X(3091) = complement of X(3522)
X(3091) = anticomplement of X(631)
X(3091) = circumcircle-inverse of X(37940)
X(3091) = homothetic center of orthocentroidal triangle and X(4)-Brocard triangle
X(3091) = homothetic center of anticomplementary triangle and cross-triangle of ABC and Euler triangle
X(3091) = homothetic center of Euler triangle and cross-triangle of ABC and Euler triangle
X(3091) = circumcenter of cross-triangle of Euler and anti-Euler triangles
X(3091) = homothetic center of X(2)-altimedial and X(4)-anti-altimedial triangles
X(3091) = homothetic center of X(20)-altimedial and X(20)-anti-altimedial triangles
X(3091) = homothetic center of X(20)-anti-altimedial and orthocentroidal triangles
X(3091) = X(7897)-of-orthic-triangle if ABC is acute
X(3091) = homothetic center of Ehrmann mid-triangle and anti-Euler triangle
X(3091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5,3090), (4,5,2), (4,3544,3090)
X(3092) lies on these lines: 2,1579 4,6 20,1578 24,1151 25,371 33,1335 34,1124 235,485 372,1593 378,1152 394,637 427,486 1377,1861 3068,3089 3069,3088
X(3092) = {X(4),X(6)}-harmonic conjugate of X(3093)
X(3093) lies on these lines: 2,1578 4,6 20,1579 24,1152 25,372 33,1124 34,1335 235,486 371,1593 378,1151 394,638 427,485 1378,1861 1905,2362 3068,3088 3069,3089
X(3093) = {X(4),X(6)}-harmonic conjugate of X(3092)
Let A2B2C2 be the 2nd Brocard triangle. Let A' = inverse-in-Brocard circle of A, and define B' and C' cyclically. Let A2' = inverse-in-circumcircle of A2, and define B2' and C2' cyclically. Let A'' = B'B2'∩C'C2', and define B'' and C'' cyclically. The lines AA", BB", CC" concur in X(3094). (Randy Hutson, December 26, 2015)
Let A'B'C' be the 3rd Brocard triangle. Then X(3094) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, December 26, 2015)
Let D = P(1), E = U(1), F = P(2), G = U(2). Let D' = X(4)-of-EFG, E' = X(4)-of-DFG, F' = X(4)-of-DEG, G' = X(4)-of-DEF. Then D'E'F'G' is a cyclic quadrilateral whose circumcenter is X(3094). (Randy Hutson, December 26, 2015)
X(3094) lies on these lines: 2,694 3,6 22,1915 69,194 76,141 99,737 262,1513 394,2056 538,599 542,1569 726,2321 1180,2979 1194,1613 1352,2549 1469,2276 2275,3056
X(3094) = midpoint of X(69) and X(194)
X(3094) = reflection of X(i) in X(j) for these (i,j): (6,39), (76,141)
X(3094) = isogonal conjugate of X(3407)
X(3094) = isotomic conjugate of X(3114)
X(3094) = Brocard-circle-inverse of X(1691)
X(3094) = 2nd-Brocard-circle-inverse of X(32)
X(3094) = crosssum of X(32) and X(182)
X(3094) = crosspoint of X(76) and X(262)
X(3094) = reflection of X(6) in line PU(1)
X(3094) = X(76)-of-1st-Brocard-triangle
X(3094) = X(6)-of-5th-Brocard-triangle
X(3094) = X(4)-of-X(3)PU(1)
X(3094) = inverse-in-circle-{{X(3102),X(3103),PU(1)}} of X(3)
X(3094) = perspector of 2nd Brocard triangle and unary cofactor triangle of 1st Brocard triangle
X(3094) = 1st-Brocard-isogonal conjugate of X(4048)
X(3094) = 1st-Brocard-isotomic conjugate of X(3734)
X(3094) = {X(371),X(372)}-harmonic conjugate of X(3398)
X(3094) = {X(1340),X(1341)}-harmonic conjugate of X(39)
Let U be the pedal triangle of the 1st Brocard point of ABC, and let V be the pedal triangle of the 2nd Brocard point. Let X' = X(39)-of-U and Y' = X(512)-of-U; let X'' = X(39)-of-V and Y'' = X(512)-of-V. Then X(3095) = X'Y'∩X''Y''. (Randy Hutson, September 5 , 2015)
Let X be a point on the 2nd Brocard circle. The locus of X(4) of triangle XPU(1) as X varies is a circle with center X(3095). This circle is the reflection of the 2nd Brocard circle in X(39). See also X(9821). (Randy Hutson, July 20, 2016)
The locus of the orthocenter in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X(3095) and segment X(3557)X(3558) as diameter. The circle also passes through X(4) and its antipode X(194). (Randy Hutson, August 29, 2018)
Let OA be the circle centered at the A-vertex of the 5th anti-Brocard triangle and passing through A; define OB and OC cyclically. X(3095) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(3095) lies on these lines: 3,6 4,147 5,76 237,3060 355,730 381,538 726,946 732,1352 1993,2001
X(3095) = midpoint of X(i) and X(j) for these (i,j): (4,194), (3557, 3558)
X(3095) = reflection of X(i) in X(j) for these (i,j): (3,39), (76,5)
X(3095) = isogonal conjugate of X(3406)
X(3095) = circumcircle-inverse of X(35375)
X(3095) = Brocard-circle-inverse of X(3398)
X(3095) = 2nd-Brocard-circle-inverse of X(182)
X(3095) = reflection of X(3) in line PU(1)
X(3095) = harmonic center of 2nd Lemoine circle and the circle {{X(371),X(372),PU(1),PU(39)}}
X(3095) = center of circle {{X(4),X(194),X(3557),X(3558)}}
X(3095) = X(4)-of-X(6)PU(1)
X(3095) = orthologic center of these triangles: Johnson to 1st Neuberg
X(3095) = X(76)-of-Johnson-triangle
X(3095) = inverse-in-circle-{X(3102),X(3103),PU(1)} of X(6)
X(3095) = intersection of lines PU(1) of pedal triangles of PU(1)
X(3095) = {X(371),X(372)}-harmonic conjugate of X(1691)
X(3096) lies on these lines: 2,32 4,3098 76,141 114,631 211,2979 316,2076
X(3096) = complement of X(7787)
X(3096) = {X(15),X(16)}-harmonic conjugate of X(32)
X(3096) = homothetic center of medial triangle and 5th Brocard triangle
X(3096) = homothetic center of ABC and cross-triangle of ABC and 5th Brocard triangle
X(3097) lies on these lines: 1,39 2,726 10,194 43,63 76,1698 165,511 262,1699 984,1575 1670,1700 1671,1701
X(3097) = reflection of X(1699) in X(262)
Let AA1A2, BB1B2, CC1C2 be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the midpoint of A1 and A2, and define B' and C' cyclically. Triangle A'B'C' is homothetic to ABC at X(3) and perspective to the Trinh triangle at X(3098).
X(3098) lies on these lines: 3,6 4,3096 20,1352 22,1495 30,141 35,1469 36,3056 69,74 184,323 186,1974 206,1511 378,1843 399,2916 550,1503 805,842
X(3098) = midpoint of X(i) and X(j) for these (i,j): (3,1350), (20,1352)
X(3098) = reflection of X(i) in X(j) for these (i,j): (182,3), (576,182), (1351,575)
X(3098) = inverse-in-Brocard-circle of X(5092)
X(3098) = {X(371),X(372)}-harmonic conjugate of X(5007)
X(3098) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(5007)
X(3098) = X(7)-of-Trinh-triangle if ABC is acute
X(3098) = Trinh-isotomic conjugate of X(3357)
X(3098) = X(182)-of-circumcevian-triangle-of-X(511)
X(3098) = X(182)-of-5th-Brocard-triangle
X(3098) = 6th-Brocard-to-5th-Brocard similarity image of X(3)
X(3099) lies on these lines: 1,32 10,2896 518,2076 672,1282 726,2959 846,902
X(3100) lies on these lines: 1,7 2,33 4,1062 11,858 21,270 22,55 30,1870 36,2071 78,280 165,3100 229,2646 238,2310 376,1060 394,2192 497,1370 518,677 522,663 651,971 655,1807 984,1253 1776,2361
X(3100) = anticomplement of X(1861)
X(3100) = crosssum of X(i) and X(j) for these (i,j): (65,1456),
(1409,2223)
X(3100) = crossdifference of every pair of points on the line
X(657)X(1400)
X(3101) lies on these lines: 2,19 8,20 22,55 27,1441 46,387 57,347 65,81 71,1654 100,1297 306,2897 329,1763 345,1760 573,1726 1172,1214 1305,1952 1370,2550 1442,1790 1770,2960 1869,2475
The 1st Brocard point (trilinears c/b : a/c : b/a) and 2nd Brocard point (b/c : c/a : a/b) are opposing vertices of a square here called the Brocard square. The other two vertices are X(3102) and X(3103). Both are on the Brocard axis X(3)X(6), and X(3102) is the closer to X(3).
X(3102) lies on these lines: 3,6 76,486 194,488 262,485 325,640 2782,3071
X(3102) = reflection of X(3103) in X(39)
X(3102) = inverse-in-2nd-Brocard-circle of X(372)
X(3103) is the vertex of the Brocard square that is farest from X(3); see X(3102).
X(3103) lies on these lines: 3,6 76,485 194,487 262,486 325,639 2782,3070
X(3103) = reflection of X(3102) in X(39)
X(3103) = inverse-in-2nd-Brocard-circle of X(371)
There are two 60-rotations of the 1st Brocard point about the 2nd; X(3104) is the one closer to X(3).
X(3104) lies on these lines: 3,6 14,76 17,262 194,616 633,2896
X(3104) = reflection of X(i) in X(j) for these (i,j): (3105,39), (3107, 3106)
X(3104) = inverse-in-2nd-Brocard-circle of X(16)
There are two 60-rotations of the 1st Brocard point about the 2nd; X(3105) is the one farther from X(3).
X(3105) lies on these lines: 3,6 13,76 18,262 194,617 634,2896
X(3105) = reflection of X(i) in X(j) for these (i,j): (3104,39), (3106, 3107)
X(3105) = inverse-in-2nd-Brocard-circle of X(15)
X(3106) is the center of the equilateral triangle introduced at X(3104).
X(3106) lies on these lines: 3,6 13,262 14,2782 18,76 194,627
The locus of the 2nd Isogonic Center in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X(3106). (Randy Hutson, August 29, 2018)
X(3106) = midpoint of X(3104) and X(3107)
X(3106) = reflection of X(i) in X(j) for these (i,j): (3105,3107), (3107, 39)
X(3106) = inverse-in-2nd-Brocard-circle of X(62)
X(3106) = X(13)-of-X(3)PU(1)
X(3106) = X(14)-of-X(6)PU(1)
X(3107) is the center of the equilateral triangle introduced at X(3105).
The locus of the X(13) in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X(3107). (Randy Hutson, August 29, 2018)
X(3107) lies on these lines: 3,6 13,2782 14,262 17,76 194,628
X(3107) = midpoint of X(3105) and X(3106)
X(3107) = reflection of X(i) in X(j) for these (i,j): (3105,3106), (3106, 39)
X(3107) = inverse-in-2nd-Brocard-circle of X(61)
X(3107) = X(14)-of-X(3)PU(1)
X(3107) = X(13)-of-X(6)PU(1)
For an introduction to the DC and CD mappings, see the discussion just before X(2979).
X(3108) lies on these lines: 6,1627 25,1180 39,251 111,1194 305,1239
X(3108) = cevapoint of X(i) and X(j) for these (i,j): (6,39), (1015,2483)
X(3108) = X(3005)-cross conjugate of X(110)
X(3109) lies on the cubic K165 and these lines: {1, 523}, {2, 3}, {11, 759}, {12, 2222}, {58, 6788}, {110, 952}, {119, 1793}, {163, 1146}, {229, 18990}, {355, 3233}, {476, 953}, {513, 13868}, {525, 24347}, {643, 1145}, {662, 10609}, {691, 2726}, {1043, 6790}, {1086, 30927}, {1290, 5253}, {1304, 2734}, {1483, 14611}, {2150, 21933}, {2617, 34586}, {3833, 34583}, {5690, 35193}, {5972, 6739}, {6789, 25079}, {10543, 15792}, {10950, 17104}, {11657, 13408}, {11809, 16332}, {12019, 24624}, {12690, 19642}, {18180, 31849}, {18653, 28160}, {25533, 34123}
X(3109) = orthogonal projection of X(1) on Euler line
X(3109) = circumcircle-inverse of X(859)
X(3109) = Conway-circle-inverse of X(5214)
X(3109) = antigonal conjugate of X(38950)
X(3109) = polar-circle-inverse of X(860)
X(3109) = orthoptic-circle-of-Steiner-inellipse-inverse of X(8229)
X(3109) = intersection, other than X(3), of the Euler line and circle O(1,3)
X(3109) = X(110)-of-X(1)-Brocard triangle
X(3109) = crossdifference of every pair of points on line {647, 2245}
X(3109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 29, 14010}, {21, 3658, 3}, {29, 7452, 4}, {423, 448, 1375}, {1113, 1114, 859}, {7424, 7478, 1325}, {11101, 13746, 5}
X(3110) lies on the circle O(1,3) and these lines: 1, 512 3,6 21,1083 60,249 691,840 741,1015 813,1500 1362,1414
X(3110) = inverse-in-circumcircle of X(3286)
X(3110) = crossdifference of every pair of points on the line X(523)X(2238)
X(3110) = X(99) of X(1)-Brocard triangle
Let A'B'C' be the medial triangle. Let A" be the reflection of X(115) in line B'C', and define B" and C" cyclically. X(3111) is the centroid of (degenerate) triangle A"B"C".
X(3111) lies on these lines: 2, 512 3,6 373,1316
X(3111) = X(99)-of-X(2)-Brocard triangle
X(3111) = Brocard axis intercept, other than X(3), of circle O(2,3)
T and G Mappings
This section presents points X(3112) to X(3118) as representative of points introduced by Quang Tuan Bui in Hyacinthos, August 5, 2006.
Suppose X is a point not on a sideline of ABC. Let
gX = isogonal conjugate of X
tX = isotomic conjugate of X
tgX = isotomic conjugate of gX
gtX = isogonal conjugate of tX
Gt = intersection of lines X(tX) and (gX)(gtX)
Tg = intersection of lines X(gX) and (tX)(tgX)
Then the points A, B, C, gX, tX, Tg, Gt are on a conic. As a circumconic, it is the image under the isogonal conjugate mapping of line X(gtX). It is also the image under the isotomic conjugate mapping of line X(tgX).
If X = x : y : z (trilinears), then
Gt = (b2 - c2)/(x(b2y2 - c2z2)) : (c2 - a2)/(y(c2z2 - a2x2)) : (a2 - b2)/(z(a2x2 - b2y2))Tg = (b2 - c2)/(x(y2 - z2)) : (c2 - a2)/(y(z2 - x2)) : (a2 - b2)/(z(x2 - y2))
X(3112) lies on these lines: 1,561 31,75 38,799 42,308 83,213 92,1973 689,741 1402,1441
X(3112) = isogonal conjugate of X(1964)
X(3112) = isotomic conjugate of X(38)
X(3112) = complement of X(21217)
X(3112) = anticomplement of X(16587)
X(3112) = cevapoint of X(1) and X(75)
X(3112) = X(i)-cross conjugate of X(j) for these (i,j): (1,82),
(661,799), (1215,2), (1580,1821)
X(3112) = crosspoint of X(1) and X(75) wrt the excentral triangle
X(3112) = trilinear pole of line X(798)X(812)
X(3113) lies on these lines: 1, 1927 31,561 75,560 313,983 321,2205 870,1441
X(3113) = isogonal conjugate of X(3116)
X(3113) = trilinear pole of line X(1577)X(1924)
Let A'B'C' be the 3rd Brocard triangle. Let La be the line through A' parallel to BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(3114). (Randy Hutson, August 29, 2018)
X(3114) lies on these lines: 6,706 32,76 183,327 264,419 313,983 870,1215
X(3114) = isogonal conjugate of X(3117)
X(3114) = isotomic conjugate of X(3094)
X(3114) = trilinear pole of line X(669)X(804)
X(3115) lies on these lines: 32,710 76,1501
X(3115) = isogonal conjugate of X(3118)
X(3116) lies on these lines: 1,1581 31,48 38,75 63,1740 99,723 210,1575 982,2887 1932,2172
X(3116) = isogonal conjugate of X(3113)
X(3116) = crosspoint of X(75) and X(2186)
X(3116) = crossdifference of every pair of points on the line
X(1577)X(1924)
Let P1' and U1' be the Steiner-circumellipse-inverses of P(1) and U(1), resp. Then X(3117) = P(1)U1'∩U(1)P1'. (Randy Hutson, January 17, 2020)
X(3117) lies on these lines: 2,39 3,695 6,694 32,184 99,707 110,737 187,353 982,2275 1334,3009
X(3117) = isogonal conjugate of X(3114)
X(3117) = crossdifference of every pair of points on line X(669)X(804)
X(3118) lies on these lines: 32,184 76,141 83,694 99,695 2076,2916
X(3118) = isogonal conjugate of X(3115)
Danneels Perspectors
Suppose A1B1C1 is the cevian triangle
of a point X.
Let LAB be the line through B parallel to
A1B1, and let LAC be the line through
C parallel to A1C1.
Let A2 = LAB∩LAC, and define
B2 and C2 cyclically. Let
A3 = BB2∩CC2, B3 = CC2∩AA2, C3 = AA2∩BB2.
Eric Danneels proves in "A Simple Perspectivity," Forum Geometricorum 6 (2006) 199-203, that the triangles A3B3C3 and ABC are perspective. If X = x : y : z (barycentrics), then the Danneels perspector P(X) is given by
P(X) = x(y - z)2 : y(z - x)2 : z(x - y)2.If X' denotes the isotomic conjugate of X, then P(X') = P(X). A few examples selected from Danneels's article follow:
P(X(1)) = P(X(75)) = X(244)
P(X(3)) = P(X(264)) = X(2972)
P(X(4)) = P(X(69)) = X(125)
P(X(7)) = P(X(8)) = X(11).
X(3119) lies on these lines: 9,100 11,1146 125,661 281,2171 282,604 1200,1864
X(3119) = complement of X(35312)
X(3119) = X(i)-Ceva conjugate of X(j) for these (i,j): (85,522), (282,663), (1146,2310)
X(3119) = X(3022)-cross conjugate of X(2310)
X(3119) = crosspoint of X(i) and X(j) for these (i,j): (33,650), (85,522), (514,3062)
X(3119) = crosssum of X(i) and X(j) for these (i,j): (41,109), (57,934), (77,651), (101,165), (269,1461)
X(3119) = crossdifference of every pair of points on the line X(109)X(934)
X(3120) lies on these lines: 1,149 2,846 7,2648 11,244 27,2206 31,1836 38,2886 42,226 58,79 109,2006 115,125 225,1042 321,2887 334,1978 442,2292 516,902 614,990 726,3006 851,1284 899,908 946,1201 986,2476 1109,2632 1365,2611 1834,2650 2170,2969
X(3120) = reflection of X(902) in X(3011)
X(3120) = X(i)-Ceva conjugate of X(j) for these (i,j): (10,523), (27,649), (79,513), (86,514), (226,661), (313,1577), (1086,3125)
X(3120) = cevapoint of X(115) and X(2643)
X(3120) = X(2643)-cross conjugate of X(3125)
X(3120) = crosspoint of X(i) and X(j) for these (i,j): (10,523), (65,513), (86,514), (313,1577), (321,693), (1086,1111)
X(3120) = crosssum of X(i) and X(j) for these (i,j): (21,100), (42,101), (58,110), (163,2206), (692,1333), (1110,1252)
X(3120) = crossdifference of every pair of points on the line
X(101)X(110)
X(3120) = isogonal conjugate of X(4570)
X(3120) = isotomic conjugate of X(4600)
X(3120) = intersection of tangents to Steiner inellipse at X(115) and X(1086)
X(3120) = crosspoint wrt medial triangle of X(115) and X(1086)
X(3120) = complement of X(4427)
X(3120) = X(10)-isoconjugate of X(1101)
X(3120) = barycentric product X(i)*X(j) for these {i,j}: {10,1086}, {514,423}
X(3121) lies on these lines: 81,893 100,292 244,665 351,865 890,1977 1402,2205
X(3121) = isogonal conjugate of X(4601)
X(3121) = X(i)-Ceva conjugate of X(j) for these (i,j): (37,512), (213,798), (274,513), (893,649), (1015,3122), (1333,667), (1402,669), (2203,1980)
X(3121) = crosspoint of X(i) and X(j) for these (i,j): (37,512), (42,649), (213,798), (274,513), (667,1333)
X(3121) = crosssum of X(i) and X(j) for these (i,j): (81,99), (86,190), (100,213), (274,799), (314,645), (321,668)
X(3121) = crossdifference of every pair of points on the line X(99)X(100)
X(3121) = intersection of tangents to Steiner inellipse at X(1015) and X(1084)
X(3121) = crosspoint wrt medial triangle of X(1015) and X(1084)
X(3122) lies on these lines: 6,2054 11,244 86,256 190,291 351,865 674,3009 1015,1960 1400,1918 1929,2640 2092,2667 2110,2277 2642,2643
X(3122) = X(i)-Ceva conjugate of X(j) for these (i,j): (10,661), (42,512), (58,649), (244,3125), (256,513), (310,514), (1015,3121), (1400,798), (1474,1919)
X(3122) = X(3124)-cross conjugate of X(3125)
X(3122) = crosspoint of X(i) and X(j) for these (i,j): (10,661), (37,513), (42,512), (58,649), (244,1015), (310,514), (667,1402)
X(3122) = crosssum of X(i) and X(j) for these (i,j): (10,190), (58,662), (81,100), (86,99), (101,1918), (306,1331), (314,668), (333,643), (645,1043), (765,1016), (1332,1792)
X(3122) = isogonal conjugate of X(4600)
X(3122) = intersection of tangents to Steiner inellipse at X(1084) and X(1086)
X(3122) = crosspoint wrt medial triangle of X(1084) and X(1086)
X(3122) = crossdifference of every pair of points on the line X(99)X(101)
X(3123) lies on these lines: 11,244 31,1633 38,1227 350,2227 536,2228 661,1084 1423,2209
X(3123) = isotomic conjugate of isogonal conjugate of X(38986)
X(3123) = X(i)-Ceva conjugate of X(i) and X(j) for these (i,j): (76,661), (1015,244)
X(3123) = crosspoint of X(75) and X(513)
X(3123) = crosssum of X(i) and X(j) for these (i,j): (31,100), (87,932), (101,2209)
X(3123) = crossdifference of every pair of points on the line X(101,932)
X(3124) lies on the Brocard inellipse and these lines: 2,694 6,110 23,1691 25,1501 32,3457 39,373 42,2054 51,1196 115,125 237,2021 244,661 351,865 394,2987 593,2248 1193,2653 1495,1692 1506,3118 1613,3060 1994,2056 2092,3030 2501,2970 2679,3005
X(3124) = isogonal conjugate of X(4590)
X(3124) = isotomic conjugate of X(34537)
X(3124) = complement of X(4576)
X(3124) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3005), (6,512), (26,669), (76,523), (83,2514), (111,351), (755,2872), (2207,2489), (2248,649), (3125,2643)
X(3124) = X(i)-cross conjugate of X(j) for these (i,j): (1084,2971), (2679,2086)
X(3124) = crosspoint of X(i) and X(j) for these (i,j): (6,512), (25,2501), (76,523), (647,2351), (661,756), (2207,2489),
(3122,3125)
X(3124) = crosssum of X(i) and X(j) for these (i,j): (2,99), (6,1634), (32,110), (112,459), (317,648), (662,757), (6189,6190)
X(3124) = crosssum of circumcircle intercepts of line X(2)X(32)
X(3124) = X(i)-line conjugate of X(j) for these (i,j): 6,110 111,110
X(3124) = crossdifference of every pair of points on the line X(99)X(110)
X(3124) = perspector of circumconic centered at X(3005)
X(3124) = center of circumconic that is locus of trilinear poles of lines passing through X(3005) (the hyperbola {{A,B,C,X(6),X(39)}})
X(3124) = trilinear pole wrt symmedial triangle of Brocard axis
X(3124) = intersection of tangents to Steiner inellipse at X(115) and X(1084)
X(3124) = crosspoint wrt medial triangle of X(115) and X(1084)
X(3124) = bicentric difference of PU(105)
X(3124) = PU(105)-harmonic conjugate of X(351)
X(3124) = pole of de Longchamps line wrt Kiepert hyperbola
X(3124) = perspector of unary cofactor triangles of 1st and 2nd Brocard triangles
X(3124) = crosspoint of X(5639) and X(5638)
X(3124) = barycentric square of X(661)
X(3124) = polar conjugate of isotomic conjugate of X(20975)
X(3124) = X(63)-isoconjugate of X(18020)
X(3125) lies on these lines: 1,1929 6,1718 10,762 37,1018 38,1573 65,213 115,125 244,665 257,274 335,668 614,1572 758,2238 910,2251 918,1086 1333,2160 1411,1415 1575,1739 2092,2294 2642,2643 3271,4560
X(3125) = isogonal conjugate of X(4567)
X(3125) = isotomic conjugate of X(4601)
X(3125) = X(i)-Ceva conjugate of X(j) for these (i,j): (28,667), (37,661), (65,512), (81,513), (244,3122), (257,514), (274,2530), (321,523), (1086,3120), (2160,649)
X(3125) = cevapoint of X(2643) and X(3124)
X(3125) = X(i)-cross conjugate of X(j) for these (i,j): (2643,3120), (3124,3122)
X(3125) = crosspoint of X(i) and X(j) for these (i,j): (10,514), (37,661), (81,513), (244,1086), (321,523), (649,1400), (1880,2501)
X(3125) = crosssum of X(i) and X(j) for these (i,j): (37,100), (58,101), (72,906), (81,662), (110,1333), (190,333), (643,2287), (692,2205), (765,1252), (1331,2327)
X(3125) = intersection of tangents to Steiner inellipse at X(115) and X(1015)
X(3125) = crosspoint wrt medial triangle of X(115) and X(1015)
X(3125) = trilinear product of extraversions of X(37)
X(3125) = crossdifference of every pair of points on the line X(100)X(110)
X(3125) = polar conjugate of isotomic conjugate of X(18210)
X(3125) = X(63)-isoconjugate of X(5379)
X(3126) lies on these lines: 1,905 2,885 3,667 9,513 10,514 119,120 142,522 442,1577 650,1027 656,2092 665,1642 1025,1026
X(3126) = center of hyperbola {{A,B,C,X(11),X(100),X(650),X(693)}}
X(3127) lies on these lines: 2,3 1162,1165 1587,1899 1853,3070
X(3127) = X(4)-Ceva conjugate of X(1162)
As a point on the Euler line, X(3128) has Shinagawa coefficients (F,E + F - S).
X(3128) lies on these lines: 2,3 1163,1164 1588,1899 1853,3071
X(3128) = X(4)-Ceva conjugate of X(1163)
As a point on the Euler line, X(3129) has Shinagawa coefficients (31/2F - S, -31/2E - 31/2F + S).
X(3129) lies on these lines: 2,3 6,3458 13,1605 15,1495 51,62 61,184 619,2926 1251,1953
X(3129) = isogonal conjugate of X(2992)
X(3129) = circumcircle-inverse of X(32461)
X(3129) = X(13)-Ceva conjugate of X(6)
X(3129) = pole wrt circumcircle of trilinear polar of X(13) (line X(395)X(523))
X(3129) = crosspoint of circumcircle intercepts of inner Napoleon circle
As a point on the Euler line, X(3130) has Shinagawa coefficients (31/2F + S, -31/2E - 31/2F - S).
X(3130) lies on these lines: 2,3 6,3457 14,1606 16,1495 51,61 62,184 618,2925
X(3130) = isogonal conjugate of X(2993)
X(3130) = circumcircle-inverse of X(32460)
X(3130) = X(14)-Ceva conjugate of X(6)
X(3130) = pole wrt circumcircle of trilinear polar of X(14) (line X(396)X(523))
X(3130) = crosspoint of circumcircle intercepts of outer Napoleon circle
As a point on the Euler line, X(3131) has Shinagawa coefficients (F - 31/2S, -E - F + 31/2S).
X(3131) lies on these lines: 2,3 15,184 16,51 17,1607
X(3131) = X(17)-Ceva conjugate of X(6)
X(3131) = circumcircle-inverse of X(37974)
X(3131) = {X(2),X(37457)}-harmonic conjugate of X(3132)
X(3131) = {X(3),X(25)}-harmonic conjugate of X(3132)
As a point on the Euler line, X(3132) has Shinagawa coefficients (F + 31/2S, -E - F - 31/2S).
X(3132) lies on these lines: 2,3 15,51 16,184 18,1608
X(3132) = X(18)-Ceva conjugate of X(6)
X(3132) = circumcircle-inverse of X(37975)
X(3132) = {X(2),X(37457)}-harmonic conjugate of X(3131)
X(3132) = {X(3),X(25)}-harmonic conjugate of X(3131)
As a point on the Euler line, X(3133) has Shinagawa coefficients ((E + 2F)(E - 2F) - 4S2,(E + 2F)2 + 4S2).
X(3133) lies on these lines: 2,3 571,1147
X(3133) = X(24)-Ceva conjugate of X(52)
As a point on the Euler line, X(3134) has Shinagawa coefficients ((7E - 2F)F - 2S2,(E + F)(E - 2F) - 2S2).
X(3134) lies on these lines: 2,3 122.136 125,526 2970,2972
X(3134) = inverse-in-nine-point circle of X(3154)
X(3134) = X(74)-Ceva conjugate of X(523)
X(3134) = crosspoint of X(264) and X(2394)
X(3134) = crosssum of X(184) and X(2420)
As a point on the Euler line, X(3135) has Shinagawa coefficients ((E + 2F)F + 2S2, -(E + F)(E + 2F) - 2S2).
X(3135) lies on these lines: 2,3 51,570 160,184 161,2351
X(3135) = X(96)-Ceva conjugate of X(6)
X(3135) = crosssum of X(338) and X(924)
As a point on the Euler line, X(3136) has Shinagawa coefficients ($aSASB$-$aSCSA$ +$aSC$F-$aSB$F, $aSC2$ -$aSB2$).
X(3136) lies on these lines: 2,3 12,42 71,1213 118,125 310,325 672,1901 1211,2886 1214,1893 1230,3006 1836,2245 2238,2911
X(3136) = inverse-in-orthocentroidal circle of X(1011)
X(3136) = X(102)-Ceva conjugate of X(523)
X(3136) = complement of X(4148)
X(3136) = crosspoint of X(10) and X(264)
X(3136) = crosssum of X(58) and X(184)
As a point on the Euler line, X(3137) has Shinagawa coefficients ($aSA6$ -$aSA3$[(E+F)3-3ES2] +$aSBSC$(E+F)FS2-$a$F2S4, -$aSA4$S2 +$aSA3$(E+F)S2+$aSA$FS4 -$a$(E+F)FS4).
X(3137) lies on these lines: 2,3 124,125
X(3137) = X(102)-Ceva conjugate of X(523)
As a point on the Euler line, X(3138) has Shinagawa coefficients ($aSA5$ -$aSA2$[(E+F)3-3ES2] +$aSBSC$ES2 +$aSA$(E+F)FS2-$a$(E+F)2FS2, -$aSA3$S2 +$aSA2$(E+F)S2 -$aSBSC$(E+F)S2+$a$FS4).
X(3138) lies on these lines: 2,3 116,125 2972,2973
X(3138) = X(103)-Ceva conjugate of X(523)
As a point on the Euler line, X(3139) has Shinagawa coefficients (2$aSBSC$+3$aSA$F-$a$S2, $aSA$(E+F)-$a$S2).
X(3139) lies on these lines: 2,3 11,125 123,136 2968,2970 2969,2972
X(3139) = X(104)-Ceva conjugate of X(523)
As a point on the Euler line, X(3140) has Shinagawa coefficients ($abSC3$+$abSASB$(E+F) -$abSC$[(E+F)2-2S2]-$ab$FS2, 3$abSC$S2-$ab$(E+F)S2).
X(3140) lies on these lines: 2,3 11,115 125,2775 339,2969 2968,2971
X(3140) = X(105)-Ceva conjugate of X(523)
X(3140) = nine-point-circle-inverse of X(37986)
As a point on the Euler line, X(3141) has Shinagawa coefficients ($aSA4$ -$aSA2$[(E+F)2-2S2] +2$aSBSC$S2 +2$aSA$FS2-2$a$(E+F)FS2, 3$aSA2$S2 -$a$[(E+F)2-2S2]S2).
X(3141) lies on these lines: 2,3 125,2776
X(3141) = X(106)-Ceva conjugate of X(523)
As a point on the Euler line, X(3142) has Shinagawa coefficients ($aSA2SB$ -$aSCSA2$-$aSASB$F +$aSCSA$F, $aSB2SC$ -$aSBSC2$-$aSB$S2 +$aSC$S2).
X(3142) lies on these lines: 2,3 11,1193 12,73 117,125 226,1425 1211,1329 1213,2183 1214,1867 1400,1901
X(3142) = X(109)-Ceva conjugate of X(523)
X(3142) = crosspoint of X(226) and X(264)
X(3142) = crosssum of X(184) and X(284)
As a point on the Euler line, X(3143) has Shinagawa coefficients (2(E + F)2F - (E - 2F)S2,2(E + F)3 - (7E - 2F)S2).
X(3143) lies on these lines: 2,3 115,804 125,2780 339,2971
X(3143) = X(111)-Ceva conjugate of X(523)
X(3143) = crossdifference of every pair of points on the line
X(647)X(1634)
As a point on the Euler line, X(3144) has Shinagawa coefficients (2FS2, -ES2 - $bcSBSC$).
X(3144) lies on these lines: 2,3 34,978 158,2752 225,1247 242,1838 1068,1148 1193,1870 1714,3186
X(3144) = X(i)-Ceva conjugate of X(j) for these (i,j): (225,4),
(1940,1148), (2907,1046)
X(3144) = X(2907)-cross conjugate of X(4)
Let P be a point in the plane of ABC and not on the circumcircle. Let (OA) be the circle tangent to the circumcircle at A and passing through P. Let A' be the antipode of A in (OA). Let LA be the tangent to (OA) at A'. Define LB and LC cyclically. Let TA = LB∩LC, and define TB, TC cyclically. Triangle TATBTC is homothetic to the tangential triangle. When P = X(1), the center of homothety is X(3145). (Randy Hutson, June 7, 2019)
X(3145) lies on these lines: 1,1283 2,3 35,228 41,2276 45,198 51,580 55,976 56,1626 109,1425 184,581 283,511 500,1437 692,2594 1284,1486 1400,1950 1495,2360
X(3145) = X(226)-Ceva conjugate of X(6)
X(3145) = crosspoint of X(109) and X(250)
X(3145) = crosssum of X(125) and X(522)
X(3145) = tangential isogonal conjugate of X(23361)
X(3145) = orthic-to-tangential similarity image of X(407)
As a point on the Euler line, X(3146) has Shinagawa coefficients (1, -4).
The point X(3146) exemplifies a theorem based on a discussion in Tran Quang Hung and Peter Moses, Hyacinthos 25668. The theorem (Peter Moses, April 10, 2017) is as follows.
Suppose that P is a point on a circumhyperbola H. Let A'B'C' be the cevian triangle of P and let O' be the circumcircle of A'B'C'. Let A'' be the point other than A' that lies on both O' and line BC, and define B'' and C'' cyclically. The orthocenter of A"B"C" lies on the isogonal conjugate of H.
Writing P' for the orthocenter of of A"B"C", examples of the mapping P → P' include the following:
Jerabek hyperbola → Euler line
Feuerbach hyperbola → X(1)X(3)
Kiepert hyperbola → Brocard axis.
The image of a point P = a^2 / (S^2+k SB SC) : : on the Jerabek hyperbola is the following point on the Euler line:
(S^2 SA SB+S^2 SB^2+S^2 SA SC+4 S^2 SB SC+3 k SA SB^2 SC+S^2 SC^2+3 k SA SB SC^2+2 k SB^2 SC^2) (S^4 SA^3+S^4 SA^2 SB+S^4 SA SB^2-2 k S^2 SA^3 SB^2-k^2 SA^4 SB^3+S^4 SA^2 SC+7 S^4 SA SB SC-2 k S^2 SA^3 SB SC+2 S^4 SB^2 SC+8 k S^2 SA^2 SB^2 SC-3 k^2 SA^4 SB^2 SC+2 k S^2 SA SB^3 SC+k^2 SA^3 SB^3 SC+S^4 SA SC^2-2 k S^2 SA^3 SC^2+2 S^4 SB SC^2+8 k S^2 SA^2 SB SC^2-3 k^2 SA^4 SB SC^2+14 k S^2 SA SB^2 SC^2+3 k^2 SA^3 SB^2 SC^2+2 k S^2 SB^3 SC^2+7 k^2 SA^2 SB^3 SC^2-k^2 SA^4 SC^3+2 k S^2 SA SB SC^3+k^2 SA^3 SB SC^3+2 k S^2 SB^2 SC^3+7 k^2 SA^2 SB^2 SC^3+5 k^2 SA SB^3 SC^3)
For P = X(69), the parameter k = = -SW S^2 / (SA SB SC), and P' = X(3146)
For P = X(68), see X(13322).
X(3146) lies on the cubics K117, K127, K347, K425, K461, K558, K841, and these lines: {1,5556}, {2,3}, {7,950}, {8,144}, {10,9778}, {11,5265}, {12,5281}, {13,5366}, {14,5365}, {33,4296}, {34,3100}, {35,10590}, {36,10591}, {40,3219}, {51,10574}, {52,6241}, {55,5229}, {56,5225}, {61,5335}, {62,5334}, {63,5175}, {65,9961}, {69,11469}, {74,12295}, {78,1750}, {84,3218}, {107,5896}, {145,515}, {146,6193}, {147,7900}, {148,2794}, {149,2829}, {150,10725}, {151,10726}, {152,10727}, {153,5840}, {154,5893}, {165,9780}, {185,3060}, {193,1503}, {226,4313}, {265,12244}, {279,4872}, {280,5081}, {315,10513}, {316,3926}, {317,6527}, {323,11441}, {329,3984}, {346,7270}, {347,7282}, {355,4678}, {385,2996}, {388,390}, {389,11002}, {393,3284}, {485,6453}, {486,6454}, {489,1271}, {490,1270}, {497,3304}, {498,4324}, {499,4316}, {511,5921}, {517,3621}, {519,9589}, {527,12625}, {528,12632}, {542,8596}, {543,7758}, {576,6776}, {578,11003}, {671,10991}, {938,3586}, {942,11220}, {944,3623}, {946,3622}, {971,3868}, {990,5262}, {1056,9655}, {1058,9668}, {1105,1629}, {1125,9779}, {1131,3068}, {1132,3069}, {1151,2671}, {1152,2672}, {1173,4846}, {1181,1994}, {1219,4514}, {1285,5305}, {1327,8960}, {1350,3620}, {1351,12174}, {1352,7929}, {1376,8165}, {1467,8544}, {1478,3746}, {1479,4293}, {1493,12254}, {1495,11449}, {1498,1993}, {1539,12121}, {1587,6419}, {1588,6420}, {1614,9545}, {1697,8545}, {1698,12512}, {1699,3616}, {1737,4333}, {1836,3486}, {1837,3474}, {1853,5894}, {1870,9538}, {1891,4329}, {1968,10313}, {2548,7756}, {2549,7747}, {2777,3448}, {2883,11206}, {2979,5907}, {3047,9934}, {3055,11742}, {3058,9657}, {3070,3592}, {3071,3594}, {3085,3585}, {3086,3583}, {3087,5158}, {3101,11471}, {3180,5869}, {3181,5868}, {3241,4301}, {3306,9841}, {3329,5395}, {3346,10152}, {3431,5944}, {3436,6253}, {3476,12701}, {3488,11036}, {3579,5818}, {3580,5925}, {3590,10147}, {3591,10148}, {3601,5226}, {3618,10541}, {3624,12571}, {3672,5716}, {3679,5493}, {3681,7957}, {3785,7802}, {3812,5918}, {3817,5550}, {3818,10519}, {3869,12688}, {3870,12651}, {3873,12680}, {3876,5927}, {3877,9856}, {3935,6769}, {4295,10572}, {4298,10580}, {4304,5703}, {4305,12047}, {4308,12053}, {4309,5270}, {4311,9614}, {4312,6738}, {4314,5290}, {4317,4857}, {4325,10072}, {4330,10056}, {4355,6744}, {4440,11851}, {4855,5748}, {4907,7273}, {5007,5286}, {5012,11424}, {5032,8550}, {5057,12679}, {5080,5537}, {5128,7285}, {5204,10589}, {5217,10588}, {5218,10895}, {5250,11372}, {5251,12511}, {5254,5304}, {5260,5584}, {5284,8273}, {5302,7964}, {5319,11648}, {5328,5438}, {5434,9670}, {5435,9581}, {5446,5890}, {5480,7864}, {5596,9968}, {5609,7728}, {5640,9729}, {5643,9815}, {5656,9716}, {5658,11015}, {5663,6243}, {5698,5794}, {5734,5882}, {5763,9963}, {5787,12246}, {5795,11530}, {5806,10167}, {5842,12667}, {5870,6280}, {5871,6279}, {5889,6000}, {6054,10992}, {6194,6248}, {6256,10528}, {6321,9862}, {6337,7773}, {6427,7581}, {6428,7582}, {6482,12818}, {6483,12819}, {6515,12324}, {6519,8981}, {6564,9540}, {6759,9544}, {6800,11425}, {7288,10896}, {7710,7783}, {7712,11430}, {7738,7745}, {7785,8721}, {7787,12203}, {7795,7842}, {7928,10516}, {7989,10164}, {8117,8122}, {8118,8121}, {9140,10990}, {9580,9785}, {9613,10624}, {9730,9781}, {9748,12110}, {9799,9965}, {10404,11038}, {10446,10454}, {10453,12545}, {10625,11459}, {10711,10993}, {10738,12248}, {10749,12253}, {11004,11456}, {11411,12293}, {11412,11455}, {11416,11470}, {11417,11473}, {11418,11474}, {11420,11475}, {11421,11476}, {11432,11820}, {11523,12536}, {11750,12897}, {12160,12315}, {12219,12292}, {12220,12294}, {12223,12298}, {12224,12299}, {12226,12300}, {12282,12290}, {12317,12902}, {12684,12690}
X(3146) = midpoint of X(i) in X(j) for these {i,j}: {382,5073}, {3529,11541}
X(3146) = reflection of X(i) in X(j) for these (i,j): (2,3543), (3,3627), (4,382), (8,5691), (20,4), (74,12295), (145,962), (146,10721), (147,10722), (148,10723), (149,10724), (150,10725), (151,10726), (152,10727), (153,10728), (376,3830), (550,3853), (944,12699), (1657,5), (3448,10733), (3529,3), (3869,12688), (5059,20), (5189,10296), (5925,6247), (5984,148), (6225,5895), (6240,7553), (6241,52), (6361,355), (9862,6321), (9961,65), (10430,3586), (10575,5446), (11001,381), (11411,12293), (11412,12162), (11750,12897), (12103,12102), (12111,11381), (12121,1539), (12219,12292), (12220,12294), (12221,12296), (12222,12297), (12223,12298), (12224,12299), (12225,1885), (12226,12300), (12244,265), (12246,5787), (12248,10738), (12253,10749), (12279,185), (12317,12902), (12383,7728), (12536,11523)
X(3146) = isogonal conjugate of X(3532)
X(3146) = isotomic conjugate of X(35510)
X(3146) = isotomic conjugate of polar conjugate of X(33630)
X(3146) = complement of X(5059)
X(3146) = anticomplement of X(20)
X(3146) = X(253)-Ceva conjugate of X(2)
X(3146) = crosspoint of X(1131) and X(1132)
X(3146) = crosspoint of PU(i) for these i: 116, 117
X(3146) = crosssum of X(1151) and X(1152)
X(3146) = orthocentroidal-circle-inverse of X(3832)
X(3146) = intersection of tangents to Kiepert hyperbola at X(1131) and X(1132)
X(3146) = orthocenter of cevian triangle of X(253)
X(3146) = X(11531)-of-orthic-triangle if ABC is acute
X(3146) = polar conjugate of X(38253)
X(3146) = pole wrt polar circle of trilinear polar of X(38253) (line X(523)X(13473))
X(3146) = circumcircle-inverse of X(37941)
X(3146) = X(19)-isoconjugate of X(36609)
X(3146) = exsimilicenter of polar circle and de Longchamps circle
X(3146) = Ehrmann-mid-to-Johnson similarity image of X(382)
X(3146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,3832), (2,20,3522), (2,3832,5068), (2,3854,5), (2,5059,20), (2,6872,11106), (3,4,3091), (3,5,3525), (3,381,3628), (3,382,3627), (3,546,3090), (3,1657,12103), (3,3090,10303), (3,3091,2), (3,3525,3523), (3,3529,20), (3,3627,4), (3,3628,631), (3,3843,5079), (3,5072,632), (3,5076,546), (3,5079,140), (3,5198,1995), (3,7530,3518), (3,12103,376), (4,5,3839), (4,20,2), (4,21,6870), (4,24,6623), (4,376,5), (4,377,6894), (4,382,3543), (4,411,6871), (4,550,5056), (4,631,381), (4,1593,7378), (4,1657,3523), (4,3090,546), (4,3522,5068), (4,3523,3854), (4,3524,3855), (4,3528,3545), (4,3529,3), (4,3533,3858), (4,3534,7486), (4,3545,3843), (4,3575,6995), (4,3651,6843), (4,3855,3845), (4,5059,3522), (4,6240,7487), (4,6622,10151), (4,6815,7394), (4,6836,5046), (4,6850,6839), (4,6851,6840), (4,6865,6957), (4,6868,6837), (4,6869,6838), (4,6875,6866), (4,6876,6867), (4,6897,6849), (4,6899,6893), (4,6903,6929), (4,6906,6844), (4,6909,5187), (4,6916,6835), (4,6925,2475), (4,6934,6848), (4,6938,6847), (4,6942,6968), (4,6948,6953), (4,6988,7548), (4,7580,5177), (4,7714,1906), (4,10299,3850), (4,10431,6895), (4,10996,6997), (4,11001,631), (4,11111,10883), (4,11541,3529), (5,376,3523), (5,550,12100), (5,1657,376), (5,3523,2), (5,3627,12102), (5,3830,4), (5,3839,3854), (5,10124,1656), (5,12103,3), (20,1559,2060), (20,3091,3), (20,3523,376), (20,3543,4), (20,3830,3854), (20,3839,3523), (20,4208,7411), (20,5056,10304), (20,7396,11413), (20,7486,3528), (20,10304,550), (21,5177,2), (22,7378,2), (23,12086,3), (24,12085,2071), (25,7396,2), (26,3520,10298), (55,5229,5261), (56,5225,5274), (140,3534,3528), (140,3545,7486), (140,3843,3545), (140,3857,5079), (140,3860,5), (140,7486,2), (376,1657,20), (376,3525,3), (376,3830,3839), (376,3839,2), (376,12100,10304), (376,12102,3091), (377,452,2), (377,11114,452), (378,7387,7488), (381,550,631), (381,631,5056), (381,3853,4), (381,10304,2), (381,11001,10304), (382,1657,3830), (382,11541,3091), (388,6284,390), (404,6919,2), (405,4208,2), (411,1012,4189), (427,10565,2), (428,7386,7398), (443,5129,2), (443,11113,5129), (489,12323,1271), (490,12322,1270), (497,7354,3600), (546,632,5072), (546,3090,3091), (546,3627,5076), (546,5076,4), (546,12103,12108), (546,12108,5), (548,1656,3524), (548,3845,1656), (549,3851,5067), (549,3861,3851), (550,631,10304), (550,3628,3), (550,3853,381), (550,11001,20), (631,5056,2), (631,11001,550), (631,12100,3523), (632,5072,3090), (858,4232,2), (946,5731,3622), (950,9579,7), (962,6223,5905), (1006,6993,2), (1327,9681,8960), (1370,6995,2), (1479,10483,4293), (1532,6943,5154), (1597,11414,7503), (1656,3845,3855), (1657,3830,5), (1657,3854,3522), (1657,3860,3528), (1657,12102,3525), (1699,4297,3616), (1885,12173,4), (1907,6823,5133), (2043,2044,3524), (2071,6623,2), (2475,6872,2), (2478,6904,2), (2979,11439,5907), (3060,12279,185), (3070,6459,7585), (3071,6460,7586), (3090,10303,2), (3091,10303,3090), (3091,11541,5059), (3149,6909,4188), (3151,6994,2), (3152,7518,2), (3515,10151,6622), (3518,7464,3), (3522,3832,2), (3523,3839,5), (3523,10303,12108), (3524,3855,1656), (3525,3529,12103), (3525,12102,3839), (3525,12108,10303), (3526,3850,5071), (3526,8703,10299), (3528,3545,140), (3528,3843,7486), (3529,3627,3091), (3529,5076,10303), (3529,12102,3523), (3530,3858,5055), (3530,5055,3533), (3530,12101,3858), (3534,3830,3860), (3534,3843,140), (3534,5079,3), (3543,3839,3830), (3543,5056,3853), (3543,5059,3832), (3543,10303,5076), (3545,3857,3091), (3583,4299,3086), (3585,4302,3085), (3586,4292,938), (3616,10248,1699), (3627,5073,11541), (3627,11541,20), (3627,12102,3830), (3817,7987,5550), (3839,3854,3832), (3843,5079,3857), (3850,8703,3526), (3853,11001,5056), (3855,12811,3091), (3857,5079,3545), (4188,5187,2), (4189,6871,2), (4190,5046,2), (4304,9612,5703), (4314,5290,10578), (5004,5005,9909), (5056,10304,631), (5071,10299,3526), (5189,7519,2), (5446,10575,5890), (6039,6040,10011), (6284,12943,388), (6825,6888,2), (6826,6992,2), (6831,6932,5141), (6833,6960,2), (6834,6972,2), (6837,6908,2), (6838,6847,2), (6839,6987,2), (6848,6890,2), (6850,7491,6987), (6868,6923,6908), (6869,6938,20), (6884,6889,2), (6885,6929,6964), (6891,6979,2), (6917,6930,6846), (6926,6953,2), (6928,6948,6926), (6999,7406,2), (7000,7374,1513), (7354,12953,497), (7379,7390,2), (7386,7398,2), (7391,7500,2), (7503,11414,6636), (7517,12084,186), (7526,12083,7512), (7527,12082,7492), (7737,7748,5286), (7802,11185,3785), (9580,10106,9785), (11412,11455,12162), (12102,12103,5), (12102,12108,546), (32614, 32615, 110)
As a point on the Euler line, X(3147) has Shinagawa coefficients (4F, -E - 2F).
X(3147) lies on these lines: 2,3 323,2904 1068,1940 1249,3003
X(3147) = isogonal conjugate of X(38260)
X(3147) = X(254)-Ceva conjugate of X(4)
As a point on the Euler line, X(3148) has Shinagawa coefficients ((E + F )F - S2, -(E + F)2 + S2).
X(3148) lies on these lines: 2,3 6,157 32,51 39,184 206,570 216,1974 574,1495 577,1843 878,1640 1993,2001
X(3148) = inverse-in-orthocentroidal circle of X(2450)
X(3148) = X(262)-Ceva conjugate of X(6)
X(3148) = crossdifference of every pair of points on the line
X(647)X(2799)
As a point on the Euler line, X(3149) has Shinagawa coefficients ($aSA$ - abc, -$aSA$ - abc).
X(3149) lies on these lines: 1,227 2,3 35,1699 40,936 55,946 56,515 57,1071 64,1715 78,517 84,1728 100,962 221,1771 222,1745 282,2270 355,956 573,1437 578,1437 581,940 603,2635 908,1259 938,944 971,1445 1155,1158 1454,1858 1498,1754 1617,3086 1735,1854 1741,1903
X(3149) = X(285)-Ceva conjugate of X(1)
X(3149) = X(21)-gimel conjugate of X(3)
As a point on the Euler line, X(3150) has Shinagawa coefficients (2(E + F)(2E - F)F - (E + 2F)S2, -2(E + F)2F + (E - 2F)S2).
X(3150) lies on these lines: 2,3 115,122 125,127 339,2972 879,2435
X(3150) = X(i)-Ceva conjugate of X(j) for these (i,j): (290,525), (1297,523)
X(3150) = crossdifference of X(i) and X(j) for every pair of points on the line X(647)X(1624)
X(3150) = nine-point-circle-inverse of X(37987)
As a point on the Euler line, X(3151) has Shinagawa coefficients (E + $bc$, -2E - 2F - 2$bc$).
X(3151) lies on these lines: 2,3 63,2893 71,1654 144,2895 152,2822 306,1330
X(3151) = reflection of X(27) in X(440)
X(3151) = isogonal conjugate of X(34440)
X(3151) = complement of X(31292)
X(3151) = anticomplement of X(27)
X(3151) = X(i)-Ceva conjugate of X(j) for these (i,j): (306,2),
(1330,1654)
As a point on the Euler line, X(3152) has Shinagawa coefficients ((E + F + $bc$)$aSBSC$ + ($a$F + $(b+c)$)S2, 2$a$(E + F + $bc$)S2).
X(3152) lies on these lines: 2,3 40,151 78,1330 145,347 307,2893 1654,3177 2654,3100
X(3152) = complement of X(31294)
X(3152) = anticomplement of X(29)
X(3152) = X(307)-Ceva conjugate of X(2)
As a point on the Euler line, X(3153) has Shinagawa coefficients (E, -2E - 8F).
X(3153) lies on these lines: 2,3 110,1568 146,1531 265,1154 827,2697 1287,1297
X(3153) = refection of X(i) in X(j) for these (i,j): (20,2071), (23,403), (110,1568), (186,2072), (2070,5), (2071,858)
X(3153) = anticomplement of X(186)
X(3153) = inverse-in-de-Longchamps-circle of X(3)
X(3153) = inverse-in-Johnson-circle of X(5)
X(3153) = homothetic center of dual of orthic triangle and Ehrmann vertex-triangle
X(3153) = X(484)-of-Ehrmann-vertex-triangle
Trilinears u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z), where x : y : z
= X(477) and u : v : w = X(523)
X(3154) = 3 X[2] + X[17511], 3 X[125] - X[6070], X[476] - 5 X[15059], X[477] + 3 X[14644], X[1553] - 3 X[36518], 5 X[1656] - X[36193], 5 X[3091] - X[36172], 3 X[3258] + X[6070], 2 X[3258] + X[12079], 3 X[5627] + 2 X[41522], 2 X[6070] - 3 X[12079], 3 X[9140] + X[14480], X[10733] + 3 X[38701], 2 X[12068] + X[17511], X[13471] + 3 X[14893], 3 X[14644] - X[34150], 11 X[15025] + X[38678], 3 X[15061] + X[20957], 2 X[16340] + X[36169], X[18319] - 3 X[21315], 3 X[23515] - X[25641], X[30714] - 3 X[31378], 5 X[30745] - X[36188], 3 X[34128] - X[38609].
As a point on the Euler line, X(3154) has Shinagawa coefficients (9(E - 2F)F - 2S2, -(E + F)(E + 10 F) + 6S2).
For a construction see Antreas Hatzipolakis and Kadir Altintas, euclid 1791.
X(3154) lies on these lines: {2, 3}, {110, 31945}, {115, 9209}, {122, 16221}, {125, 523}, {136, 13611}, {265, 14934}, {476, 15059}, {477, 14644}, {525, 16280}, {1112, 12052}, {1503, 16319}, {1553, 36518}, {2452, 26869}, {3233, 5972}, {3448, 14611}, {5627, 41522}, {6723, 22104}, {8901, 15470}, {9140, 14480}, {10113, 38610}, {10733, 38701}, {14220, 15453}, {15025, 38678}, {15061, 20957}, {15359, 22264}, {16168, 20304}, {16178, 35968}, {17702, 31379}, {18319, 21315}, {20417, 32417}, {23515, 25641}, {30714, 31378}, {34128, 38609}}.
X(3154) = anticomplement of X(12068)
X(3154) = complement of X(7471)
X(3154) = midpoint of X(i) and X(j) for these {i,j}: {3, 36184}, {4, 36164}, {5, 16340}, {125, 3258}, {265, 14934}, {477, 34150}, {3448, 14611}, {7471, 17511}, {10113, 38610}
X(3154) = reflection of X(i) in X(j) for these {i,j}: {110, 31945}, {1112, 12052}, {3233, 5972}, {7471, 12068}, {12079, 125}, {22104, 6723}, {36169, 5}
X(3154) = nine-point-circle-inverse of X(3134)
X(3154) = polar-circle-inverse of X(4240)
X(3154) = orthoptic-circle-of-Steiner-inellipe-inverse of X(7422)
X(3154) = X(i)-complementary conjugate of X(j) for these (i,j): {656, 42424}, {15453, 10}, {32710, 8062}
X(3154) = X(477)-Ceva conjugate of X(523)
X(3154) = crosssum of X(110) and X(15055)
X(3154) = crossdifference of every pair of point on the line X(647)X(2420)
X(3154) = orthogonal projection of X(125) on Euler line
X(3154) = Euler line intercept of Simpson line of X(74)
X(3154) = barycentric product X(338)*X(15035)
X(3154) = barycentric quotient X(15035)/X(249)
X(3154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 858, 36170}, {2, 7471, 12068}, {2, 17511, 7471}, {2, 36166, 468}, {477, 14644, 34150}, {868, 36189, 14120}, {1312, 1313, 3134}, {1650, 35235, 3134}, {14120, 37987, 868}
As a point on the Euler line, X(3155) has Shinagawa coefficients (F - S, -E - F + S).
X(3155) lies on these lines: 2,3 51,372 154,1151 157,590 184,371 485,2351 493,1976
X(3155) = X(i)-Ceva conjugate of X(j) for these (i,j): (485,6), (2351,3156)
As a point on the Euler line, X(3156) has Shinagawa coefficients (F + S, -E - F - S).
X(3156) lies on these lines: 2,3 51,371 154,1152 157,615 184,372 486,2351 494,1976
X(3156) = X(i)-Ceva conjugate of X(j) for these (i,j): (486,6), (2351,3155)
X(3157) lies on these lines: 1,90 3,73 6,169 12,68 31,1066 40,1419 46,1079 55,500 56,215 65,921 72,394 221,517 495,611 602,1458 916,2293 971,1498 999,1201 1062,1071 1064,1496 1068,3193 1092,1425 1339,1616 1433,1807 1480,3057 1745,1936 1854,2771 1870,1993
\
X(3157) = isogonal conjugate of X(7040)
X(3157) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,3), (3173,3211), (3193,46)
X(3157) = crosspoint of X(i) and X(j) for these (i,j): (1,46), (1800,3193)
X(3157) = crosssum of X(1) and X(90)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3158) lies on these lines: 1,474 9,55 31,678 40,758 41,728 42,1449 57,100 78,1697 145,1420 165,518 528,1699 612,1962 1743,3052 1998,2078 2340,3208
X(3158) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,9), (145,1743)
X(3158) = crosspoint of X(i) and X(j) for these (i,j): (1,1743), (145,3161)
X(3158) = crosssum of X(513) and X(3020)
X(3158) = centroid of antipedal triangle of X(84)
X(3158) = crossdifference of every pair of points on line X(3669)X(4394)
X(3158) = intouch-to-ABC barycentric image of X(2)
X(3158) = {X(1),X(63)}-harmonic conjugate of X(968)
X(3159) lies on these lines: 10,321 37,39 58,190 72,519 192,386
X(3159) = midpoint of X(72) and X(2901)
X(3159) = reflection of X(596) in X(1125)
X(3159) = X(1)-Ceva conjugate of X(10)
X(3159) = crosspoint of X(1) and X(3216)
X(3160) = perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C and has center X(7)
X(3160) lies on these lines: 1,7 3,934 8,348 9,2124 144,1419 220,651 241,2275 738,1697 944,1565 1358,1388
X(3160) = midpoint of X(175) and X(176)
X(3160) = X(2)-Ceva conjugate of X(7)
X(3160) = cevapoint of X(i) and X(j) for these (i,j): (1,2124),(165,1419)
X(3160) = X(165)-cross conjugate of X(144)
X(3160) = crosspoint of X(2) and X(144)
X(3160) = crosssum of X(663) and X(3022)
X(3160) = X(99)-beth conjugate of X(8)
X(3160) = center of circumconic that is locus of trilinear poles of lines passing through X(7)
X(3161) = perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C and has center X(8)
X(3161) lies on these lines: 2,2415 6,644 8,9 37,2275 45,1213 145,1743 190,344 268,1809 329,440 355,537 404,1696
X(3161) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,8), (1222,200)
X(3161) = X(3158)-cross conjugate of X(145)
X(3161) = crosspoint of X(2) and X(145)
X(3161) = crosssum of X(649) and X(1357)
X(3161) = center of circumconic that is locus of trilinear poles of lines passing through X(8); this conic is the isotomic conjugate of the Gergonne line.
X(3161) = complement of X(4373)
X(3161) = anticomplement of X(4859)
X(3162) lies on hyperbola {{X(2),X(6),X(216),X(233),X(1249),X(1560),X(3162)}}. This hyperbola is a circumconic of the medial triangle, and the locus of perspectors of circumconics centered at a point on the Euler line. The hyperbola is tangent to Euler line at X(2). (Randy Hutson, June 7, 2019)
X(3162) lies on these lines: X(3162) lies on the cubics K177 and K1162 and these lines: {2, 2138}, {3, 19615}, {4, 5359}, {6, 66}, {19, 614}, {22, 112}, {24, 1627}, {25, 32}, {132, 19149}, {154, 22075}, {197, 4548}, {204, 612}, {216, 1033}, {230, 8746}, {232, 1609}, {233, 7539}, {305, 648}, {378, 1180}, {394, 15595}, {426, 9475}, {468, 1611}, {607, 40962}, {608, 40961}, {653, 30918}, {1181, 37074}, {1194, 1968}, {1249, 7386}, {1370, 23115}, {1435, 40180}, {1498, 1529}, {1560, 30771}, {1783, 10327}, {1993, 41363}, {2211, 42295}, {3053, 21213}, {3163, 31152}, {5064, 5309}, {5094, 8792}, {5354, 6995}, {5523, 7391}, {6353, 8744}, {6997, 41370}, {7378, 34482}, {7396, 45245}, {7485, 39575}, {7735, 8745}, {8770, 44467}, {8891, 36879}, {9308, 11324}, {15487, 36103}, {16063, 41366}, {17903, 24605}, {21621, 40940}, {22120, 34609}, {23050, 40181}, {26283, 36414}, {28701, 30737}, {31101, 40583}, {34131, 39071}, {35477, 38862}, {39951, 40124}, {40191, 40192}, {40825, 44077}
X(3162) = reflection of X(i) in X(j) for these {i,j}: {25, 42820}, {15255, 46737}
X(3162) = complement of X(13575)
X(3162) = polar conjugate of X(40009)
X(3162) = complement of the isogonal conjugate of X(159)
X(3162) = complement of the isotomic conjugate of X(1370)
X(3162) = isogonal conjugate of the isotomic conjugate of X(41361)
X(3162) = isogonal conjugate of the polar conjugate of X(41766)
X(3162) = polar conjugate of the isotomic conjugate of X(159)
X(3162) = medial-isogonal conjugate of X(23300)
X(3162) = crosspoint of X(i) and X(j) for these (i,j): {2, 1370}, {8793, 40357}, {41361, 41766}
X(3162) = crosssum of X(i) and X(j) for these (i,j): {6, 34207}, {69, 28708}
X(3162) = crossdifference of every pair of points on line {3265, 8673}
X(3162) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 23300}, {6, 36907}, {31, 25}, {48, 14376}, {159, 10}, {1370, 2887}, {3162, 226}, {8793, 1215}, {18596, 141}, {18629, 17046}, {21582, 626}, {23115, 18589}, {41361, 20305}, {47125, 21253}
X(3162) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 25}, {1370, 455}, {2138, 17807}, {8743, 6}, {40357, 17407}, {41361, 159}
X(3162) = X(i)-isoconjugate of X(j) for these (i,j): {3, 39733}, {48, 40009}, {63, 13575}, {304, 34207}, {34055, 39129}, {39172, 46244}
X(3162) = barycentric product X(i)*X(j) for these {i,j}: {3, 41766}, {4, 159}, {6, 41361}, {19, 18596}, {20, 33584}, {22, 17407}, {25, 1370}, {112, 47125}, {393, 23115}, {427, 8793}, {455, 13575}, {607, 18629}, {1973, 21582}, {2207, 28419}, {40357, 40938}
X(3162) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 40009}, {19, 39733}, {25, 13575}, {159, 69}, {455, 1370}, {1184, 40185}, {1370, 305}, {1843, 39129}, {1974, 34207}, {8793, 1799}, {17407, 18018}, {17409, 40358}, {18596, 304}, {20968, 39172}, {21582, 40364}, {23115, 3926}, {33578, 33579}, {33584, 253}, {36417, 40144}, {41361, 76}, {41766, 264}, {46766, 40404}, {47125, 3267}
X(3162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 3172, 17409}, {427, 16318, 13854}, {1184, 2207, 25}, {1196, 14581, 36417}, {1196, 36417, 25}, {8743, 8879, 40144}, {14580, 17409, 25}, {15255, 15264, 25}
X(3162) = perspector of circumconic centered at X(25)
X(3162) = center of circumconic that is locus of trilinear poles of lines passing through X(25); this conic is the polar conjugate of the de Longchamps line.
X(3163) lies on the Steiner inellipse and on these lines: 2,648 6,13 30,1990 216,549 233,547 376,577 553,1086 1084,1196 1100,1146 1636,1637 2482,2799
X(3163) = midpoint of X(2) ane X(648)
X(3163) = isotomic conjugate of X(31621)
X(3163) = complement of X(1494)
X(3163) = isogonal conjugate of isotomic conjugate of X(36789)
X(3163) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,30), (30,3081)
X(3163) = crosspoint of X(36298) and X(36299)
X(3163) = X(3081)-cross conjugate of X(30)
X(3163) = crosspoint of X(2) and X(30)
X(3163) = crosssum of X(6) and X(74)
X(3163) = crossdifference of every pair of points on the line X(74)X(526)
X(3163) = center of circumconic that is locus of trilinear poles of lines parallel to Euler line (i.e. lines that pass through X(30))
X(3163) = perspector of circumconic centered at X(30) (parabola {A,B,C,X(30),X(476)})
X(3163) = perspector of ABC and medial triangle of cevian triangle of X(30)
X(3163) = barycentric square of X(30)
X(3163) = X(12034)-of-orthic-triangle if ABC is acute
X(3164) lies on these lines: 2,216 6,401 20,185 22,385 69,1972 160,523 206,1632 237,3186 577,648 1976,2998 3101,3187
X(3164) = reflection of X(264) in X(216)
X(3164) = isogonal conjugate of X(1988)
X(3164) = isotomic conjugate of the isogonal conjugate of X(32445)
X(3164) = complement of isotomic conjugate of X(38256)
X(3164) = complement of polar conjugate of X(38264)
X(3164) = anticomplement of X(264)
X(3164) = X(3)-Ceva conjugate of X(2)
X(3164) = anticomplementary isotomic conjugate of X(4)
X(3164) = antipedal isotomic conjugate of X(4)
X(3164) = complement of isogonal conjugate of X(36617)
X(3164) = complementary conjugate of complement of X(36617)
X(3165) lies on these lines: 15,186 16,184 35,1094 54,62 577,3165
X(3165) = X(3)-Ceva conjugate of X(15)
X(3166) lies on these lines: 15,184 16,186 35,1095 54,61 577,3165
X(3166) = X(3)-Ceva conjugate of X(16)
X(3167) lies on these lines: 3,49 6,1196 22,323 25,110 68,1656 154,511 193,459 195,973 392,912 542,1853 999,1201 1200,2280 1362,1397 1384,1501 1611,1692 1994,1995
X(3167) = isogonal conjugate of X(34208)
X(3167) = crossdifference of every pair of points on line X(2501)X(3566)
X(3167) =X(i)-Ceva conjugate of X(j) for these (i,j): (6,3), (193,3053)
X(3167) = crosspoint of X(i) and X(j) for these {i,j}: {6, 3053}, {193, 6337}
X(3167) = crosssum of X(i) and X(j) for these (i,j): {2, 2996}, {523, 8754}, {2501, 5139}, {8770, 14248}
X(3167) = Thomson-isogonal conjugate of X(20)
X(3167) = perspector of unary cofactor triangles of outer and inner Vecten triangles
X(3167) = centroid of anticevian triangle of X(3), which is also the antipedal triangle of X(64) and the tangential triangle of the MacBeath circumconic
X(3168) lies on these lines: {2,1972}, {4,51}, {6,436}, {107,184}, {232,800}, {262,459}, {450,1993}, {1148,1870}
X(3168) = X(6)-Ceva conjugate of X(4)
X(3168) = polar conjugate of the isogonal conjugate of X(32445)
X(3169) lies on these lines: 1,2092 6,979 8,9 42,1449 43,2300 100,604 145,1400 284,2319 519,573 1018,1743
X(3169) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,9), (3210,978)
X(3170) lies on these lines: 15,323 16,184 32,3171 202,2308 1994,2005
X(3170) = X(6)-Ceva conjugate of X(15)
X(3171) lies on these lines: 15,184 16,323 32,3170 61,110 203,2308 1994,2004
X(3171) = X(6)-Ceva conjugate of X(16)
X(3172) lies on these lines: 3,112 6,64 19,1104 20,1249 24,1384 25,32 31,607 41,3195 204,3198 232,3053 608,1042 1907,3087
X(3172) = isogonal conjugate of X(34403)
X(3172) = crossdifference of every pair of points on line X(3265)X(8057)
X(3172) = X(i)-Ceva conjugate of X(j) for these (i,j): (1249,154), (2332,1973)
X(3172) = crosspoint of X(i) and X(j) for these (i,j): (6,154), (204,3213)
X(3172) = crosssum of X(2) and X(253)
X(3173) lies on these lines: 1,90 3,1794 6,226 7,2982 55,916 63,77 109,3190 221,758 223,2323 278,651 515,1498 1362,1397 1708,2911 1936,2947 2192,2801
X(3173) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,3),
(2982,1214)
X(3173) = cevapoint of X(3157) and X(3211)
X(3174) lies on these lines: 1,142 9,55 40,518 65,2136 78,390 100,1445 516,1490 527,2951 528,1537 936,1001 2324,2340 2801,2950
X(3174) = midpoint of X(i) and X(j) for these (i,j): (2136,3174), (2550,3183)
X(3174) = crosssum of X(663) and X(3020)
X(3174) = X(7)-Ceva conjuate of X(9)
X(3174) = X(66)-of-excentral-triangle
X(3174) = intangents-to-extangents similarity image of X(9)
X(3174) = intouch-to-ABC barycentric image of X(9)
X(3175) lies on these lines: 2,37 72,519 190,1999 354,726 428,528 1211,2321
X(3175) = X(7)-Ceva conjugate of X(10)
X(3175) = X(2)-of-inverse(n(Medial)*n(Incentral))-triangle
X(3176) lies on these lines: 1,281 4,65 8,1034 20,653 40,1712 84,1767 92,938 207,1490 243,1788 278,1210 451,3085 1863,1902
X(3176) = X(8)-Ceva conjugate of X(4)
X(3177) lies on these lines: 2,85 63,194 105,330 144,145 220,664 329,1655 672,3212 894,2263 1654,3152
X(3177) = reflection of X(85) in X(1212)
X(3177) = anticomplement of X(85)
X(3177) = X(9)-Ceva conjugate of X(2)
X(3177) = isotomic conjugate of isogonal conjugate of X(20995)
X(3177) = polar conjugate of isogonal conjugate of X(20793)
X(3177) = anticomplementary conjugate of X(21285)
X(3177) = perspector of Gemini triangle 35 and cross-triangle of Gemini triangles 35 and 37
X(3178) lies on this line: 1,2
X(3178) = X(12)-Ceva conjugate of X(10)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
X(3179) lies on these lines: 1,15 9,46 13,484 18,1653 57,1081
X(3179) = X(13)-Ceva conjugate of X(1)
X(3179) = X(202)-cross conjugate of X(1)
X(3180) lies on these lines: 2,6 13,533 61,634 62,619 148,531 194,617 383,1351
X(3180) = reflection of X(i) in X(j) for these (i,j): (298,396), (616,15), (621,13), (3181,385)
X(3180) = anticomplement of X(298)
X(3180) = X(13)-Ceva conjugate of X(2)
X(3180) = inner-Napoleon-circle-inverse of X(32525)
X(3180) = {X(2),X(193)}-harmonic conjugate of X(3181)
X(3180) = {X(6),X(7837)}-harmonic conjugate of X(3181)
X(3181) lies on these lines: 2,6 14,532 16,533 61,618 62,633 148,530 194,616 1080,1351
X(3181) = reflection of X(i) in X(j) for these (i,j): (299,395),
(617,16), (622,14), (3180,385)
X(3181) = anticomplement of X(299)
X(3181) = X(14)-Ceva conjugate of X(2)
X(3181) = outer-Napoleon-circle-inverse of X(32525)
X(3181) = {X(2),X(193)}-harmonic conjugate of X(3180)
X(3181) = {X(6),X(7837)}-harmonic conjugate of X(3180)
X(3182) lies on the Darboux cubic and these lines: 1,64 3,223 4,57 20,3347 40,3346 579,2270 658,1097 1044,1716 1490,3348 1498,3354 3183,3473
X(3182) = reflection of X(3345) in X(3)
X(3182) = X(20)-Ceva conjugate of X(1)
X(3182) = isogonal conjugate of X(3347)
X(3182) = excentral isogonal conjugate of X(7992)
X(3182) = perspector of excentral triangle and 3rd extouch triangle
X(3183) lies on the Darboux cubic and these lines: 1,196 3,1033 4,64 19,84 20,3348 40,1712 376,1075 393,1192 1490,3354 1498,2131 1620,1990 3182,3473
X(3183) = reflection of X(3346) in X(3)
X(3183) = anticomplement of X(33546)
X(3183) = X(20)-Ceva conjugate of X(4)
X(3183) = crosspoint of X(20) and X(2060)
X(3183) = isogonal conjugate of X(3348)
X(3183) = orthocenter of cevian triangle of X(20)
X(3183) = perspector of hexyl triangle and anticevian triangle of X(1712)
X(3184) lies on these lines: 3,113 20,107 30,133 112,376
X(3184) = inverse-in-circumcircle of X(2935)
X(3184) = midpoint of X(20) and X(107)
X(3184) = reflection of X(122) in X(3)
X(3184) = X(20)-Ceva conjugate of X(30)
X(3184) = center of conic {{A,B,C,X(20),X(107)}}
X(3185) = isogonal conjugate of X(2995)
X(3185) = X(21)-Ceva conjugate of X(6)
X(3185) = crosspoint of X(100) and X(2149)
X(3185) = crosssum of X(123) and X(525)
X(3185) = crossdifference of every pair of points on the line
X(905)X(1577)
X(3186) lies on these lines: {1,242}, {2,3505}, {4,69}, {6,419}, {19,2319}, {24,1075}, {25,385}, {141,5117}, {232,800}, {237,3164}, {393,694}, {427,3314}, {571,1632}, {648,1974}, {1714,3144}, {3462,3542}
X(3186) = isogonal conjugate of X(3504)
X(3186) = X(25)-Ceva conjugate of X(4)
X(3186) = X(1613)-cross conjugate of X(194)
X(3186) = crossdifference of every pair of points on the line X(2524)X(3049)
X(3186) = pole wrt polar circle of trilinear polar of X(2998) (line X(512)X(625))
X(3186) = polar conjugate of X(2998)
X(3187) lies on these lines: 1,2 7,1943 31,740 75,81 92,1172 100,2352 193,1839 335,2606 1724,2901 3101,3164
X(3187) = anticomplement of X(306)
X(3187) = X(27)-Ceva conjugate of X(2)
X(3187) = crosspoint of X(648) and X(1016)
X(3187) = crosssum of X(647) and X(1015)
X(3187) = crossdifference of every pair of points on the line
X(649)X(838)
X(3188) lies on these lines: 1,7 3,1446 21,85 28,272 241,379 348,377 514,1729 917,934 958,1441
X(3188) = X(27)-Ceva conjugate of X(7)
X(3189) lies on these lines: 1,142 4,2900 8,21 20,518 40,476 65,145 71,3169 78,497 100,1788 200,950 210,452 281,2332 346,2264 380,2321 390,960 528,962 938,1376 997,1058 1118,1897
X(3189) = reflection of X(2550) in X(3174)
X(3189) = X(27)-Ceva conjugate of X(9)
X(3189) = crosssum of X(1357) and X(1459)
X(3189) = intangents-to-extangents similarity image of X(8)
X(3190) lies on these lines: 1,2 6,1260 9,2318 25,101 33,2900 57,1818 58,1259 63,991 72,581 100,1754 109,3173 197,1630 209,579 212,2323 219,284 228,573 318,2901 516,2947 517,3198 518,1214 674,3185 1331,1993 1612,1724 1897,2052 2172,2187
X(3190) = X(29)-Ceva conjugate of X(9)
X(3190) = crosspoint of X(1252) and X(1897)
X(3190) = crosssum of X(1086) and X(1459)
X(3190) = crossdifference of every pair of points on the line
X(649)X(2504)
X(3191) lies on these lines: 1,6 28,101 33,2901 40,228 201,758 329,581 568,1153 943,2328 1331,3193 1824,2910 1896,1897
X(3191) = X(29)-Ceva conjugate of X(10)
X(3192) lies on these lines: 1,406 4,386 6,25 19,2258 24,58 31,1182 33,42 34,1193 53,430 55,3195 73,208 162,2185 199,577 216,1011 235,1834 387,3089 461,1249 475,3216 581,3194 995,1870 2207,2271
X(3192) = X(29)-Ceva conjugate of X(19)
X(3193) lies on these lines: 1,21 4,155 28,110 29,1069 40,1790 46,453 57,1819 225,651 323,2475 377,394 517,1437 648,1896 1068,3157 1331,3191 1442,1444 1816,1936 2287,2323
X(3193) = X(29)-Ceva conjugate of X(21)
X(3193) = cevapoint of X(i) and X(j) for these (i,j): (1,155), (46,3157)
X(3193) = X(i)-cross conjugate of X(j) for these (i,j): (1800,21), (3157,1800)
X(3193) = crosspoint of X(1) and X(155) wrt both the excentral and tangential triangles
X(3194) lies on these lines: 1,204 4,6 28,34 29,81 40,2331 73,108 112,972 196,221 223,2360 240,1046 329,3195 580,1715 581,3192 937,1474 1104,1870 1193,1430 1203,1838 1741,1778 1828,2203 1875,2194 2266,2332
X(3194) = X(i)-Ceva conjugate of X(j) for these (i,j): (29,28),
(81,1172)
X(3194) = cevapoint of X(i) and X(j) for these (i,j): (6,204),
(208,221), (2331,3195)
X(3194) = X(221)-cross conjugate of X(2360)
X(3195) lies on these lines: 6,33 25,31 41,3172 55,3192 108,222 154,478 208,221 213,607 329,3194 611,1957 614,1876 1193,1593 1201,1398 2187,3209
X(3195) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,607),
(33,25), (208,3209), (3194,2331)
X(3195) = crosspoint of X(i) and X(j) for these (i,j): (6,221),
(208,2331)
X(3195) = crosssum of X(i) and X(j) for these (i,j): (2,280),
(905,2968)
X(3196) lies on these lines: 6,101 9,1030 36,44 45,55 198,2265 909,3207
X(3196) = isogonal conjugate of X(8046)
X(3196) = X(i)-Ceva conjugate of X(j) for these (i,j): (36,55), (44,6)
X(3196) = crosssum of X(650) and X(3025)
X(3196) = crossdifference of every pair of poins on the line X(900)X(1387)
X(3197) lies on the cubics K179 and K750, and on these lines:
{3,1630}, {6,19}, {9,6001}, {22,11445}, {24,11460}, {25,3611}, {37,1854}, {40,219}, {48,55}, {64,71}, {101,1604}, {109,15905}, {155,8141}, {159,20468}, {184,11406}, {218,2270}, {222,18725}, {268,12330}, {281,5776}, {394,3101}, {517,3211}, {910,2911}, {958,15656}, {966,20306}, {1030,3207}, {1181,6197}, {1190,2266}, {1191,21770}, {1376,10174}, {1407,26934}, {1503,2550}, {1615,7964}, {1802,12335}, {1853,3925}, {1903,7079}, {1993,9536}, {2093,18594}, {2183,7355}, {2259,11051}, {2289,10310}, {2301,4254}, {2343,8602}, {3196,5036}, {3198,3990}, {3779,9924}, {4258,11434}, {5415,17819}, {5416,17820}, {5928,30686}, {6237,17834}, {6252,17840}, {6253,17845}, {6255,17846}, {6404,17843}, {6759,10306}, {6769,22153}, {7291,23144}, {7688,10606}, {7719,9119}, {7724,17835}, {7957,7973}, {8251,17814}, {8539,17813}, {8802,17833}, {8804,12779}, {9537,11441}, {9816,17825}, {10119,17847}, {10319,17811}, {10636,17826}, {10637,17827}, {10902,17821}, {11206,17784}, {11428,17809}, {11435,17810}, {11471,15811}, {12417,17836}, {12661,17838}, {12662,17839}, {12663,17842}, {13041,17841}, {13042,17844}, {13567,18921}, {15509,24310}, {17837,22840}, {18405,18406}, {18451,18453}, {18598,22132}, {19132,19133}, {19180,19181}, {19430,19432}, {19431,19433}, {21778,21779}, {26908,26909}, {26952,26953}, {26957,26958}
X(3197) = isogonal conjugate of the polar conjugate of X(3176)
X(3197) = X(i)-Ceva conjugate of X(j) for these (i,j): {40, 55}, {219, 6}, {610, 198}
X(3197) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3345}, {57, 1034}, {63, 7149}, {75, 7152}, {81, 8806}, {85, 7037}, {189, 3342}, {333, 8811}, {348, 7007}, {8064, 17896}
X(3197) = crosspoint of X(100) and X(23984)
X(3197) = crosssum of X(i) and X(j) for these (i,j): {9, 6769}, {514, 16596}, {650, 3318}
X(3197) = crossdifference of every pair of points on line {521, 14837}
X(3197) = barycentric product X(i)*X(j) for these {i,j}: {1, 1490}, {3, 3176}, {8, 1035}, {40, 3341}, {55, 5932}, {65, 13614}, {72, 8885}, {78, 207}, {109, 14302}
X(3197) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 7149}, {31, 3345}, {32, 7152}, {42, 8806}, {55, 1034}, {207, 273}, {1035, 7}, {1402, 8811}, {1490, 75}, {2175, 7037}, {2187, 3342}, {2212, 7007}, {3176, 264}, {3341, 309}, {5932, 6063}, {8885, 286}, {13614, 314}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 21767, 221}, {19, 2261, 2264}, {19, 19350, 6}, {48, 2272, 1436}, {55, 10536, 154}, {607, 1409, 6}, {610, 1498, 7152}, {2187, 2357, 55}, {5584, 6254, 64}, {10536, 11190, 55}
X(3198) lies on these lines: 10,440 12,1869 19,25 31,2264 40,64 42,65 100,1297 204,3172 210,1903 212,2182 517,3190 1011,1212 1071,1715 1108,2352 1260,1766
X(3198) = X(i)-Ceva conjugate of X(j) for these (i,j): (40,71),
(72,37), (200,42)
X(3198) = crosspoint of X(20) and X(610)
X(3198) = crosssum of X(64) and X(2184)
X(3199) lies on these lines: 4,39 5,53 6,1598 24,187 25,32 33,1500 34,1015 51,217 52,1626 97,2984 115,235 297,626 393,800 427,1506 574,1593 1504,3092 1505,3093 1574,1861 1692,1974 1843,2211
X(3199) = isogonal conjugate of X(34386)
X(3199) = polar conjugate of X(34384)
X(3199) = crosspoint of X(i) and X(j) for these {i,j}: {6, 32319}, {25, 393}
X(3199) = crosssum of X(i) and X(j) for these (i,j): {2, 20477}, {6, 11206}, {69, 394}
X(3199) = crossdifference of every pair of points on line X(3265)X(15414)
X(3199) = X(53)-Ceva conjugate of X(51)
X(3199) = crosspoint of X(25) and X(393)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
X(3200) lies on these lines: 3,3205 6,3201 13,110 16,184 18,54 49,62 61,1147 202,215 3105,3202
X(3200) = X(54)-Ceva conjugate of X(15)
X(3200) = crosspoint of X(15) and X(62)
X(3200) = crosssum of X(13) and X(18)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b), where f(a,b,c) = u(-u/x + v/y + w/z)
X(3201) lies on these lines: 3,3206 6,3200 14,110 15,184 17,54 49,61 62,1147 203,215 3104,3202
X(3201) = X(i)-Ceva conjugate of X(j) for these (i,j): (54,16),
(2981,50)
X(3201) = crosspoint of X(16) and X(61)
X(3201) = crosssum of X(14) and X(17)
Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X(3202) = X(39)-of-A'B'C'. (Randy Hutson, January 15, 2019)
X(3202) lies on these lines: 6,3203 26,206 39,184 49,3095 76,110 156,2782 1916,3044 3104,3201 3105,3200 3106,3206 3107,3205
X(3202) = X(54)-Ceva conjugate of X(32)
X(3202) = crosspoint of X(1670) and X(1671)
X(3202) = crosssum of X(1676) and X(1677)
X(3203) lies on these lines: 6,3202 32,184 83,110 140,141
X(3203) = X(i)-Ceva conjugate of X(j) for these (i,j): (54,39),
(110,3050)
X(3203) = crosspoint of X(1342) and X(1343)
X(3204) lies on these lines: 6,101 9,2174 41,584 45,284 218,583 220,2301 1953,2246 2176,2220
X(3204) = X(54)-Ceva conjugate of X(55)
X(3205) lies on these lines: 3,3200 6,3206 13,156 14,54 15,1147 17,110 49,61 62,184 203,2477 3107,3202
X(3205) = X(54)-Ceva conjugate of X(61)
X(3206) lies on these lines: 3,3201 6,3205 13,54 14,156 16,1147 18,110 49,62 61,184 202,2477 3106,3202
X(3206) = X(54)-Ceva conjugate of X(62)
X(3207) lies on these lines: 1,910 3,101 6,41 31,1191 36,218 45,2182 169,1385 595,1384 909,3196 944,1146 1030,3197 1100,2270 1190,1617 1319,2082 1334,2272 1376,2329 1388,2170 1604,1630 1616,1914 1696,2268 2053,2110 2176,2223 2242,2271
X(3207) = isogonal conjugate of X(10405)
X(3207) = polar conjugate of isotomic conjugate of X(22117)
X(3207) = X(55)-Ceva conjugate of X(6)
X(3207) = crosspoint of X(i) and X(j) for these (i,j): (101,1262), (165,1419)
X(3207) = crosssum of X(514) and X1146)
X(3207) = trilinear pole wrt tangential triangle of antiorthic axis
X(3208) lies on these lines: 1,39 8,9 43,2176 145,672 192,1423 2340,3158 3057,3061
X(3208) = X(i)-Ceva conjugate of X(j) for these (i,j): (55,9), (192,43), (196,221), (208,3195)
X(3208) = {X(1),X(1018)}-harmonic conjugate of X(3501)
X(3209) lies on these lines: 19,56 25,1096 108,281 196,347 198,208 604,608 607,1400 1593,2285 2187,3195
X(3209) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,608),
(56,25), (196,221), (208,3195)
X(3209) = crosspoint of X(19) and X(2331)
X(3209) = X(112)-beth conjugate of X(6)
X(3210) lies on these lines: 2,37 8,38 43,726 57,1999 63,194 81,330 100,1403 226,1266 384,2221 664,1407 740,982 3011,3021 3101,3164
X(3210) = isotomic conjugate of isogonal conjugate of X(21769)
X(3210) = anticomplement of X(312)
X(3210) = anticomplementary conjugate of X(21286)
X(3210) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,2), (978,3169)
X(3210) = crosssum of X(663) and X(1977)
X(3210) = polar conjugate of isogonal conjugate of X(20805)
X(3210) = perspector of Gemini triangle 36 and cross-triangle of Gemini triangles 36 and 38
X(3210) = complement of isogonal conjugate of X(36619)
X(3210) = complementary conjugate of complement of X(36619)
X(3211) lies on these lines: 3,48 6,169 57,3173 155,610 579,2178 651,1119 1069,2164 1200,2280 1467,1743
X(3211) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,3),
(3173,3157)
X(3211) = crosssum of X(650) and X(2969)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3212) lies on these lines: 1,1447 2,257 7,8 57,239 241,2275 273,1829 279,291 348,1788 607,653 672,3177 1400,2998 1431,1966 1445,2082
X(3212) = isogonal conjugate of X(2053)
X(3212) = complement of X(20535)
X(3212) = anticomplement of X(3061)
X(3212) = X(57)-Ceva conjugate of X(7)
X(3212) = cevapoint of X(43) and X(1423)
X(3212) = X(43)-cross conjugate of X(192)
X(3212) = crosspoint of X(57) and X(1423)
X(3212) = crosssum of X(9) and X(2319)
X(3213) lies on these lines: 6,208 19,56 33,2285 34,604 48,2331 57,1172 154,204 196,380 608,1407 610,1249
X(3213) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,34),
(1172,608)
X(3213) = X(3172)-cross conjugate of X(204)
X(3213) = crosspoint of X(57) and X(1394)
X(3213) = X(112)-beth conjugate of X(48)
X(3214) lies on these lines: 1,2 44,71 65,2137 209,2390 210,2292 405,2177 902,1724 1334,2238 1376,1468 1458,1788 1475,1575 1834,2318 2246,2333
X(3214) = reflection of X(1201) in X(3216)
X(3214) = X(57)-Ceva conjugate of X(37)
X(3215) lies on these lines: 3,73 6,1195 31,65 34,46 41,2253 221,2361 224,1331 225,1754 283,1038 377,1935 582,1465 602,1457 1496,2646 1708,1780 1724,1877 1726,1825 2199,2245
X(3215) = X(57)-Ceva conjugate of X(48)
Let A'B'C' be the medial triangle and A''B''C'' then cevian triangle of X(1). Let U be the circumcircle of A, A', A'', and define V and W cyclically. Then X(3216) is the radical center of U, V, W. See Antreas Hatzipolakis and Peter Moses, Hyacinthos 267292.
X(3216) lies on on the Feuerbach of the tangential triangle and these lines: {1, 2}, {3, 1724}, {4, 5400}, {6, 474}, {9, 4261}, {21, 4256}, {35, 238}, {36, 5247}, {38, 3678}, {39, 2238}, {44, 3916}, {46, 2390}, {56, 4551}, {58, 404}, {65, 1739}, {72, 3670}, {73, 3911}, {100, 595}, {140, 5396}, {155, 6911}, {171, 1203}, {213, 1575}, {216, 1713}, {244, 3874}, {267, 2640}, {284, 7523}, {392, 4646}, {405, 4255}, {411, 1780}, {475, 3192}, {500, 549}, {517, 3987}, {518, 3953}, {579, 610}, {580, 6905}, {581, 631}, {602, 6796}, {662, 849}, {748, 5248}, {872, 6533}, {960, 4424}, {966, 5105}, {970, 1764}, {979, 2163}, {982, 5904}, {986, 5692}, {991, 3523}, {992, 2092}, {1015, 3780}, {1018, 2176}, {1045, 5506}, {1046, 1054}, {1050, 5541}, {1064, 6684}, {1104, 5440}, {1191, 5687}, {1213, 5153}, {1376, 5264}, {1450, 10106}, {1453, 5438}, {1457, 4848}, {1491, 4040}, {1498, 1754}, {1574, 2295}, {1654, 13571}, {1730, 10974}, {1738, 12047}, {1740, 2228}, {1757, 4283}, {1788, 10571}, {1834, 4187}, {1964, 4974}, {2077, 3073}, {2108, 6196}, {2234, 4672}, {2276, 3294}, {2292, 10176}, {2594, 5433}, {2650, 5883}, {2901, 4358}, {2915, 2916}, {2940, 2959}, {3191, 3772}, {3264, 3875}, {3290, 3970}, {3338, 3751}, {3454, 4202}, {3555, 4694}, {3570, 7760}, {3666, 5044}, {3684, 5299}, {3725, 4647}, {3740, 4719}, {3876, 4850}, {3878, 4642}, {3915, 8715}, {4003, 4005}, {4188, 4257}, {4205, 5241}, {4210, 4278}, {4300, 10164}, {4306, 5435}, {4413, 5711}, {4507, 9433}, {4653, 5047}, {4674, 5903}, {5255, 5315}, {5398, 6924}, {5706, 6918}, {5710, 9709}, {5713, 6854}, {5718, 8728}, {5721, 6922}, {9363, 13370}, {9548, 10882}, {9549, 12435}, {9567, 10441}
X(3216) = midpoint of X(1201) and X(3214)
X(3216) = reflection of X(1) in X(1201)
X(3216) = X(i)-Ceva conjugate of X(j) for these (i,j): {58, 1}, {404, 3}
X(3216) = X(3159)-cross conjugate of X(1)
X(3216) = crosspoint of X(662) and X(7035)
X(3216) = crossdifference of every pair of points on line (649, 3726)
X(3216) = crosssum of X(i) and X(j) for these (i,j): {513, 8054}, {661, 3248}
X(3216) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 191}, {6, 1045}, {21, 20}, {28, 1714}, {58, 3216}, {81, 2}, {100, 4427}, {174, 1762}, {259, 2938}, {266, 1046}, {365, 846}, {366, 1761}, {509, 1781}, {662, 3882}, {6727, 3}
X(3216) = X(3699)-beth conjugate of X(3216)
X(3216) = X(741)-he conjugate of X(6)
X(3216) = X(i)-zayin conjugate of X(j) for these (i,j): {56, 1724}, {58, 3216}, {667, 4040}, {1193, 1}, {1203, 3293}, {2260, 1743}, {2308, 43}, {2309, 87}, {3122, 9359}, {3733, 3737}, {4057, 513}, {10457, 58}
X(3216) = barycentric product X(81)*X(3159)
X(3216) = barycentric quotient X(3159)/X(321)
X(3216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 43, 3293), (1, 6048, 3679), (2, 386, 1), (2, 9534, 10479), (3, 4383, 1724), (8, 995, 1), (10, 1193, 1), (42, 1125, 1), (43, 978, 1), (72, 3752, 3670), (614, 3811, 1), (899, 1193, 10), (936, 2999, 1), (975, 5256, 1), (1046, 1054, 3336), (1149, 3244, 1), (1698, 5313, 1), (3624, 5312, 1), (3751, 11512, 3338), (4202, 5741, 3454), (9534, 10479, 3679)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3217) lies on these lines: 6,1201 9,21 19,1830 37,2280 44,48 101,604 169,2171 198,672 218,1400 219,2347 220,2269 391,2329 644,3169 1802,2264 2174,2267 2183,2911
X(3217) = X(i)-Ceva conjugate of X(j) for these (i,j): (58,55),
(983,1253)
X(3217) = crosssum of X(514) and X(3020)
This point occurs in a Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, "Another cubic associated with the triangle," Journal for Geometry and Graphics 11 (2007) 15-26.
Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the antiorthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the antiorthic axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(908) and centroid X(3218); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)
X(3218) lies on these lines: 1,89 2,7 8,46 11,1776 21,942 22,1473 31,982 36,214 37,2666 38,171 40,145 44,88 65,2975 72,404 75,1150 80,535 81,593 97,1214 100,518 104,517 110,2651 149,516 153,1512 189,2994 191,1125 222,1993 229,1098 238,244 239,514 240,1430 241,1252 278,1748 291,2239 320,2245 323,1443 324,1947 333,2160 335,675 354,1621 388,1454 394,1407 411,1071 484,519 614,1707 651,1465 750,984 799,1921 899,1054 903,2161 919,1814 962,1158 986,1468 990,2000 1012,2095 1046,1193
X(3218) = reflection of X(i) in X(j) for these (i,j): (100,1155), (153,1512), (3257,44)
X(3218) = isogonal conjugate of X(2161)
X(3218) = isotomic conjugate of X(18359)
X(3218) = complement of X(17484)
X(3218) = anticomplement of X(908)
X(3218) = X(903)-Ceva conjugate of X(1)
X(3218) = cevapoint of X(36) and X(2323)
X(3218) = X(i)-cross conjugate of X(j) for these (i,j): (36,1443), (2245,36)
X(3218) = crosspoint of X(81) and X(88)
X(3218) = crosssum of X(i) and X(j) for these (i,j): (37,44), (649,2087), (1635,2170)
X(3218) = crossdifference of every pair of points on the line X(42)X(663)
X(3218) = inverse-in-circumconic-centered-at-X(9) of X(2)
X(3218) = {X(9),X(57)}-harmonic conjugate of X(3306)
X(3218) = X(1495)-of-excentral-triangle
X(3218) = endo-homothetic center of X(3)- and X(4)-Ehrmann triangles
This point occurs in a forthcoming paper by Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, "Another cubic associated with the triangle."
X(3219) lies on these lines: 1,2308 2,7 8,90 10,191 21,72 31,984 37,81 38,238 42,846 45,940 55,1776 71,1654 100,210 101,1790 172,593 190,321 201,1935 219,1993 220,394 281,1748 306,2895 312,1150 323,1442 324,1948 612,1707 651,1214 748,982 756,896 799,1920 912,1006 938,1728 958,2099 960,1319 1082,2307 1268,2160
X(3219) = isogonal conjugate of X(2160)
X(3219) = isotomic conjugate of X(30690)
X(3219) = complement of X(17483)
X(3219) = X(1268)-Ceva conjugate of X(1)
X(3219) = cevapoint of X(9) and X(191)
X(3219) = X(35)-cross conjugate of X(1442)
X(3219) = crosssum of X(i) and X(j) for these (i,j): (649,3125), (1652,1653)
X(3219) = crossdifference of every pair of points on the line X(663)X(2520)
X(3219) = {X(9),X(57)}-harmonic conjugate of X(3305)
X(3220) is the homothetic center of the following two triangles: tangential triangles of the excentral triangle, and excentral triangle of the tangential triangle. (Note from Peter Moses, 1/24/2007)
X(3220) lies on these lines: 1,159 3,9 19,990 22,63 25,57 35,984 36,238 48,991 56,269 58,1474 100,2751 101,1818 103,2272 104,2728 154,222 165,197 184,2003 219,1350 511,2323 603,2212 759,2722 1040,1763
X(3220) = crosspoint of X(58) and X(103)
X(3220) = crosssum of X(i) and X(j) for these (i,j): (10,516), (71,2340)
X(3220) = crossdifference of every pair of points on the line X(37)X(2509)
Trilinears for X(3221) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(647). As a symbolic substitution, this mapping takes lines onto lines. In particular, points on the Euler line, X(2)X(3), with coefficients given by X(647), are mapped to the line X(6)X(194), with coefficients given by X(3221). Symbolic substitutions are introduced in the following article:
C. Kimberling, "Symbolic substitutions in the transfigured plane of a triangle," Aequationes Mathematicae 73 (2007) 156-171.
As the isogonal conjugate of a point on the circumcircle, X(3221) lies on the line at infinity.
X(3221) lies on these (parallel) lines: 30,511 669,2451 882,1843
X(3221) = isogonal conjugate of X(3222)
X(3221) = perspector of hyperbola {{A,B,C,X(2),X(194)}}
X(3221) = bicentric difference of PU(154)
X(3221) = ideal point of PU(154)
X(3222) lies on the circumcircle and these lines: 2,699 111,2998 645,803 729,1078 733,1799 799,932
X(3222) = isogonal conjugate of X(3221)
X(3222) = cevapoint of X(2) and X(669)
X(3222) = X(670)-cross conjugate of X(99)
X(3222) = trilinear pole of line X(6)X(194)
X(3222) = Ψ(X(6), X(194))
X(3222) = circumcircle intercept, other than A, B, C, of conic {{A,B,C,PU(148)}}
X(3223) lies on these lines: 1,1965 31,1582 42,192 43,213 63,1967 1045,2258
X(3223) = isogonal conjugate of X(1740)
X(3223) = isotomic conjugate of X(17149)
X(3223) = complement of anticomplementary conjugate of X(21299)
X(3223) = anticomplement of complementary conjugate of X(21257)
X(3223) = X(75)-cross conjugate of X(1)
X(3223) = trilinear product of PU(154)
X(3224) lies on these lines: 6,194 32,1613 43,213 419,2207 729,1078 1691,1968
X(3224) = isogonal conjugate of X(194)
X(3224) = complement of X(32548)
X(3224) = cevapoint of X(2) and X(2998)
X(3224) = X(2)-cross conjugate of X(6)
X(3224) = isotomic conjugate of X(6374)
X(3224) = trilinear pole of line X(669)X(2451)
X(3224) = barycentric product of PU(154)
Trilinears for X(3225) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(98). As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X(3225).
X(3225) lies on the Steiner circumellipse and these lines: 6,670 32,99 190,1918 290,2422 648,1974 729,886
X(3225) = isogonal conjugate of X(3229)
X(3225) = isotomic conjugate of X(698)
X(3225) = cevapoint of X(i) and X(j) for these (i,j): (6,385), (192,2238)
X(3225) = Steiner-circumellipse-X(6)-antipode of X(670)
X(3225) = X(694)-cross conjugate of X(98)
X(3225) = trilinear pole of PU(148) (line X(2)X(669))
Trilinears for X(3226) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(105). As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X(3226).
X(3226) lies on the Steiner circumellipse and these lines: 1,668 6,190 56,664 58,99 75,87 86,670 239,292 648,1474 666,1438 1027,2481
X(3226) = isogonal conjugate of X(3009)
X(3226) = isotomic conjugate of X(726)
X(3226) = complement of X(39354)
X(3226) = anticomplement of X(20532)
X(3226) = cevapoint of X(1) and X(329)
X(3226) = X(i)-cross conjugate of X(j) for these (i,j): (291,673), (350,86), (659,190)
X(3226) = trilinear pole of line X(2)X(649)
X(3226) = Steiner-circumellipse-X(1)-antipode of X(668)
X(3226) = Steiner-circumellipse-X(6)-antipode of X(190)
Trilinears for X(3227) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(106). As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X(3227).
X(3227) lies on the Steiner circumellipse and these lines: 1,190 2,668 28,648 57,664 81,99 88,239 105,666 274,670 291,519 335,2087 671,2787 903,1022 957,1992
X(3227) = reflection of X(i) in X(j) for these (i,j): (2,1015), (668,2)
X(3227) = isogonal conjugate of X(3230)
X(3227) = isotomic conjugate of X(536)
X(3227) = complement of X(39360)
X(3227) = anticomplement of X(13466)
X(3227) = Steiner-circumellipse-X(1)-antipode of X(190)
X(3227) = X(1646)-cross conjugate of X(513)
X(3227) = Steiner-circumellipse-antipode of X(668)
X(3227) = projection from Steiner inellipse to Steiner circumellipse of X(1015)
X(3227) = the point of intersection, other than A, B, and C, of the Steiner circumellipse and hyperbola {{A,B,C,X(1),X(2)}}
X(3227) = antipode of X(2) in hyperbola {{A,B,C,X(1),X(2)}}
X(3227) = trilinear pole of line X(2)X(513)
Trilinears for X(3228) are obtained by applying the symbolic substitution (a,b,c) → (bc,ca,ab) to trilinear coordinates of X(111). As a symbolic substitution, this mapping takes circumconics to circumconics. In particular, it maps circumcircle onto the Steiner circumellipse, which passes through X(3228).
X(3228) lies on the Steiner circumellipse and these lines: 6,99 25,648 37,668 42,190 111,385 263,1992 290,2395 538,886 671,804
X(3228) = reflection of X(i) in X(j) for these (i,j): (2,1084), (670,2)
X(3228) = isogonal conjugate of X(3231)
X(3228) = isotomic conjugate of X(538)
X(3228) = complement of X(39361)
X(3228) = anticomplement of X(35073)
X(3228) = X(1645)-cross conjugate of X(512)
X(3228) = crossdifference of every pair of points on the line X(887)X(888)
X(3228) = Steiner-circumellipse-antipode of X(670)
X(3228) = Steiner-circumellipse-X(6)-antipode of X(99)
X(3228) = projection from Steiner inellipse to Steiner circumellipse of X(1084)
X(3228) = the point of intersection, other than A, B, and C, of the Steiner circumellipse and hyperbola {{A,B,C,X(2),X(6)}}
X(3228) = antipode of X(2) in hyperbola {{A,B,C,X(2),X(6)}}
X(3228) = trilinear pole of line X(2)X(512)
X(3229) lies on these lines: 1,893 2,39 32,1613 141,706 187,237 232,420 511,694 32746,32747
X(3229) = isogonal conjugate of X(3225)
X(3229) = isotomic conjugate of the isogonal conjugate of X(32748)
X(3229) = X(i)-Ceva conjugate of X(j) for these (i,j): (385,511), (699,6)
X(3229) = crosspoint of X(i) and X(j) for these (i,j): (2,694), (6,699)
X(3229) = crosssum of X(i) and X(j) for these (i,j): (2,698), (6,385), (192,2238), (523,2086)
X(3229) = crossdifference of every pair of points on the line X(2)X(669)
X(3229) = complement of X(3978)
X(3229) = crossdifference of PU(148)
X(3229) = X(2)-Ceva conjugate of X(39080)
X(3229) = perspector of hyperbola {{A,B,C,X(6),X(670),X(25424)}}
X(3229) = intersection of line X(2)X(39)[X(194)] and line through X(2)-Ceva conjugate of X(194) and X(194)-Ceva conjugate of X(2)
X(3230) is the point of intersection of the line X(1)X(6) and the trilinear polar of X(6); c.f. X(1323). (Randy Hutson, February 10, 2016)
X(3230) lies on these lines: 1,6 31,2242 39,1201 41,2241 99,2106 101,1914 106,292 111,2703 172,595 187,237 239,1016 519,2238 672,1015 739,898 869,2177 992,2321 995,2276 1018,1575 1125,2295 1193,1500 1197,1621
X(3230) = reflection of X(3125) in X(3290)
X(3230) = isogonal conjugate of X(3227)
X(3230) = X(i)-Ceva conjugate of X(j) for these (i,j): (739,6), (898,667)
X(3230) = crosspoint of X(i) and X(j) for these (i,j): (6,739), (898,1016)
X(3230) = crosssum of X(i) and X(j) for these (i,j): (2,536), (513,1646), (891,1015)
X(3230) = crossdifference of every pair of points on the line X(2)X(513)
X(3230) = bicentric sum of PU(26)
X(3230) = complement of anticomplementary conjugate of X(39360)
X(3230) = perspector of hyperbola {{A,B,C,X(6),X(100)}}
X(3230) = PU(26)-harmonic conjugate of X(667)
X(3230) = inverse-in-Parry-isodynamic-circle of X(5040); see X(2)
X(3231) lies on these lines: 2,6 23,2076 110,1691 111,694 187,237 468,2211
X(3231) = reflection of X(3124) in X(3291)
X(3231) = isogonal conjugate of X(3228)
X(3231) = isotomic conjugate of X(34087)
X(3231) = isotomic conjugate of isogonal conjugate of X(33875)
X(3231) = complement of anticomplementary conjugate of X(39361)
X(3231) = X(2)-Ceva conjugate of X(38998)
X(3231) = perspector of hyperbola {{A,B,C,X(6),X(99)}}
X(3231) = X(729)-Ceva conjugate of X(6)
X(3231) = crosssum of X(i) and X(j) for these (i,j): (2,538), (512,1645), (888,1084)
X(3231) = crossdifference of every pair of points on the line X(2)X(512)
X(3231) = trilinear pole of line X(887)X(888)
X(3231) = inverse-in-Parry-isodynamic-circle of X(5027); see X(2)
There exist points A', B', C' on segments BC, CA, AB, respectively, such that B'C + C'B = C'A + A'C = A'B + B'A = (a + b + c)/3, and the lines AA', BB', CC' concur in X(3232). Near the beginning of the 21st century, trilinears x : y : z were found for X(3232) in terms of the unique real root of a cubic polynomial related to the cubic polynomial shown at X(369), the 1st trisected perimeter point. The proof is given in Sadi Abu-Saymeh, Mawaffaq Hajja, and Hellmuth Stachel, "Another cubic associated with the triangle," forthcoming in Journal for Geometry and Graphics.
If you have The Geometer's Sketchpad, you can view X(3232).
X(3232) = isotomic conjugate of X(369)
Contributed by Fred Lang, April 1, 2007.
If you have The Geometer's Sketchpad, you can view Kiepert Inscribed Parabola.
X(3233) lies on these lines: 30,113 99,1304 110,476 114,468 355,3109 1302,1576
X(3233) = intercept of pedal and antipedal lines of X(110)
X(3234) lies on the curve Q078, the Yff parabola, and these lines: {100, 3239}, {101, 514}, {118, 516}, {220, 31852}, {908, 24582}, {929, 28879}, {1145, 8074}, {1633, 2736}, {5845, 15634}, {9057, 32739}, {14116, 17044}
Properties of the inscribed Yff parabola:
center = X(514)
perspector = X(190)
focus = X(101)
directrix = X(4)X(9), the trilinear pole of X(1897)
axis = X(101)X(514)
Simson line of focus = X(118)X(516)
pass-througth points: X(i) for these i: 514, 649, 3234, 3239, 4024, 4375, 6544, 24979, 31182
points on axis: X(i) for these i: 101, 514, 664, 666, 927, 3234, 3732, 4564, 14545, 14612, 34805, 34906, 36146
dual of Yff parabola: the circumconic {{A, B, C, X(2), X(7)}}
square of latus recturm = ((b - c)^2*(c - a)^2*(a - b)^2*(a - b - c)^2*(a + b - c)^2*(a - b + c)^2*(a + b + c)^2)/(a^4 - a^3*b - a*b^3 + b^4 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3 + c^4)^3
Contributed by Peter Moses, April 6, 2007 and February 4, 2010.
Yff parabola animated over the 3-periodics in an elliptic billiard (i.e., the circumconic with center X(9). Contributed by Dan Reznik, February 12, 2020.
If you have The Geometer's Sketchpad, you can view Yff Inscribed Parabola.
X(3234) = midpoint of X(101) and X(34805)
X(3234) = reflection of X(14116) in X(17044)
X(3234) = X(190)-Ceva conjugate of X(2398)
X(3234) = X(i)-isoconjugate of X(j) for these (i,j): {{911, 2400}, {2424, 36101}, {15634, 36039}}
X(3234) = crosspoint of X(190) and X(2398)
X(3234) = crosssum of X(649) and X(2424)
X(3234) = barycentric product X(i)*X(j) for these {i,j}: {{100, 24014}, {190, 23972}, {516, 2398}, {1331, 21665}, {1360, 3699}, {2426, 35517}}
X(3234) = barycentric quotient X(i)/X(j) for these {i,j}: {{516, 2400}, {676, 15634}, {1360, 3676}, {2398, 18025}, {2426, 103}, {23972, 514}, {24014, 693}}
X(3235) lies on these lines: 1,32 3,2007 6,2008 55,1688 56,1687 1124,1342 1335,1343
X(3236) lies on these lines: 1,32 3,2008 6,2007 55,1687 56,1688 1124,1343 1335,1342
X(3237) lies on these lines: 1,256 3,1673 6,1672 55,1671 56,1670 371,3236 372,3235 1124,1689 1335,1690 1664,1675 1665,1674 2007,3103 2008,3102
X(3238) lies on these lines: 1,256 3,1672 6,1673 55,1670 56,1671 371,3235 372,3236 1124,1690 1335,1689 1664,1674 1665,1675 2007,3102 2008,3103
X(3239) lies on the Yff parabola and these lines: 2,2400 9,652 100,3234 101,1309 111,2760 514,661
X(3239) = isogonal conjugate of X(1461)
X(3239) = isotomic conjugate of X(658)
X(3239) = complement of X(4025)
X(3239) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,2968), (190,8), (341,2310), (346,1146), (644,2321), (2399,2804)
X(3239) = X(i)-cross conjugate of X(j) for these (i,j): (1146,346), (2310,341), (3119,200)
X(3239) = crosspoint of X(i) and X(j) for these (i,j): (2,1897), (8,190), (644,2321)
X(3239) = crosssum of X(i) and X(j) for these (i,j): (6,1459), (56,649), (513,2262)
X(3239) = crossdifference of every pair of points on the line X(31)X(56)
X(3239) = trilinear pole of line X(1146)X(2310)
X(3239) = polar conjugate of X(36118)
X(3239) = pole wrt polar circle of trilinear polar of X(36118) (line X(19)X(57))
X(3239) = perspector of circumconic centered at X(2968) (hyperbola {{A,B,C,X(8),X(75)}})
X(3239) = center of circumconic that is locus of trilinear poles of lines passing through X(2968)
X(3239) = intersection of trilinear polars of X(8) and X(75)
X(3239) = excentral-to-ABC barycentric image of X(650)
X(3239) = perspector of Mandart hyperbola wrt extouch triangle
X(3240) lies on these lines: 1,2 6,100 44,751 45,2238 81,1376 88,1002 89,291 192,872 218,1005 238,2177 1013,1783
X(3240) = crossdifference of every pair of points on the line X(649)X(891)
X(3240) = {X(42),X(43)}-harmonic conjugate of X(2)
X(3241) lies on these lines: 1,2 4,1392 6,644 7,528 30,944 149,1478 192,537 346,1449 376,517 381,952 390,527 518,1992 529,2098 956,1621 1000,2320 1222,2334
X(3241) = midpoint of X(2) and X(145)
X(3241) = reflection of X(i) in X(j) for these (i,j): (2,1), (8,2)
X(3241) = isotomic conjugate of X(36588)
X(3241) = complement of X(31145)
X(3241) = anticomplement of X(3679)
X(3241) = anticomplementary conjugate of X(21291)
X(3241) = harmonic center of incircle and AC-incircle
X(3241) = polar conjugate of isogonal conjugate of X(23073)
X(3242) lies on these lines: 1,6 38,55 56,976 69,145 210,614 354,612 674,1469 982,1054 990,1350 1086,2550 1219,1257
X(3242) = midpoint of X(69) and X(145)
X(3242) = reflection of X(i) in X(j) for these (i,j): (6,1), (8,141), (69,145)
X(3242) = {X(1),X(9)}-harmonic conjugate of X(1279)
Let A'B'C' be the reflection of ABC in X(1). ABC and A'B'C' intersect at 6 points, which lie on an ellipse centered at X(1) with perspector X(10390). Let A" be the intersection of the tangents to this ellipse at the points where it intersects BC, and define B" and C" cyclically. (i.e., A"B"C" is the polar triangle of the ellipse.) A'B'C' and A"B"C" are perspective at X(3243). (Randy Hutson, August 29, 2018)
X(3243) lies on these lines: 1,6 7,145 57,100 65,2136 390,527 516,944 942,1706 971,1482 1056,2550
X(3243) = midpoint of X(7) and X(145)
X(3243) = reflection of X(i) in X(j) for these (i,j): (8,142), (9,1),
(2136,3174)
Let OA be the circle centered at the A-vertex of the excenters-reflections triangle and passing through A; define OB and OC cyclically. X(3244) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(3244) lies on these lines: 1,2 6,2325 9,2137 58,643 65,1317 79,1320 382,515 516,944 517,550 546,946 950,2098 996,2334 1018,1475 1100,2321 1126,1222
X(3244) = midpoint of X(1) and X(145)
X(3244) = reflection of X(i) in X(j) for these (i,j): (8,1125), (10,1)
X(3244) = {X(1),X(2)}-harmonic conjugate of X(3636)
X(3244) = {X(1),X(8)}-harmonic conjugate of X(1125)
X(3245) lies on these lines: 1,3 8,535 80,516
X(3245) = reflection of X(i) in X(j) for these (i,j): (1,1155), (36,484)
X(3245) = X(23)-of-reflection-triangle-of-X(1)
X(3245) = X(10295)-of-excentral-triangle
X(3245) = endo-homothetic center of Ehrmann vertex-triangle and X(2)-Ehrmann triangle; the homothetic center is X(7574)
X(3246) lies on these lines: {1, 6}, {8, 17342}, {10, 6687}, {31, 3742}, {36, 16686}, {63, 4906}, {88, 105}, {89, 30653}, {171, 3848}, {239, 4702}, {320, 3616}, {354, 17127}, {513, 1960}, {516, 15251}, {519, 4422}, {528, 3008}, {536, 4432}, {537, 4759}, {551, 4364}, {595, 3812}, {614, 4640}, {651, 1319}, {678, 899}, {748, 3740}, {752, 1125}, {896, 3999}, {902, 16610}, {1086, 28534}, {1149, 3248}, {1193, 27637}, {1443, 1456}, {1621, 17012}, {2239, 30950}, {2243, 3290}, {2308, 4883}, {2325, 28503}, {2999, 4428}, {3011, 5087}, {3052, 5272}, {3058, 26723}, {3271, 9037}, {3416, 29579}, {3550, 16602}, {3617, 17358}, {3622, 20072}, {3624, 31151}, {3626, 17765}, {3634, 3823}, {3683, 7191}, {3685, 17160}, {3689, 37680}, {3696, 16816}, {3745, 5284}, {3748, 32911}, {3749, 37679}, {3752, 8616}, {3791, 4891}, {3836, 19862}, {3844, 3883}, {3880, 3939}, {3912, 28538}, {3915, 5836}, {4009, 20045}, {4346, 5698}, {4395, 28580}, {4421, 23511}, {4423, 4682}, {4645, 5550}, {4660, 17356}, {4670, 24331}, {4679, 26228}, {4719, 5248}, {4849, 17715}, {4860, 36277}, {4887, 17768}, {4896, 25557}, {4914, 33157}, {4974, 28581}, {5211, 26070}, {5263, 16815}, {5269, 8167}, {5308, 38025}, {5880, 16020}, {7262, 21342}, {8301, 9324}, {8583, 25891}, {9780, 17371}, {13624, 15310}, {14997, 21870}, {15808, 28570}, {16694, 20470}, {17013, 37593}, {17237, 29660}, {17325, 25055}, {17335, 36534}, {17348, 32941}, {17357, 33076}, {17450, 21747}, {17605, 29681}, {21214, 28362}, {25351, 28562}, {26626, 38023}, {28352, 28375}, {28370, 28395}, {28555, 32922}, {30148, 31445}, {36289, 37600}
X(3246) = midpoint of X(i) and X(j) for these {i,j}: {1, 44}, {238, 1279}, {239, 4702}, {1155, 14190}, {1757, 4864}
X(3246) = reflection of X(i) in X(j) for these {i,j}: {10, 6687}, {3823, 31289}, {3834, 1125}
X(3246) = X(514)-isoconjugate of X(6017)
X(3246) = crosspoint of X(1016) and X(9089)
X(3246) = crosssum of X(1015) and X(9032)
X(3246) = crossdifference of every pair of points on line {45, 513}
X(3246) = barycentric product X(i)*X(j) for these {i,j}: {100, 6009}, {668, 8658}
X(3246) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 6017}, {6009, 693}, {8658, 513}
X(3246) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 238, 44}, {1, 15601, 5220}, {44, 1279, 1}, {748, 3744, 3740}, {1001, 1386, 15569}, {1001, 7290, 1386}, {3624, 31151, 31243}, {16672, 38315, 1}
X(3247) lies on these lines: 1,6 2,2321 187,2959 226,347 281,1886 519,966 574,988 579,1334 594,1698 612,1962 902,968 1068,1826 1255,1796
X(3247) = crosssum of X(6) and X(3303)
X(3247) = anticomplement of X(32087)
X(3247) = {X(1),X(9)}-harmonic conjugate of X(1449)
X(3248) lies on these lines: 1,190 6,292 31,692 42,678 75,87 86,2665 239,2234 244,659 341,979 560,604 662,741 757,1178 798,1084 890,1977 922,2210 1015,1960 1100,2309
X(3248) = X(i)-Ceva conjugate of X(j) for these (i,j):
(1,649), (6,798), (31,667), (87,513), (604,1919), (649,3249),
(873,1019), (979,650), (2297,2484), (2665,659)
X(3248) = isogonal conjugate of X(7035)
X(3248) = X(i)-cross conjugate of X(j) for these (i,j): (3121,1015), (3249,649)
X(3248) = crosspoint of X(i) and X(j) for these (i,j): (1,649), (31,667), (873,1019), (1015,1357)
X(3248) = crosssum of X(i) and X(j) for these (i,j): (1,190), (75,668), (341,646), (872,1018)
X(3248) = crossdifference of every pair of points on the line X(190)X(646)
X(3248) = trilinear pole wrt incentral triangle of Nagel line
X(3249) lies on these lines: 1,649 32,1919 213,667 330,514
X(3249) = X(i)-Ceva conjugate of X(j) for these (i,j): (649,3248), (1919,1977)
X(3249) = crosspoint of X(i) and X(j) for these (i,j): (649,3248), (1919,1977)
X(3249) = crossdifference of every pair of points on the line X(350)X(899)
X(3250) lies on these lines: 37,513 187,237 514,661 789,795 798,795
Let L be the line P(6)U(6) = X(37)X(513). Let M be the trilinear polar of the cevapoint of PU(6), this point being X(256). Let V = P(6)-Ceva conjugate of U(6) and W = U(6)-Ceva conjugate of P(6). The lines L, M, VW concur in X(3250). (Randy Hutson, December 26, 2015)
X(3250) = reflection of X(649) in X(665)
X(3250) = isotomic conjugate of X(37133)
X(3250) = complement of anticomplementary conjugate of X(39345)
X(3250) = X(2)-Ceva conjugate of X(38995)
X(3250) = perspector of hyperbola {{A,B,C,X(6),X(75)}}
X(3250) = X(i)-Ceva conjugate of X(j) for these (i,j): (825,6), (2186,2170)
X(3250) = crosspoint of X(6) and X(825)
X(3250) = crosssum of X(2) and X(824)
X(3250) = crossdifference of every pair of points on the line X(2)X(31)
X(3250) = isogonal conjugate of X(4586)
X(3251) lies on these lines: 1,513 42,663 55,667 678,1635 926,1642
X(3251) = reflection of X(1635) in X(1960)
X(3251) = isogonal conjugate of X(4618)
X(3251) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,2087), (100,44)
X(3251) = crosspoint of X(i) and X(j) for these (i,j): (1,1023), (44,100)
X(3251) = crosssum of X(i) and X(j) for these (i,j): (1,1022), (88,513), (100,3257)
X(3251) = crossdifference of every pair of points on the line X(44)X(88)
X(3252) lies on these lines: 6,292 7,192 9,660 37,513 59,604 291,1002 813,840 984,2113
X(3252) = X(292)-Ceva conjugate of X(672)
X(3252) = crossdifference of every pair of points on the line
X(238)X(812)
X(3253) lies on these lines: 2,2109 6,190 238,874 727,789 765,2209
X(3254) lies on these lines: 1,528 4,2801 7,149 9,11 80,518 100,142 104,516 294,2323 320,2481 390,2320 527,1156 943,1125 1000,2550
X(3254) = midpoint of X(7) and X(149)
X(3254) = reflection of X(i) in X(j) for these (i,j): (9,11), (100,142)
X(3254) = isogonal conjugate of X(2078)
X(3254) = antigonal conjugate of X(9)
X(3255) lies on these lines: 105,551 390,1392 527,2346 758,1000
X(3255) = isogonal conjugate of X(3256)
X(3256) lies on these lines: 1,3 42,109 100,226 1174,1200 1259,1706
X(3256) = isogonal conjugate of X(3255)
X(3257) lies on these lines: 44,88 100,513 106,238 320,908 527,666 658,1275 662,1019 758,1168 897,1757
Let P and Q be the intersections of line BC and circle {X(3),2R}. Let A' be the circumcenter of triangle PQX(3), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3257). (Randy Hutson, December 26, 2015)
Let A'B'C' and A"B"C" be the Ursa-minor and Ursa-major triangles, resp. Let A* be the reflection of A' in line B"C", and define B* and C* cyclically. Let A** be the reflection of A" in line B'C', and define B** and C** cyclically. Let A# = B*B**/\C*C**, and define B# and C# cyclically. The lines AA#, BB#, CC# concur in X(3257). (Randy Hutson, June 27, 2018)
X(3257) = reflection of X(i) in X(j) for these (i,j): (320,908), (3218,44)
X(3257) = isogonal conjugate of X(1635)
X(3257) = cevapoint of X(i) and X(j) for these (i,j): (1,1635), (88,1022), (100,1023), (649,1149)
X(3257) = X(i)-cross conjugate of X(j) for these (i,j): (44,765), (661,1168), (1022,88), (1023,100), (1635,1), (1769,75)
X(3257) = crossdifference of every pair of points on the line X(2087)X(3251)
X(3257) = isotomic conjugate of X(3762)
X(3257) = perspector of conic {A,B,C,PU(28)}
X(3257) = intersection of trilinear polars of P(28) and U(28)
X(3257) = trilinear pole of line X(1)X(88)
Let P = X(477) and H=X(4); let HA be the orthocenter of the triangle BCP. Define HB and HC cyclically. Then H, HA, HB, HC are the vertices of a quadrilateral that is homothetic to the cyclic quadrilateral having vertices A, B, C, P. The center of homothety is X(3258). Moreover, X(3258) is the anticenter of the quadrilateral ABCP. (Randy Hutson, 9/23/2011)
Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Euler line of A'B'C' (line X(30)X(74)). The triangle A"B"C" is homothetic to ABC, with center of homothety X(3258); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)
X(3258) lies on the nine-point circle, the cevian circle of X(30), and these lines: 2,476 4,477 30,113 114,858 115,647 125,523 131,2072 132,468 133,403 232,1560 868,1649
X(3258) = midpoint of X(4) and X(477)
X(3258) = reflection of X(i) in X(j) for these (i,j); (125,3154), (1553,113)
X(3258) = isogonal conjugate of X(15395)
X(3258) = complement of X(476)
X(3258) = complementary conjugate of X(526)
X(3258) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1637), (4,526), (1138, 523)
X(3258) = crosspoint of X(2) and X(3268)
X(3258) = crosssum of X(74) and X(110)
X(3258) = crossdifference of every pair of points on the line X(2420)X(2433)
X(3258) = inverse-in-polar-circle of X(1304)
X(3258) = inverse-in-{circumcircle, nine-point circle}-inverter of X(842)
X(3258) = inverse-in-Moses-radical-circle of X(115)
X(3258) = X(477)-of-Euler-triangle
X(3258) = reflection of X(125) in Euler line
X(3258) = perspector of circumconic centered at X(1637)
X(3258) = center of circumconic that is locus of trilinear poles of lines passing through X(1637)
X(3258) = intersection, other than X(126), of the nine-point circle of ABC and the Parry circle of the X(2)-Brocard triangle
X(3259) is the center of the rectangular hyperbola that passes through the points A, B, C, and X(56). (Randy Hutson, 9/23/2011)
X(3259) lies on the Sherman line (3259,3326); see http://forumgeom.fau.edu/FG2012volume12/FG201220.pdf)
A construction see Aris Pavlakis and Peter Moses, Hyacinthos 28741.
X(3259) lies on the nine-point circle, the Yff contact circle, and these lines: {2, 901}, {4, 953}, {11, 513}, {12, 13756}, {25, 10016}, {36, 855}, {56, 17101}, {114, 1281}, {115, 661}, {116, 3835}, {119, 517}, {120, 5087}, {121, 3814}, {133, 5146}, {149, 14513}, {153, 14511}, {226, 24201}, {244, 6615}, {1319, 1846}, {1566, 6544}, {1647, 5516}, {1878, 22835}, {2969, 20620}, {3120, 7336}, {3326, 10017}, {4370, 5513}, {4404, 24026}, {5954, 5993}, {5957, 11792}, {7951, 23153}, {8286, 8819}, {15614, 21252}, {17036, 20096}, {17605, 23152}, {24250, 25760}
X(3259) = midpoint of X(i) and X(j) for these {i,j}: {4, 953}, {56, 17101}, {149, 14513}, {153, 14511}
X(3259) = complement of X(901)
X(3259) = anticomplement of X(22102)
X(3259) = reflection of X(i) and X(j) for these {i,j}: {901, 22102}, {3937, 14115}, {6073, 119}, {6075, 11}
X(3259) = reflection of X(3937) in the line X(1)X(3)
X(3259) = {X(2),X(901)}-harmonic conjugate of X(22102)
X(3259) = polar circle inverse of X(1309)
X(3259) = orthoptic circle of the Steiner inellipe inverse of X(2726)
X(3259) = complement of the isogonal of X(900)
X(3259) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 900}, {2, 4928}, {6, 3960}, {31, 3310}, {42, 21894}, {44, 514}, {244, 1647}, {513, 519}, {514, 3834}, {519, 513}, {522, 5123}, {649, 16610}, {661, 3936}, {667, 8610}, {678, 6544}, {693, 21241}, {765, 6550}, {876, 25351}, {900, 10}, {902, 650}, {1019, 4395}, {1023, 4422}, {1319, 522}, {1404, 905}, {1635, 2}, {1639, 3452}, {1647, 11}, {1877, 521}, {1960, 37}, {2087, 1086}, {2161, 21198}, {2251, 6586}, {2325, 20317}, {2429, 25097}, {3251, 4370}, {3264, 21260}, {3285, 14838}, {3669, 17067}, {3689, 4521}, {3762, 141}, {3911, 4885}, {3943, 4129}, {4120, 1211}, {4358, 3835}, {4448, 17793}, {4530, 26932}, {4730, 1213}, {4768, 1329}, {4893, 27751}, {4895, 9}, {5440, 20315}, {6544, 16594}, {14407, 16589}, {14408, 6376}, {14427, 6554}, {14429, 21530}, {14437, 13466}, {14584, 3738}, {16704, 4369}, {17780, 24003}, {21805, 661}, {22086, 1214}, {23344, 24036}, {23703, 3035}, {23757, 119}, {23838, 3036}, {24004, 27076}
X(3259) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3310}, {4, 900}, {11, 1647}, {513, 6550}, {1145, 23757}
X(3259) = crosssum of X(100) and X(104)
X(3259) = crossdifference of every pair of points on the line X(2423)X(2427)
X(3259) = crosspoint of X(i) and X(j) for these (i,j): {513, 517}, {900, 14584}, {1145, 23757}
X(3259) = X(i)-isoconjugate of X(j) for these (i,j): {104, 9268}, {765, 10428}, {909, 5376}
X(3259) = inverse-in-polar-circle of X(1309)
X(3259) = perspector of circumconic centered at X(3310)
X(3259) = orthopole of line X(3)X(8)
X(3259) = center of circumconic that is locus of trilinear poles of lines passing through X(3310); this conic is a rectangular circumhyperbola that is isogonal conjugate of line X(3)X(8)
X(3259) = barycentric product X(i) X(j) for these {i,j}: {514, 23757}, {900, 10015}, {908, 1647}, {1086, 1145}, {1769, 3762}, {1846, 26932}, {2087, 3262}, {2397, 6550}, {4120, 23788}, {4530, 22464}, {17205, 21942}
X(3259) = barycentric quotient X(i) / X(j) for these {i,j}: {517, 5376}, {900, 13136}, {1015, 10428}, {1145, 1016}, {1769, 3257}, {2087, 104}, {2183, 9268}, {2397, 6635}, {2427, 6551}, {2804, 4582}, {3310, 901}, {6550, 2401}, {8661, 2423}, {10015, 4555}, {23757, 190}, {23788, 4615}
X(3260) lies on these lines: 4,69 94,323 98,2855 99,477 183,1995 290,892 298,300 298,301 324,394 325,523 328,1494 338,524 1138,1272
X(3260) = isotomic conjugate of X(74)
X(3260) = anticomplement of X(3003)
X(3260) = X(328)-Ceva conjugate of X(311)
X(3260) = cevapoint of X(i) and X(j) for these (i,j): (2,146), (69,1272), (323,2071)
X(3260) = X(113)-cross conjugate of X(2)
X(3260) = crosspoint of X(300) and X(301)
X(3260) = crossdifference of every pair of points on the line X(32)X(3049)
X(3260) = trilinear pole of line X(1637)X(5664)
X(3260) = pole wrt polar circle of trilinear polar of X(8749) (line X(25)X(512))
X(3260) = polar conjugate of X(8749)
X(3261) lies on these lines: 2,2412 75,522 86,1459 99,2690 100,2860 325,523 514,1921 798,802 824,1577
X(3261) = isogonal conjugate of X(32739)
X(3261) = isotomic conjugate of X(101)
X(3261) = complement of X(21225)
X(3261) = anticomplement of X(6586)
X(3261) = X(i)-Ceva conjugate of X(j) for these (i,j): (561,1111), (668,1233), (670,1269), (1978,76)
X(3261) = cevapoint of X(i) and X(j) for these (i,j): (2,150), (850,1577)
X(3261) = X(i)-cross conjugate of X(j) for these (i,j): (116,2), (1111,561), (1577,693), (1978,76)
X(3261) = crosspoint of X(76) and X(1978)
X(3261) = crosssum of X(i) and X(j) for these (i,j): (32,1919), (213,3063), (1980,2205)
X(3261) = crossdifference of every pair of points on the line X(32)X(560)
X(3261) = trilinear pole of line X(1111)X(3120)
X(3261) = pole wrt polar circle of trilinear polar of X(8750) (line X(25)X(41))
X(3261) = polar conjugate of X(8750)
X(3262) lies on these lines: 7,8 92,345 99,2687 100,2861 105,2865 264,1969 286,1792 311,313 314,1389 325,523 1111,1266 1232,1269 1332,1944
X(3262) = isogonal conjugate of X(34858)
X(3262) = isotomic conjugate of X(104)
X(3262) = cevapoint of X(2) and X(153)
X(3262) = X(119)-cross conjugate of X(2)
X(3262) = crosssum of X(2175) and X(2251)
X(3262) = crossdifference of every pair of points on the line X(32)X(3063)
X(3262) = X(19)-isoconjugate of X(14578)
X(3263) lies on these lines: 2,37 8,304 10,1930 85,341 99,1325 100,2862 104,2865 109,2866 274,1390 305,561 313,1233 314,2346 325,523 442,1228 742,2238
X(3263) = isotomic conjugate of X(105)
X(3263) = anticomplement of X(3290)
X(3263) = X(120)-cross conjugate of X(2)
X(3263) = crosspoint of X(75) and X(334)
X(3263) = crosssum of X(31) and X(2210)
X(3263) = crossdifference of every pair of points on the line X(32)X(667)
X(3263) = trilinear pole of line X(918)X(4437)
X(3263) = pole wrt polar circle of trilinear polar of X(8751) (line X(25)X(884))
X(3263) = polar conjugate of X(8751)
X(3264) lies on these lines: 10,75 86,1222 99,2758 100,2863 190,2183 320,668 325,523 350,899 730,2234 1240,1268
X(3264) = isotomic conjugate of X(106)
X(3264) = X(121)-cross conjugate of X(2)
X(3264) = crossdifference of every pair of points on the line X(32)X(1919)
X(3264) = anticomplement of X(8610)
X(3264) = trilinear pole of line X(3762)X(4120)
X(3264) = pole wrt polar circle of trilinear polar of X(8752) (line X(25)X(8643))
X(3264) = polar conjugate of X(8752)
X(3265) lies on the Kiepert Parabola and these lines: 2,2419 99,1304 325,523 441,525 520,4131 2501,2799
X(3265) = midpoint of X(647) and X(2525)
X(3265) = isogonal conjugate of X(32713)
X(3265) = isotomic conjugate of X(107)
X(3265) = complement of X(33294)
X(3265) = X(i)-Ceva conjugate of X(j) for these (i,j): (99,69), (3267,525)
X(3265) = X(i)-cross conjugate of X(j) for these (i,j): (122,2), (520,525), (2972,394)
X(3265) = crosspoint of X(69) and X(99)
X(3265) = crosssum of X(i) and X(j) for these (i,j): (6,2485), (25,512), (1974,2489)
X(3265) = crossdifference of every pair of points on the line X(25)X(32)
X(3265) = anticomplement of X(6587)
X(3265) = perspector of hyperbola {{A,B,C,X(69),X(76)}} (the isotomic conjugate of the van Aubel line)
X(3265) = pole wrt polar circle of trilinear polar of X(6529) (line X(25)X(393))
X(3265) = polar conjugate of X(6529)
X(3265) = trilinear pole of line X(3269)X(15526)
X(3266) lies on these lines: 2,39 23,99 98,2858 110,2868 325,523 1975,1995
X(3266) = isogonal conjugate of X(32740)
X(3266) = isotomic conjugate of X(111)
X(3266) = anticomplement of X(3291)
X(3266) = trilinear pole of line X(690)X(5181)
X(3266) = pole wrt polar circle of trilinear polar of X(8753) (X(25)X(2489))
X(3266) = polar conjugate of X(8753)
X(3266) = crossdifference of every pair of points on line X(32)X(669)
X(3267) is the trilinear pole of the line X(125)X(339. Randy Hutson, August 15, 2013
X(3267) lies on these lines: 2,2485 99,935 100,2859 325,523
X(3267) = isotomic conjugate of X(112)
X(3267) = anticomplement of X(2485)
X(3267) = X(i)-Ceva conjugate of X(j) for these (i,j): (305,339), (670,1502)
X(3267) = X(i)-cross conjugate of X(j) for these (i,j): (127,2), (339,305()
X(3267) = crosspoint of X(i) and X(j) for these (i,j): (99,1799), (670,1502)
X(3267) = crosssum of X(i) and X(j) for these (i,j): (32,3049), (512,1843), (669,1501), (2489,3199)
X(3267) = crossdifference of every pair of points on the line X(32)X(682)
X(3267) = polar conjugate of X(32713)
X(3268) lies on these lines: 2,1637 99,110 147,2793 325,523 339,2972 525,1636 1297,2373 2394,2986
X(3268) = isogonal conjugate of X(14560)
X(3268) = isotomic conjugate of X(476)
X(3268) = anticomplement of X(1637)
X(3268) = X(3258)-cross conjugate of X(2)
X(3268) = crosspoint of X(99) and X(1494)
X(3268) = crosssum of X(512) and X(1495)
X(3268) = crossdifference of every pair of points on the line
X(32)X(3124)
X(3269) lies on the Brocard inellipse, the inconic with perspector X(2052), and on these lines: 3,248 4,1987 6,74 32,1204 39,185 64,2207 99,287 115,125 184,574 186,1971 339,525 1425,1500 1636,2972 1899,2549
X(3269) = reflection of X(3331) in X(232)
X(3269) = isogonal conjugate of X(23582)
X(3269) = isotomic conjugate of polar conjugate of X(20975)
X(3269) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,647), (64,512), (66,3005), (287,684), (394,520), (1177,351), (2052, 523)
X(3269) = crosspoint of X(6) and X(647)
X(3269) = crosssum of X(i) and X(j) for these (i,j): (2,648), (3,1625), (4,112), (6,1624), (99,315), (107,393), (110,577), (162,2326)
X(3269) = crossdifference of every pair of points on the line X(107)X(110)
X(3269) = X(19)-isoconjugate of X(18020)
X(3269) = X(92)-isoconjugate of X(250)
X(3269) = crosssum of X(2479) and X(2480)
X(3269) = barycentric square of X(656)
X(3270) lies on the orthic inconic and on these lines: 1,185 3,1813 11,125 25,2192 33,51 55,184 56,1204 64,1398 103,1461 217,1500 497,1899 511,3100 651,2808 926,2170 1069,1092 1409,2293 1827,2262 2637,2638 2876,3056
X(3270) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,647), (3,652), (4,650), (6,657), (64,649), (74,654), (2192,663)
X(3270) = crosspoint of X(i) and X(j) for these (i,j): (1,1021), (3,652), (4,650), (6,1459), (1146,2968)
X(3270) = crosssum of X(i) and X(j) for these (i,j): (1,1020), (2,1897), (3,651), (4,653), (7,934), (100,329), (108,278)
X(3270) = crossdifference of every pair of points of the line X(651)X(653)
X(3270) = orthic-isogonal conjugate of X(650)
X(3270) = perspector of orthic triangle and tangential triangle of Feuerbach hyperbola
X(3270) = X(63)-isoconjugate of X(23984)
X(3270) = X(92)-isoconjugate of X(1262)
X(3270) = polar conjugate of isotomic conjugate of X(35072)
X(3271) lies on the Mandart inellipse and on these lines: 1,2810 6,692 8,646 9,3056 11,124 21,1682 25,1397 31,51 44,674 55,2316 56,1461 65,2835 100,3030 105,651 215,2194 238,511 244,1357 373,750 389,3073 512,2643 513,1086 614,1401 926,2170 1015,1960 1083,1332 1361,1411 1365,2969 1428,3220 1843,2212 1977,3124 2092,2309 2161,2875 2183,2223 2293,2347 3125,4560
X(3271) = isogonal conjugate of X(4998)
X(3271) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,3063), (8,650), (25,667), (31,512), (42,2488), (55,663), (56,649), (105,665), (244,1015), (263,788), (1411,3310), (2218,647)
X(3271) = crosspoint of X(i) and X(j) for these (i,j): (6,513), (55,663), (56,651), (244,2170)
X(3271) = crosssum of X(i) and X(j) for these (i,j): (2,100), (7,664), (8,190), (56,651), (480,644), (1259,1332)
X(3271) = crossdifference of every pair of points of the line X(190)X(644)
X(3271) = X(59)-isoconjugate of X(75)
X(3271) = trilinear square of X(6729)
Points Associated with Equilateral Triangles
The points X(3272) to X(3283) are associated with six equilateral triangles related to the (1st) Morley triangle; trilinears for vertices of these triangles are given just below. Received from Milorad R. Stevanovic, Nov. 24, 2007 and Dec. 25, 2007.
1st Morley triangle, M1M2M3,
where
M1 = 1 : 2 cos C/3 : 2 cos B/3
M2 = 2 cos C/3 : 1 : 2 cos A/3
M3 = 2 cos B/3 : 2 cos A/3 : 1
2nd Morley triangle, P1P2P3,
where
P1 = 1 : 2 cos(C/3 - 2π/3) : 2
cos(B/3 - 2π/3)
P2 = 2 cos(C/3 - 2π/3) : 1 : 2
cos(A/3 - 2π/3)
P3 = 2 cos(B/3 - 2π/3) : 2
cos(A/3 - 2π/3) : 1
3rd Morley triangle, S1S2S3,
where
S1 = 1 : 2 cos(C/3 - 4π/3) : 2
cos(B/3 - 4π/3)
S2 = 2 cos(C/3 - 4π/3) : 1 : 2
cos(A/3 - 4π/3)
S3 = 2 cos(B/3 - 4π/3) : 2
cos(A/3 - 4π/3) : 1
The triangles M1M2M3,
P1P2P3,
S1S2S3 are discussed in TCCT, page
165-166. The three are pairwise homothetic. See also
1st
Morley triangle at MathWorld
2nd
Morley triangle at MathWorld
3rd
Morley triangle at MathWorld
Unique equilateral triangle inscribed in ABC and homothetic to
the 1st Morley triangle, J1J2J3,
where
J1 = 0 : sin(A/3 - C/3 + π/3) :
sin(A/3 - B/3 + π/3)
J2 = sin(B/3 - C/3 + π/3) : 0 :
sin(B/3 - A/3 + π/3)
J3 = sin(C/3 - B/3 + π/3) :
sin(C/3 - A/3 + π/3) : 0
Circumtangential triangle, inscribed in the circumcircle of ABC and homothetic to the 1st Morley triangle, T1T2T3. For trilinears, of the vertices, see TCCT, page 166, or MathWorld. Trilinears found by M. Stevanovic:
T1 = sin(B/3 - A/3 + π/3)
sin(C/3 - A/3 + π/3) : sin(C/3 - B/3) sin(B/3 - A/3 + π/3) :
sin(B/3 - C/3) sin(C/3 - A/3 + π/3)
T2 = sin(C/3 - A/3) sin(A/3 - B/3 +
π/3) : sin(C/3 - B/3 + π/3) sin(A/3 - B/3 + π/3) : sin(A/3 -
C/3) sin(C/3 - B/3 + π/3)
T3 = sin(B/3 - A/3) sin(A/3 - C/3 +
π/3) : sin(A/3 - B/3) sin(B/3 - C/3 + π/3) : sin(A/3 - C/3 +
π/3) sin(B/3 - C/3 + π/3)
Circumnormal triangle inscribed in the circumcircle of ABC and homothetic to the 1st Morley triangle, N1N2N3. For trilinears, of the vertices, see TCCT, page 166, or MathWorld. Trilinears found by M. Stevanovic:
N1 = - cos(B/3 - A/3 + π/3)
cos(C/3 - A/3 + π/3) : cos(B/3 - C/3) cos(B/3 - A/3 + π/3) :
cos(B/3 - C/3) cos(C/3 - A/3 + π/3)
N2 = cos(C/3 - A/3) cos(A/3 - B/3 +
π/3) : - cos(C/3 - B/3 + π/3) cos(A/3 - B/3 + π/3) : cos(A/3 -
A/3) cos(C/3 - B/3 + π/3)
N3 = cos(A/3 - B/3) cos(A/3 - C/3 +
π/3) : cos(A/3 - B/3) cos(B/3 - C/3 + π/3) : - cos(A/3 - C/3 +
π/3) cos(B/3 - C/3 + π/3)
See also X(3602)-X(3609) and
Jean-Pierre Ehrmann and Bernard Gibert, "A Morley Configuration," Forum Geometricorum 1 (2001) 51-58. (Click here to download a PDF of this article.)
Peter Moses notes (April, 14, 2008) that, regarding the reference triangle R and the 6 equilateral triangles (here abbreviated as M, P, S, T, N, J), there are 21 pairs, and each pair except {R,T}, {R,N}, {R, J} are perspective. Perspectors for the remaining 18 pairs are given by this table:
| Pair | Perspector |
|---|---|
| R, M | X(357) |
| R, P | X(1136) |
| R, S | X(1134) |
| M, P | X(358) |
| M, S | X(1135) |
| M, T | X(3278) |
| M, N | X(3279) |
| M, J | X(3273) |
| P, S | X(1137) |
| P, T | X(3280) |
| P, N | X(3281) |
| P, J | X(3274) |
| S, T | X(3282) |
| S, N | X(3283) |
| S, J | X(3275) |
| T, N | X(3) |
| T, J | X(3334) |
| N, J | X(3335) |
Centers of the six equilateral triangles are as follows:
| Triangle | Center |
|---|---|
| M | X(356) |
| P | X(3276) |
| S | X(3277) |
| T | X(3) |
| N | X(3) |
| J | X(3272) |
Let A1B1C1, A2B2C2, A3B3C3 be the 1st Morley, 2nd Morley and 3rd Morley triangles of ABC, respectively. Then {A, A1, A2, A3} are concyclic on a circle, A0.
{B, B1, B2, B3} are concyclic on a circle, B0.
{C, C1, C2, C3} are concyclic on a circle C0.
The radical center of A0, B0, C0 is X(3272). (César Lozada, June 16, 2018)
X(3272) lies on these lines: 3,3334 356,3273 357,1135 358,1136 396,523 1134,1137 3274,3276 3275,3277
X(3272) = {X(357),X(3603)}-harmonic conjugate of X(1135)
Let A'B'C' be the 1st Morley triangle. Let La be the line through A' parallel to BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(3273). (Randy Hutson, July 20, 2016)
X(3273) lies on these lines: 16,358 356,3272 357,1136
X(3273) = crosssum of X(1135) and X(3274)
X(3273) = isogonal conjugate of X(3604)
X(3273) = perspector of ABC and unary cofactor triangle of 3rd Morley triangle
Let A'B'C' be the 1st Morley triangle. Let La be the trilinear polar of A', and define Lb, Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(3274). (Randy Hutson, July 20, 2016)
X(3274) lies on these lines: 16,358 1134,1136 3272,3276
X(3274) = isogonal conjugate of X(3602)
X(3274) = crosssum of X(358) and X(3275)
X(3274) = crossdifference of every pair of points on the perspectrix of ABC and 1st Morley triangle
X(3274) = perspector of ABC and unary cofactor triangle of 1st Morley triangle
X(3274) = perspector of ABC and unary cofactor triangle of 1st Morley adjunct triangle
X(3275) lies on these lines: 16,358 357,1134 3272,3277
X(3275) = isogonal conjugate of X(3603)
X(3275) = perspector of ABC and unary cofactor triangle of 2nd Morley triangle
X(3275) = perspector of ABC and unary cofactor triangle of 2nd Morley adjunct triangle
X(3275) = crosssum of X(1137) and X(3273)
X(3276) lies on these lines: 3,3280 356,357 1136,1137 3272,3274
X(3276) = isogonal conjugate of X(3606)
X(3277) lies on these lines: 3,3282 356,1134 1136,1137 3272,3275
X(3277) = isogonal conjugate of X(3607)
X(3278) lies on these lines: 3,356 16,358
X(3279) lies on these lines: 3,356 358,3281 1135,3283
X(3280) lies on these lines: 3,3276 16,358
X(3281) lies on these lines: 3,3276 358,3279 1137,3283
X(3282) lies on these lines: 3,3277 16,358
X(3283) lies on these lines: 3,3277 1135,3279 1137,3281
Let L denote the line through X(4) perpendicular to the Euler line. Coefficients for a trilinear equation for L are the trilinears for X(3284).
X(3284) lies on these lines: 2,340 3,6 23,232 30,1990 112,2693 231,2072 248,895 393,3146 401,648 441,425 520,647
X(3284) = midpoint of X(401) and X(648)
X(3284) = isogonal conjugate of X(16080)
X(3284) = complement of X(340)
X(3284) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1511), (2420,1636)
X(3284) = X(1636)-cross conjugate of X(2420)
X(3284) = crosssum of X(i) and X(j) for these (i,j): (4,1990), (6,186), (125,1637), (281,860), (470,471)
X(3284) = crosspoint of X(2) and X(265)
X(3284) = crossdifference of every pair of points on the line X(4)X(523)
X(3284) = X(92)-isoconjugate of X(74)
X(3284) = perspector of circumconic centered at X(1511)
X(3284) = center of circumconic that is locus of trilinear poles of lines passing through X(1511)
X(3284) = MacBeath-circumconic-inverse of X(3)
X(3284) = Moses-circle-inverse of X(32761)
X(3284) = {X(61),X(62)}-harmonic conjugate of X(389)
X(3284) = {X(39377),X(39378)}-harmonic conjugate of X(34329)
Let L denote the line through X(10) perpendicular to the Euler line. Coefficients for a trilinear equation for L are the trilinears for X(3285).
X(3285) lies on these lines: 3,6 21,45 44,2251 55,2206 81,89 110,2384 112,953 649,834
X(3285) = isogonal conjugate of X(4080)
X(3285) = cevapoint of X(902) and X(2251)
X(3285) = crosspoint of X(81) and X(759)
X(3285) = crosssum of X(37) and X(758)
X(3285) = crossdifference of every pair of points on the line X(10)X(523)
Let L denote the line through X(37) perpendicular to the Euler line. Coefficients for a trilinear equation for L are the trilinears for X(3286).
X(3286) lies on these lines: 3,6 7,21 28,277 31,2274 36,238 55,81 63,2352 110,840 141,1009 198,1778 241,1876 283,1037 333,1376 518,2223 672,1818 741,813 759,1308 940,1011 958,1010
X(3286) = isogonal conjugate of X(13576)
X(3286) = inverse-in-circumcircle of X(3110)
X(3286) = cevapoint of X(672) and X(2223)
X(3286) = X(2254)-cross conjugate of X(2283)
X(3286) = crosspoint of X(58) and X(741)
X(3286) = crosssum of X(10) and X(740)
X(3286) = crossdifference of every pair of points on the line X(37)X(523)
Let L denote the line through X(1) parallel to the Brocard axis, X(3)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3287).
X(3287) lies on these lines: 6,523 37,2605 44,513
X(3287) = isogonal conjugate of X(37137)
X(3287) = cevapoint of X(3907) and X(30584)
X(3287) = crosssum of X(i) and X(j) for these {i,j}: {1, 3287}, {57, 7180}, {514, 30097}, {650, 3666}, {665, 34253}, {4841, 21471}, {20284, 24533}
X(3287) = crossdifference of every pair of points on line X(1)X(256)
X(3287) = X(i)-Ceva conjugate of X(j) for these (i,j): (83,11), (99,55), (666,385)
X(3287) = crosspoint of X(i) and X(j) for these {i,j}: {1, 37137}, {9, 645}, {651, 2298}
X(3287) = bicentric difference of PU(88)
X(3287) = PU(88)-harmonic conjugate of X(1284)
Let L denote the line through X(2) parallel to the Brocard axis, X(3)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3288).
X(3288) lies on these lines: 6,523 111,2698 187,237 323,401 352,1499 419,2501
X(3288) = reflection of X(i) in X(j) for these (i,j): (2451,3049), (3049,3050)
X(3288) = crosspoint of X(99) and X(3114)
X(3288) = crosssum of X(512) and X(3117)
X(3288) = crossdifference of every pair of points on the line X(2)X(51)
X(3288) = X(2)-Ceva conjugate of X(38997)
X(3288) = perspector of hyperbola {{A,B,C,X(6),X(95),X(98)}}
X(3288) = radical center of {circumcircle, Brocard circle, orthosymmedial circle}
Let L denote the line through X(4) perpendicular to the Brocard axis, X(3)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3289).
X(3289) lies on these lines: 2,6 3,217 30,1625 110,1971 112,2706 115,1568 184,418 232,511 520,647 571,1501
X(3289) = reflection of X(3331) in X(1625)
X(3289) = isogonal conjugate of isotomic conjugate of X(36212)
X(3289) = X(i)-Ceva conjugate of X(j) for these (i,j): (287,3), (2421,684)
X(3289) = crosspoint of X(3) and X(287)
X(3289) = crosssum of X(4) and X(232)
X(3289) = crossdifference of every pair of points on the line X(4)X(512)
Let L denote the line through X(3) perpendicular to the line X(1)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3290).
X(3290) lies on these lines: 1,2271 2,37 6,354 9,982 19,1611 25,1841 65,2176 105,910 111,1290 120,1738 172,1104 213,942 230,231 241,292 244,672 518,2238 1018,1739 1149,2170 1212,2275
X(3290) = midpoint of X(3125) and X(3230)
X(3290) = isogonal conjugate of X(2991)
X(3290) = complement of X(3263)
X(3290) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,120), (666,513)
X(3290) = crosspoint of X(2) and X(105)
X(3290) = crosssum of X(6) and X(518)
X(3290) = crossdifference of every pair of points on the line X(3)X(667)
X(3290) = perspector of circumconic centered at X(120)
X(3290) = center of circumconic that is locus of trilinear poles of lines passing through X(120)
X(3290) = PU(4)-harmonic conjugate of X(6591)
X(3290) = X(63)-isoconjugate of X(15344)
X(3290) = polar conjugate of isotomic conjugate of X(34381)
X(3290) = crossdifference of PU(44)
Let L denote the line through X(3) perpendicular to the line X(2)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3291).
X(3291) lies on these lines: 2,39 6,373 23,111 25,1611 32,1995 51,1613 110,1692 115,858 230,231 325,1570
X(3291) = midpoint of X(3124) and X(3231)
X(3291) = complement of X(3266)
X(3291) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,126), (892,512)
X(3291) = crosssum of X(i) and X(j) for these (i,j): (6,524), (525,1648)
X(3291) = crossdifference of every pair of points on the line X(3)X(669)
X(3291) = crosspoint of X(2) and X(111)
X(3291) = X(92)-isoconjugate of X(290)
X(3291) = perspector of circumconic centered at X(126)
X(3291) = center of circumconic that is locus of trilinear poles of lines passing through X(126)
X(3291) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)} at X(2) and X(111)
X(3291) = polar conjugate of isotomic conjugate of X(8681)
X(3291) = X(63)-isoconjugate of X(2374)
X(3291) = PU(4)-harmonic conjugate of X(2489)
Let L denote the line through X(4) perpendicular to the line X(2)X(6). Coefficients for a trilinear equation for L are the trilinears for X(3292).
Let O* be the circle with segment X(15)X(16) as diameter (and center X(187). Let P be the perspector of O*. Then X(3292) is the trilinear pole of the polar of P with respect to O*. See X(5642) for a similar property involving the segment X(13)X(14). (Randy Hutson, July 18, 2014)
X(3292) lies on these lines: 2,575 3,49 6,237 23,110 51,576 112,2763 352,2030 450,648 520,647 539,2072 542,858 1209,1493
X(3292) = midpoint of X(110) and X(323)
X(3292) = reflection of X(1495) in X(1)
X(3292) = X(i)-Ceva conjugate of X(j) for these (i,j): (524,187), (895,3)
X(3292) = isotomic conjugate of isogonal conjugate of X(23200)
X(3292) = isotomic conjugate of polar conjugate of X(187)
X(3292) = crosspoint of X(3) and X(895)
X(3292) = crosssum of X(4) and X(468)
X(3292) = X(19)-isoconjugate of X(671)
X(3292) = crossdifference of every pair of points on the line X(4)X(1499)
X(3293) lies on these lines: 1,2 37,762 40,209 55,1724 58,100 71,380 81,1126 191,1045 213,1018 484,1046 740,872 942,1739
X(3293) = reflection of X(1) in X(1193)
X(3293) = X(i)-Ceva conjugate of X(j) for these (i,j): (81,37),
(83,3294), (1126,1)
X(3294) lies on these lines: 1,6 21,101 190,274 573,962 672,1125 846,2664
X(3294) = X(i)-Ceva conjugate of X(j) for these (i,j): (83,3293), (86,42)
X(3295) lies on these lines: 1,3 4,390 5,497 11,498 12,381 20,1056 21,145 30,388 33,1598 34,1597 73,1480 77,1059 149,2476 172,1384 212,1497 221,500 284,2256 355,950 378,1398 382,1478 474,1387 496,1058 595,1126 601,1496 602,1263 902,1468 916,2293 943,1260 944,1012 1036,1807
X(3295) = midpoint of X(1) and X(1697)
X(3295) = isogonal conjugate of X(3296)
X(3295) = crosssum of X(1) and X(3338)
X(3295) = crossdifference of every pair of points of the line X(650)X(2423)
X(3295) = extangents-to-intangents similarity image of X(3)
X(3295) = center of circle that is locus of crosssums of incircle antipodes
X(3295) = X(1595)-of-excentral-triangle
X(3295) = X(1593)-of-2nd-circumperp-triangle
X(3295) = X(11414)-of-1st-circumperp-triangle
X(3295) = excentral-to-2nd-circumperp similarity image of X(1697)
X(3295) = homothetic center of inner Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(3295) = {X(i),X(j)}-harmonic conjugate of X(k) for thewe (i,j,k): (1,3,999), (1,40,942), (55,56,35)
Let A' be the midpoint of X(1) and the A-intouch point. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(3296). (Randy Hutson, January 15, 2019)
X(3296) lies on these lines: 1,376 4,354 7,1058 21,999 65,1000 79,497 80,388 443,942 1039,1870
X(3296) = isogonal conjugate of X(3295)
X(3296) = cevapoint of X(1) and X(3338)
X(3296) = polar conjugate of isotomic conjugate of X(30679)
X(3296) = trilinear pole of line X(650)X(2423)
X(3297) lies on these lines: 1,6 371,999 485,496 486,495 605,1496 1151,2066
X(3298) lies on these lines: 1,6 372,999 485,495 486,496 606,1496 1151,2067
X(3299) lies on these lines: 1,6 35,372 36,371 42,589 46,1702 47,605 482,651
X(3299) = isogonal conjugate of X(3300)
X(3299) = {X(1),X(6)}-harmonic conjugate of X(3301)
X(3300) lies on these lines: 1,615 57,1373 498,1336 499,1123
For a construction of the point X(3300) see Ercole Suppa, euclid 5207.
X(3300) = isogonal conjugate of X(3299)
X(3301) lies on these lines: 1,6 35,371 36,372 42,588 46,1703 47,606 481,651
X(3301) = isogonal conjugate of X(3302)
X(3301) = {X(1),X(6)}-harmonic conjugate of X(3299)
X(3302) lies on these lines: 1,590 57,1374 498,1123 499,1336
X(3302) = isogonal conjugate of X(3301)
X(3303) lies on these lines: 1,3 8,344 11,1058 12,497 42,1191 145,958 198,1953 221,2293 377,528 388,390 474,551 495,546 496,498 499,632 500,1480 575,613 576,611 939,1411 943,1000 950,954 1193,1616 1201,2177
X(3303) = crosssum of X(1) and X(3333)
X(3303) = {X(1),X(40)}-harmonic conjugate of X(354)X(3304) lies on these lines: 1,3 7,1476 11,153 12,1056 31,1616 33,1398 106,386 144,1001 145,1376 198,1100 474,519 495,499 496,546 529,2478 575,611 576,613 1149,1191 1201,1696
X(3304) = midpoint of X(1) and X(3338)
X(3304) = crosssum of X(1) and X(1697)
X(3304) = {X(1),X(56)}-harmonic conjugate of X(55)
X(3304) = {X(55),X(56)}-harmonic conjugate of X(5204)
X(3305) lies on these lines: 1,748 2,7 10,1479 19,469 21,936 43,968 78,405 81,1743 169,857 200,1621 210,1001 306,344 614,984 750,1707 975,1724
X(3305) = {X(2),X(9)}-harmonic conjugate of X(63)
X(3305) = {X(9),X(57)}-harmonic conjugate of X(3219)
X(3306) lies on these lines: 1,88 2,7 46,1125 77,1465 81,2999 85,658 92,1435 145,1706 165,1621 171,614 354,1376 377,1210 612,982 748,1707 940,1100 1001,1155 1038,1393
X(3306) = crossdifference of every pair of points on the line X(663)X(1635)
X(3306) = anticomplement of X(5316)
X(3306) = {X(2),X(7)}-harmonic conjugate of X(908)
X(3306) = {X(2),X(57)}-harmonic conjugate of X(63)
X(3306) = {X(9),X(57)}-harmonic conjugate of X(3218)
As the isogonal conjugate of a point on the circumcircle, X(3307) lies on the line at infinity.
The asymptotes of the Feuerbach hyperbola meet the infinity line in X(3307) and X(3308). See the note at X(2574).
X(3307) lies on these (parallel) lines: 11,2446 30,511 100,1381 104,1382
X(3307) = isogonal conjugate of X(1381)
As the isogonal conjugate of a point on the circumcircle, X(3308) lies on the line at infinity.
X(3308) lies on these (parallel) lines: 11,2447 30,511 100, 1382 104, 1381
X(3308) = isogonal conjugate of X(1382)
As the isogonal conjugate of a point on the circumcircle, X(3309) lies on the line at infinity.
X(3309) lies on these (parallel) lines: 1,3669 3,667 4,885 30,511 74,2752 98,2711 99,2704 100,2742 101,2736 102,2751 103,2725 104,840 109,2730 110,2691 218,2440 644,1292 650,1734 663,905 764,1482 1293,2748 1294,2749 1295,2750 1296,2753 1297,2754
X(3309) = isogonal conjugate of X(1292)
X(3309) = ideal point of PU(44)
X(3309) = bicentric difference of PU(44)
X(3309) = Thomson-isogonal conjugate of X(105)
X(3309) = Lucas-isogonal conjugate of X(105)
X(3309) = Cundy-Parry Psi transform of X(14267)
Barycentric Products of Perpendicular Directions
Suppose P and U are points on the line at infinity, given in barycentric coordinates by P = p : q : r and U = u : v : w. We call P and U perpendicular directions if for every point X not in the line at infinity, the lines XP and XU are perpendicular. (It is well known that if P and U are an antipodal pair on the circumcircle, then their isogonal conjugates are perpendicular directions.)
Theorem: If P and U are perpendicular directions, then their barycentric product lies on the orthic axis.
An outline of a proof follows. For given P on the line at infinity, the direction perpendicular to P is given by
a/d1 : b/d2 : c/d3,
where
d1 = a2qr cos B cos C - p(b2r +
c2q) cos A
d2 = b2rp cos C cos A - q(c2p +
a2r) cos B
d3 = c2pq cos A cos B - r(a2q +
b2p) cos C,
and the barycentric product P*U is the point x : y : z given by
ap/d1 : bq/d2 : cr/d3.
For P to lie on the line at infinity means that r = - p - q. Substitution for r, a computer quickly shows that
x cot A + y cot B + z cot C = 0,
as desired.
The orthic axis is perpendicular to the Euler line. For this and other properties, visit MathWorld. The appearance of (I, J, K) in the following list means that X(K) is the barycentric product of perpendicular directions X(i) and X(j).
(30,1511,1637)
(511,523,2491)
(514,516,676)
(2574,2575,647)
(513,517,3310)
X(3310) lies on these lines: 6,654 37,1639 42,926 230,231 244,665 649,854 909,2423 1024,1945
X(3310) = isogonal conjugate of X(13136)
X(3310) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3259), (1411,3271), (2397,517), (2401,513), (2427,2183)
X(3310) = crosspoint of X(i) and X(j) for these (i,j): (2,901), (106,1461), (513,2401), (517,2397), (650,2432), (2183,2427)
X(3310) = crosssum of X(i) and X(j) for these (i,j): (6,900), (100,2427), (104,2423), (519,3239), (651,2406)
X(3310) = crossdifference of every pair of points on the line X(3)X(8)
X(3310) = polar conjugate of isogonal conjugate of X(23220)
X(3310) = polar conjugate of isotomic conjugate of X(8677)
X(3310) = complement of complement of polar conjugate of isogonal conjugate of X(23184)
X(3310) = X(63)-isoconjugate of X(1309)
X(3310) = perspector of hyperbola {A,B,C,X(4),X(56)} (circumconic centered at X(3259) and the isogonal conjugate of line X(3)X(8))
X(3310) = center of circumconic that is locus of trilinear poles of lines passing through X(3259)
X(3310) = intersection of trilinear polars of X(4) and X(56)
X(3310) = barycentric product of circumcircle intercepts of Sherman line
X(3311) is the perspector of each of the following pairs of triangles:
Lucas central triangle and the symmedial triangle (the cevian triangle of X(6))
Lucas tangents triangle and the Lucas(-1:1) central triangle
Lucas(2:3) central triangle and the circumsymmedial triangle.
Moreover, X(3311), is the radical center of the Lucas(4:1) circles. See X(371) and X(3312). (Randy Hutson,
9/23/2011)
X(3311) is the perspector of each of the pair of the following four triangles:
symmedial triangle. Lucas central triangle, Lucas(-1) secondary central triangle, 1st Lucas(-1) secondary tangents triangle. (Randy Hutson, September 14,. 2016)
X(3311) lies on these lines: 3,6 4,1131 5,1588 25,588 30,1587 381,485 486,590 488,1992 517,1702 591,641 640,1991 999,1124
X(3311) = isogonal conjugate of X(3316)
X(3311) = inverse-in-Brocard-circle of X(3312)
X(3311) = {X(6),X(371)}-harmonic conjugate of X(3)
X(3311) = {X(6),X(1151)}-harmonic conjugate of X(372)
X(3311) = {X(61),X(62)}-harmonic conjugate of X(3592)
X(3311) = {X(371),X(372)}-harmonic conjugate of X(1151)
X(3311) = {X(372),X(1151)}-harmonic conjugate of X(3)
X(3312) is the perspector of each of the following pairs of triangles:
Lucas(-1:1) central triangle and the symmedial triangle
Lucas(-1:1) tangents triangle and the Lucas central triangle
Lucas(-2:3) central triangle and the circumsymmedial triangle.
Moreover, X(3311), is the radical center of the Lucas(-4:1) circles. See X(371) and X(3311). (Randy Hutson, 9/23/2011)
X(3312) is the perspector of each of the pair of the following four triangles:
symmedial triangle, Lucas(-1) central triangle, Lucas secondary central triangle, 1st Lucas secondary tangents triangle (Randy Hutson, September 14, 2016)
X(3312) lies on these lines: 3,6 4,1132 5,1587 25,589 30,1588 381,486 487,1992 517,1703 591,639 615,1656 642,1991 999,1335
X(3312) = isogonal conjugate of X(3317)
X(3312) = inverse-in-Brocard-circle of X(3311)
X(3312) = {X(6),X(372)}-harmonic conjugate of X(3)
X(3312) = {X(6),X(1152)}-harmonic conjugate of X(371)
X(3312) = {X(61),X(62)}-harmonic conjugate of X(3594)
X(3312) = {X(371),X(1152)}-harmonic conjugate of X(3)
X(3312) = {X(371),X(372)}-harmonic conjugate of X(1152)
X(3313) lies on these lines: 3,6 22,206 66,69 141,427 159,394 1205,2854 1216,1352
X(3313) = reflection of X(i) in X(j) for these (i,j): (52,182), (1352,1216), (1843,141)
X(3313) = isotomic conjugate of isogonal conjugate of X(23208)
X(3313) = X(69)-Ceva conjugate of X(141)
X(3313) = crosspoint of X(22) and X(315)
X(3313) = crosssum of X(66) and X(2353)
X(3314) lies on these lines: 2,6 3,147 76,115 315,384 982,2887
X(3314) = complement of X(7766)
X(3314) = anticomplement of X(7792)
X(3314) = isotomic conjugate of X(3407)
X(3314) = crosspoint of X(327) and X(1502)
X(3314) = {X(2),X(193)}-harmonic conjugate of X(16989)
X(3315) lies on these lines: 1,88 2,1280 105,110 149,1086 291,1255 651,1421 982,1621 1283,1623
X(3315) = crosssum of X(678) and X(2087)
X(3316) lies on these lines: 3,1131 4,590 5,1132 6,3317 226,1374 376,1327 485,631
X(3316) = isogonal conjugate of X(3311)
X(3316) = X(1587)-cross conjugate of X(4)
X(3316) = {X(6),X(5067)}-harmonic conjugate of X(3317)
X(3317) lies on these lines: 3,1132 4,615 5,1131 6,3316 226,1373 376,1328 486,631
X(3317) = isogonal conjugate of X(3312)
Points on the Incircle
X(3317) = X(1588)-cross conjugate of X(4)
X(3317) = {X(6),X(5067)}-harmonic conjugate of X(3316)
Suppose that U = u : v : w (trilinears) is a point other than the symmedian point, X(6). Then the incircle transform of U is the point.
T(U) = a (a - b + c) (a + b - c) (c v - b w)^2 : : ,
which lies on the incircle. Geometrically, T(U) is the Brisse transform of the psi transform of X(6) and U. (Brisse transform is described just before X(1254), and psi transform, also called the "Psi mapping" elsewhere in ETC, and TCCT, p. 80, is defined in the preamble just before X(98).
X(3318) lies on the incircle and these lines: 1,1359 11,123 55,108 56,1295 944,1317
X(3318) = reflection of X(1359) in X(1)
X(3318) = X(108)-of-Mandart-incircle-triangleX(3319) lies on the incircle and these lines: 1,3326 11,515 55,2716 56,2222 513,1361 517,1364 522,1317 944,2720
X(3319) = reflection of X(3326) in X(1)
X(3319) = anticomplement of X(5249)
X(3320) lies on the incircle and these lines: 11,132 12,127 55,1297 56,112 65,1367
X(3320) = X(1297)-of-Mandart-incircle-triangle
X(3320) = homothetic center of intangents triangle and reflection of extangents triangle in X(1297)
X(3321) lies on the incircle and these lines: 7,11 55,934 57,1358 1122,1357 1155,1323
X(3321) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,1323), 658,1638)
X(3321) = crosspoint of X(7) and X(1323)
X(3321) = X(111)-of-intouch-triangle
X(3322) lies on the incircle and these lines: 1,3328 7,840 55,2222 56,1308 513,1362 514,1317 516,1155 1283,1284
X(3322) = reflection of X(3328) in X(1)
X(3323) lies on the incircle and these lines: 7,840 55,2736 56,2725 348,1083 514,1358 518,1362
X(3323) = trilinear pole wrt intouch triangle of line X(6)X(7)
X(3324) lies on the incircle and these lines: 12,122 55,1294 56,107 65,1363 133,1838
X(3324) = X(1294)-of-Mandart-incircle-triangle
X(3324) = homothetic center of intangents triangle and reflection of extangents triangle in X(1294)
X(3325) lies on the incircle and these lines: 12,126 55,1296 56,111 65,1366
X(3325) = X(1296)-of-Mandart-incircle-triangle
X(3325) = homothetic center of intangents triangle and reflection of extangents triangle in X(1296)
X(3326) lies on the incircle and these lines: 1,3319 36,1354 55,2222 56,2716 125,2618 513,1364 517,1361 1118,2745
X(3326) = reflection of X(3319) in X(1)
X(3326) = contact point of incircle and the Sherman line (3259,3326) (see http://forumgeom.fau.edu/FG2012volume12/FG201220.pdf)
X(3327) lies on the incircle and these lines: 12,128 55,930 56,1141 496,1263
X(3328) lies on the incircle and these lines: 1,3322 55,1308 56,2717 516,1317 517,1362
X(3328) = reflection of X(i) in X(j) for these (i,j): (1155,1323), (3322,1)
X(3328) = X(7)-Ceva conjugate of X(1638)
X(3328) = crosspoint of X(i) in X(j) for these (i,j): (7,1638), (514,1323)
X(3329) lies on these lines: 2,6 5,147 39,83 98,575 182,262 427,1031 7761,7812
X(3229) = complement of X(3978)
X(3229) = crossdifference of PU(148)
X(3329) = {X(2),X(193)}-harmonic conjugate of X(16990)
X(3329) = {X(6),X(183)}-harmonic conjugate of X(7766)
X(3330) lies on these lines: 4,6 9,1745 37,73 44,513 198,408 216,1765 651,857 1100,2654
X(3330) = X(1294)-Ceva conjugate of X(55)
X(3330) = crosssum of X(1) and X(3330)
X(3330) = crossdifference of every pair of points on the line X(1)X(520)
X(3331) lies on these lines: 4,6 30,1625 74,1987 111,2713 112,1971 187,237
X(3331) = reflection of X(i) in X(j) for these (i,j): (3269,232),
(3289,1625)
X(3331) = crossdifference of every pair of points on the line
X(2)X(520)
X(3332) lies on these lines: 1,7 2,1754 4,6 42,2947 219,2550 278,1456
X(3332) = crossdifference of every pair of points on the line X(520)X(657)
Let I be the incenter, X(1), and let KA be the symmedian point of triangle IBC; define KB, KC cyclically. Let X be the midpoint of segment AI,and define Y, Z cyclically. Then the triangles KAKBKC and XYZ are perspective, and their perspector is X(3333). Contributed by Cosmin Pohoata, April 4, 2008.
Let R be the circumradius, r the inradius, and rA the radius of the A-excircle. The trilinears given above are equivalent to 2R - rA : 2R - rB : 2R - rC. The 1st trilinear representation
(a - b + c)(a + b - c)(a + b + c) + 4abc
shows that X(3333) lies on the line IO of the incenter and the circumcenter, which is also the line X(1)X(57); viz., X(57) has 1st trilinear (a - b + c)(a + b - c) and X(1) has 1st trilinear 1, so that the above representation may be viewed as a linear combination using as coefficients the symmetric functions a + b + c and 4abc. The representation generalizes to
(a - b + c)(a + b - c)(a + b + c) + kabc,
where k is a symmetric function of degree 0 in a,b,c. Every point on the line IO necessary has such a form. (Indeed, every point on every line is given by such linear combinations of any two points on the line.) For the line IO, Peter Moses contributed (April 24, 2008) a list of points with corresponding functions k(a,b,c). Aside from inradius r and circumradius R, other symbols used are identified elsewhere in ETC as indicated, or else are formulated here: s = (a+b+c)/2; J as at X(1113); e as at X(1340); SA, SB SC as at X(1117); and sA = (-a+b+c)/2, sB = (a-b+c), sC = (a+b-c)/2.
| X | k |
|---|---|
| 1 | (k infinite) |
| 3 | k = -2(r + 2R)/R |
| 35 | k = -2(r + 3R)/R |
| 36 | k = -2(r + R)/R |
| 40 | k = -4 |
| 46 | k = -2 |
| 55 | k = -2(r + 4R)/R |
| 56 | k = -2r/R |
| 57 | k = 0 |
| 65 | k = 2r/R |
| 165 | k = -(r +4r)/R |
| 171 | k = -(r2 + 4rR - s2)/(rR) |
| 241 | k = -2s2/(rR + 4R2) |
| 354 | k = 2(r + 4R)//R |
| 484 | k = -3 |
| 559 | k = 2(31/2s//R |
| 940 | k = (a + b + c)3/(abc) |
| 942 | k = 2(r + 2R)/R |
| 980 | k = -2s2(r2 + s2)/(rR(r2 + 4rR + s2)) |
| 982 | k = -(a + b + c)(a2 + b2 + c2)/(abc) |
| 986 | k = (r2 - s2)/(rR) |
| 988 | k = -(r2 + s2)/(rR) |
| 999 | k = -2(r - 2rR)/R |
| 1038 | k = (J2 - 4r/R - 9)/2 |
| 1040 | k = - (J2 + 4r/R + 7)/2 |
| 1060 | k = J2 - 2r/R - 5 |
| 1062 | k = - (J2 + 2r/R + 3) |
| 1082 | k = -2(31/2)s/R |
| 1115 | k = -2(r + 4R)/(3R) |
| 1159 | k = 2(5r + 2R)/(3R) |
| 1214 | k = -2s2/(rR + 2R2) |
| 1319 | k = -6r/R |
| 1381 | k = 2(-r - R + (R2 - 2rR)1/2)/R |
| 1382 | k = 2(-r - R + -(R2 - 2rR)1/2)/R |
| 1385 | k = -2(3r + 2R)/R |
| 1388 | k = -10r/R |
| 1402 | k = -2r(r2 + 4rR + s2)/(R(r2 + s2)) |
| 1403 | k = -2r(r2 + 4rR + s2)/(R(r2 - 4rR + s2)) |
| 1420 | k = -4r/R |
| 1429 | k = -2r(a + b + c)2/(R(a2 + b2 + c2)) |
| 1454 | k = -4r/(2r + R) |
| 1460 | k = 2(area)(a2 + b2 + c2)/(2Rs(r2 - s2)) |
| 1466 | k = -2r(r + 2R)/(rR - 2R2) |
| 1467 | k = -r(r + 2R)/R2 |
| 1470 | k = 2r(r + R)/(R2 - rR) |
| 1482 | k = 2(3r - 2R)/R |
| 1617 | k = -2r(r + 4R)/(2R2 + rR) |
| 1697 | k = -8 |
| 1735 | k = -2 + (J2 - 1)R/(2r) |
| 1754 | k = -(4r + (7 + J2)R)/(2r + 4R2) |
| 1758 | k = -(r2 + 4rR + s2)/(2rR + R2) |
| 1764 | k = -4s2/(r2 + s2) |
| 1771 | k = -2 + (1 - J2)R/(2r) |
| 1936 | k = [s2 - (r + 2R)(4 + 4R)]/(Rr + R2) |
| 2077 | k = - 4 + 2r2 /(R2 - rR) |
| 2078 | k = -2(r2 + 4rR)/(rR + R2) |
| 2093 | k = (r - 2R)/R |
| 2095 | k = (2r2 - 8R2)/(3rR + 2R2) |
| 2098 | k = 2(3r - 4R)/R |
| 2099 | k = 6r/R |
| 2446 | k = -4(1 + (1 - 2r/R)1/2) + 2r/R |
| 2447 | k = -4(1 - (1 - 2r/R)1/2) + 2r/R |
| 2448 | k = -3 + (1 + 2r/R)1/2 |
| 2449 | k = -3 - (1 + 2r/R)1/2 |
| 2556 | k = -4 + 2r(e + E)/(Re - RE), where E = (1 - 2r/R)1/2 |
| 2557 | k = -4 + 2r(e - E)/(Re + RE), where E = (1 - 2r/R)1/2 |
| 2464 | k = -4(1 + e) - 2r/R |
| 2465 | k = -4(1 - e) - 2r/R |
| 2572 | k = -4 - 2r/(R + eR) |
| 2573 | k = -4 - 2r/(R - eR) |
| 2646 | k = -2(3r + 4R)/R |
| 3057 | k = 2(r - 4R)/R |
| 3072 | k = (R - RJ2 -4r)/(2r + 2R) |
| 3075 | k = (SASBSC2)/(abcsAsBsC2) |
| 3245 | k = -2(5R - r)/(3R) |
| 3256 | k = 2r(r + 4R)/(R - rR2) |
| 3303 | k = -2(r + 8R)/R |
| 3304 | k = -2(r - 4R)/R |
| 3333 | k = 4 |
| 3336 | k = -1 |
| 3337 | k = 1 |
| 3338 | k = 2 |
| 3339 | k = r/R |
| 3340 | k = 4r/R |
| 3361 | k = -r/R |
Let A' be the midpoint of X(1) and A-intouch point. Define B' and C' cyclically. Traingle A'B'C' is homothetic to the excentral triangle, and the center of homothety is X(3333). (Randy Hutson, December 2, 2017)
If you have The Geometer's Sketchpad, you can view Pohoata Point.
X(3333) lies on these lines: 1,3 4,1435 7,84 9,1125 10,1056 48,1449 58,2191 81,2360 109,1497 200,474 226,3086 386,1066 388,1210 495,1698 496,1699 515,938 516,1058 518,936 519,1706 580,1471 581,1458 614,1453 1106,2263 1203,3157 1387,1768 1731,2257 2136,3244
X(3333) = midpoint of X(1) and X(3339)
X(3333) = crosssum of X(55) and X(2256)
X(3333) = X(1598)-of-excentral-triangle
The equilateral triangles T1T2T3 and J1J2J3 are defined just before X(3272). The points X(3334) and X(3335), contributed April 16, 2008 by Peter Moses, complete a list (just before X(3272) of perspectors associated with equilateral triangles.
X(3334) lies on these lines: 3,3272 16,358 357,6124 1134,6125 1136,6123
X(3334) = {X(3),X(3272)}-harmonic conjugate of X(3335)
The equilateral triangles N1N2N3 and J1J2J3 are defined just before X(3272).
X(3335) lies on these lines: 3,3272 357,6121 1134,6122 1136,6120 X(3335) = {X(3),X(3272)}-harmonic conjugate of X(3334)X(3336) lies on the Napoleon cubic and these lines: 1,3 2,191 4,1768 5,79 7,498 17,1652 18,1653 54,3468 58,1325 62,2306 63,1698 109,1393 244,595 269,1079 404,758 412,1784 579,1781 583,2160 920,1445 1046,1054 1158,1699 1210,1770 1254,1772 1478,1788 1707,1775 1727,1836 3459,3460 3461,3462
X(3336) = X(i)-Ceva conjugate of X(j) for these (i,j): (5,3468), (77,1745)
X(3336) = {X(1)X(46)}-harmonic conjugate of X(484)
X(3336) = SS(A → 3A) of X(1507)
X(3336) = isogonal conjugate of X(3467)
X(3336) = Kosnita(X(484),X(3)) point
X(3336) = {X(1),X(40)}-harmonic conjugate of X(37563)
X(3337) lies on these lines: 1,3 7,499 11,79 47,1471 58,229 81,501 202,2306 553,1776 946,1768 1111,1434 1399,1421 1731,1781
X(3338) lies on these lines: 1,3 7,90 9,583 38,975 58,614 63,1125 84,1699 169,1475 226,499 255,1471 388,1737 474,518 496,1836 497,1770 553,946 584,1449 990,1717 1056,1788 1106,1448 1210,1478 1398,1905 1435,1838 1452,1870 1468,1718 1537,1768 1723,2260
X(3338) = reflection of X(1) in X(3304)
X(3338) = X(3296)-Ceva conjugate of X(1)
X(3338) = {X(1),X(57)}-harmonic conjugate of X(46)
See the note at X(3333).
X(3339) lies on these lines: 1,3 7,10 109,1451 169,1046 196,1838 208,1844 221,1203 223,2939 269,1126 386,1042 388,533 516,938 518,1706 527,2551 595,1471 758,936 959,978 1210,1699 1406,2003 1698,1788 1721,2955 1767,2956
X(3339) = reflection of X(1) in X(3333)
See the note at X(3333).
The reflection of the excentral triangle in X(1) and the intouch triangle are homothetic from X(3340. (Randy Hutson, January 29, 2015)
X(3340) lies on these lines: 1,3 7,145 8,226 9,1405 34,1126 78,1706 84,1389 109,1468 221,2003 278,1869 386,1457, 388,519 595,1451 950,962 1125,1788 1404,1449 1419,2263 1871,1887 1953,2257
X(3340) = reflection of X(1697) in X(1)
X(3340) = 2nd-extouch-to-intouch similarity image of X(8)
X(3340) = X(1593)-of-intouch-triangle
X(3340) = X(1593)-of-excenters-reflections-triangle
X(3340) = X(11414)-of-Hutson-intouch-triangle
X(3340) = excentral-to-intouch similarity image of X(1697)
X(3341) lies on the Thomson cubic and these lines: 1,1073 2,271 3,3352 4,57 6,282 9,3344 223,3349 280,938
X(3341) = isogonal conjugate of X(3342)
X(3341) = isogonal conjugate of complement of X(34162)
X(3341) = complement of X(1034)
X(3341) = anticomplement of X(20809)
X(3341) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,282), (271,84)
X(3341) = X(i)-cross conjugate of X(j) for these (i,j): (6,3352), (1035,1490)
X(3342) lies on the Thomson cubic and these lines: 1,3343 2,271 3,223 6,3351 9,1249 57,3350
X(3342) = isogonal conjugate of X(3341)
X(3342) = complement of X(34162)
X(3342) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3351), (1034,3345)
X(3342) = X(i)-cross conjugate of X(j) for these (i,j): (6,223), (208,40)
X(3342) = perspector of ABC and antipedal triangle of X(3182)
X(3342) = perspector of pedal and anticevian triangles of X(3345)
X(3342) = perspector of ABC and medial triangle of pedal triangle of X(3347)
X(3342) = perspector of circumconic centered at X(3351)
X(3342) = center of circumconic that is locus of trilinear poles of lines passing through X(3351)
X(3343) lies on the Thomson cubic and these lines: 1,3342 2,1032 3,2130 4,64 6,1073 9,3352 57,282 154,1301 1249,3356
X(3343) = isogonal conjugate of X(3344)
X(3343) = complement of X(1032)
X(3343) = X(2)-Ceva conjugate of X(1073)
X(3343) = X(i)-cross conjugate of X(j) for these (i,j): (6,3349), (1033,1498)
X(3343) = perspector of ABC and antipedal triangle of X(3348)
X(3343) = perspector of pedal and anticevian triangles of X(2130)
X(3343) = perspector of ABC and medial triangle of pedal triangle of X(3183)
X(3343) = perspector of circumconic centered at X(1073)
X(3343) = center of circumconic that is locus of trilinear poles of lines passing through X(1073)
X(3344) lies on the Thomson cubic and these lines: 1,3351 2,1032 3,1033 6,3350 9,3341
X(3344) = isogonal conjugate of X(3343)
X(3344) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3350), (1032,3346)
X(3344) = X(6)-cross conjugate of X(1249)
X(3344) = perspector of ABC and antipedal triangle of X(3183)
X(3344) = perspector of pedal and anticevian triangles of X(3346)
X(3344) = perspector of ABC and medial triangle of pedal triangle of X(3348)
X(3344) = perspector of circumconic centered at X(3350)
X(3344) = center of circumconic that is locus of trilinear poles of lines passing through X(3350)
Let A'B'C' be the cevian triangle of X(329). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3345). (Randy Hutson, July 11, 2019)
Let ABC be a triangle with incenter I = X(1), De Longchamps point L = X(20), and Bevan point Be = X(40). Let DEF = cevian triangle of L. Let A' = ABe∩ID, and define B' and C' cyclically. The triangles ABC and A'B'C' are orthologic. The center of orthology of ABC with respect to A'B'C' is X(3345) and the center of orthology of A'B'C' with respect to ABC is X(9121). (Angel Montesdeoca, October 3, 2023 *)
X(3345) lies on the Darboux cubic and these lines: 1,196 3,223 4,282 20,78 40,219 64,3472 84,2130 165,1794 283,1817 1722,2636 1753,2338
X(3345) = reflection of X(3182) in X(3)
X(3345) = isogonal conjugate of X(1490)
X(3345) = isotomic conjugate of X(33672)
X(3345) = X(1034)-Ceva conjugate of X(3342)
X(3345) = cevapoint of X(649) and X(2638)
X(3345) = X(i)-cross conjugate of X(j) for these (i,j): (34,1), (64,84), (1436,57)
X(3345) = perspector of ABC and the reflection in X(223) of the antipedal triangle of X(223)
Let A'B'C' be the cevian triangle of X(20). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3346). (Randy Hutson, November 18, 2015)
Let M be a point on the slideline BC of triangle ABC. Let (AM) be the circle with diameter AM. Let Mb be the point, other than A, in (AM)∩AC, and let Mc be the point, other than A, in (AM)∩AB. Let Tb be the line tangent to (AM) at Mb, and let Tc be the line tangent to (AM) at Mc. As M varies on BC, the locus of Tb∩Tc is a line, La. Define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABCF, and the perspector is X(3346). (Angel Montesdeoca, November 11, 2017)
X(3346) lies on the Darboux cubic, the cubic pK(X393,X2). and on these lines: 1,3353 3,1033 4,1073 20,394 40,3182 64,3355 84,3472
X(3346) = reflection of X(3183) in X(3)
X(3346) = isogonal conjugate of X(1498)
X(3346) = isotomic conjugate of X(6527)
X(3346) = X(i)-Ceva conjugate of X(j) for these (i,j): (20,3355), (1032,3344)
X(3346) = cevapoint of X(122) and X(523)
X(3346) = X(i)-cross conjugate of X(j) for these (i,j): (64,4), (393,2)
X(3346) = cyclocevian conjugate of X(6504)
X(3346) = intersection of tangents at X(20) and X(253) to Lucas cubic K007
X(3346) = perspector of ABC and the reflection in X(1249) of the antipedal triangle of X(1249)
X(3347) lies on the Darboux cubic and these lines: 1,2130 3,3353 4,3472 20,3182 40,1712 84,3355 1490,1498
X(3347) = reflection of X(3353) in X(3)
X(3347) = isogonal conjugate of X(3182)
X(3347) = X(64)-cross conjugate of X(1)
X(3348) lies on the Darboux cubic and these lines: 1,3472 3,2130 4,3355 20,3183 40,3353 1490,3182
X(3348) = reflection of X(2130) in X(3)
X(3348) = isogonal conjugate of X(3183)
X(3349) lies on the Thomson cubic and these lines: 3,2130 4,2131 9,3351 223,3341
X(3349) = isogonal conjugate of X(3350)
X(3349) = X(6)-cross conjugate of X(3343)
X(3349) = perspector of ABC and antipedal triangle of X(2130)
X(3349) = perspector of pedal and anticevian triangles of X(3348)
X(3349) = perspector of ABC and medial triangle of pedal triangle of X(2131)
X(3350) lies on the Thomson cubic and these lines: 1,3352 2,3349 3,2131 4,1073 6,3344 57,3342
X(3350) = isogonal conjugate of X(3349)
X(3350) = X(2)-Ceva conjugate of X(3344)
X(3350) = X(6)-cross conjugate of X(3356)
X(3350) = perspector of ABC and antipedal triangle of X(2131)
X(3350) = perspector of pedal and anticevian triangles of X(3355)
X(3350) = perspector of ABC and medial triangle of pedal triangle of X(2130)
X(3350) = perspector of circumconic centered at X(3344)
X(3350) = center of circumconic that is locus of trilinear poles of lines passing through X(3344)
X(3351) lies on the Thomson cubic and these lines: 1,3344 2,3352 4,282 57,1073 6,3342 9,3349 223,3356
X(3351) = isogonal conjugate of X(3352)
X(3351) = complement of isotomic conjugate of X(34162)
X(3351) = X(2)-Ceva conjugate of X(3342)
X(3351) = perspector of ABC and antipedal triangle of X(3354)
X(3351) = perspector of pedal and anticevian triangles of X(3472)
X(3351) = perspector of ABC and medial triangle of pedal triangle of X(3353)
X(3351) = perspector of circumconic centered at X(3342)
X(3351) = center of circumconic that is locus of trilinear poles of lines passing through X(3342)
X(3352) lies on the Thomson cubic and these lines: 1,3350 2,3351 3,3341 9,3343
X(3352) = isogonal conjugate of X(3351)
X(3352) = X(6)-cross conjugate of X(3341)
X(3352) = perspector of pedal and anticevian triangles of X(3347)
X(3352) = perspector of ABC and medial triangle of pedal triangle of X(3354)
X(3353) lies on the Darboux cubic and these lines: 1,3346 3,3347 4,282 20,3354 40,3348 64,84 1490,2131 1498,3473
X(3353) = reflection of X(3347) in X(3)
X(3353) = isogonal conjugate of X(3354)
X(3353) = X(20)-Ceva conjugate of X(84)
X(3354) lies on the Darboux cubic and these lines: 1,3355 3,3472 20,3353 40,2130 1490,3183 1498,3182
X(3354) = reflection of X(3472) in X(3)
X(3354) = isogonal conjugate of X(3353)
X(3354) = X(64)-cross conjugate of X(40)
X(3355) lies on the Darboux cubic and these lines: 1,3354 3,2130 4,3348 40,3473 64,3346 84,3347
X(3355) = isogonal conjugate of X(3637)
X(3355) = reflection of X(2131) in X(3)
X(3355) = X(20)-Ceva conjugate of X(3346)
X(3356) lies on the Thomson cubic and these lines: 3,2131 223,3351 1249,3343
X(3356) = X(6)-cross conjugate of X(3350)
X(3356) = perspector of ABC and antipedal triangle of X(3355)
X(3356) = perspector of pedal and anticevian triangles of X(2131)
X(3356) = perspector of ABC and medial triangle of pedal triangle of X(3637)
X(3357) lies on these lines: 64 4,74 20,2888 140,2883 185,378 195,2935 382,1853 389,1593 550,1503 576,2781 1092,2071 1192,1598
X(3357) = midpoint of X(3) and X(64)
X(3357) = reflection of X(2883) in X(140)
X(3357) = inverse-in-Jerabek-hyperbola of X(1204)
X(3357) = {X(4),X(74)}-harmonic conjugate of X(1204)
X(3357) = X(8)-of-Trinh-triangle if ABC is acute
X(3357) = Trinh-isotomic conjugate of X(3098)
X(3358) lies on these lines: 3,9 4,1445 11,57 516,1158 954,1071
X(3358) = midpoint of X(9) and X(84)
X(3359) lies on these lines: 1,3 2,1519 9,119 10,1158 63,2096 72,2057 84,355 200,912 997,2800 1103,3157 1452,1753 1722,3073 2550,3358
X(3359) = midpoint of X(40) and X(57)
X(3359) = midpoint of X(2096) and X(3421)
X(3360) lies on these lines: 1,2162 6,194 159,2076 1740,2176
X(3360) = isogonal conjugate of the isotomic conjugate of X(32747)
X(3360) = X(1613)-Ceva conjugate of X(6)
Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side ob BC opposite A. Define OB and OC cyclically. Let A' be the insimilicanter of OB and OC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3361). See the reference at X(1001).
Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the inverse-in-incircle triangle at X(3361). (Randy Hutson, July 31 2018)
X(3361) lies on these lines: 1,3 7,1125 28,1435 34,2163 58,269 200,404 222,1203 223,1451 386,1458 388,1698 610,2260 614,1448 936,1445 961,1722 978,1400 995,1042 1416,1472 1427,1453 1468,2999
X(3361) = isogonal conjugate of X(4866)X(3362) lies on the McKay cubic and these lines: 1,1075 3,1745 46,296 158,2638 283,1816 580,1795 1069,1936 1794,2947
X(3362) = isogonal conjugate of X(1745)
X(3362) = cevapoint of X(650) and X(2638)
X(3362) = X(4)-cross conjugate of X(1)
X(3362) = trilinear product of PU(126)
As a point on the Euler line, X(3363) has Shinagawa coefficients (2(E + F)2 + 3S2,9S2).
X(3363) is the symmedian point of the pedal triangle of the centroid. A second construction uses the midpoint X(597) of the symmedian point and centroid and the midpoint X(115) of the two Fermat points, X(13) and X(14); specifically, X(3363) is the point in which the line X(115)X(597) meets the Euler line. Contributed by Po-chieh Chen and Shao-cheng Liu, May 29, 2008.
If you have The Geometer's Sketchpad, you can view X(3363).
X(3363) lies on these lines: 2,3 115,597
X(3364) and related points occupy Bernard Gibert's Table 38 concerning these three loci: Brocard axis, Kiepert hyperbola, and the cubic K457. Trilinear equations for these curves, of degrees 1,2,3, respectively, in the variables a,b,c, are given as follows:
| Curve | Trilinears |
|---|---|
| Brocard axis | cos(A + t) : cos(B + t) : cos(C + t) |
| Kiepert hyperbola | sec(A + t) : sec(B + t) : sec(C + t) |
| K457 | tan(A + t) : tan(B + t) : tan(C + t) |
X(3364) lies on these lines: 3,6 14,485 17,486 18,590 202,2067 203,1124
X(3364) = isogonal conjugate of X(3366)
X(3364) = inverse-in-Brocard-circle of X(3390)
X(3364) = X(18)-Ceva conjugate of X(3365)
X(3364) = X(3206)-cross conjugate of X(3365)
X(3364) = {X(15),X(62)}-harmonic conjugate of X(3365)
X(3364) = {X(371),X(372)}-harmonic conjugate of X(3389)
X(3365) lies on these lines: 3,6 14,486 17,485 18,615 203,1335
X(3365) = isogonal conjugate of X(3367)
X(3365) = inverse-in-Brocard-circle of X(3389)
X(3365) = X(18)-Ceva conjugate of X(3364)
X(3365) = X(3206)-cross conjugate of X(3364)
X(3365) = {X(15),X(62)}-harmonic conjugate of X(3364)
X(3365) = {X(371),X(372)}-harmonic conjugate of X(3390)
X(3366) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3366) lies on the Kiepert hyperbola and these lines: 5,15 14,371 17,372 18,590
X(3366) = isogonal conjugate of X(3364)
X(3366) = X(62)-cross conjugate of X(3367)
X(3367) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3367) lies on the Kiepert hyperbola and these lines: 5,15 14,372 17,371 18,615
X(3367) = isogonal conjugate of X(3365)
X(3367) = X(62)-cross conjugate of X(3366)
X(3368) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3368).
X(3368) lies on these lines: 2,1140 3,6 4,1139
X(3368) = isogonal conjugate of X(3370)
X(3368) = inverse-in-Brocard-circle of X(3395)
X(3368) = X(3382)-Ceva conjugate of X(3369)
X(3368) = {X(371),X(372)}-harmonic conjugate of X(3396)
X(3369) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3369).
X(3369) lies on these lines: 2,1139 3,6
X(3369) = isogonal conjugate of X(1140)
X(3369) = {X(371),X(372)}-harmonic conjugate of X(3395)
X(3369) = inverse-in-Brocard-circle of X(3396)
X(3369) = X(3382)-Ceva conjugate of X(3368)
X(3369) = crosssum of X(3393) and X(3394)
X(3370) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
Let A' be the center of the regular pentagon AA2A3A4A5, with opposite orientation than ABC and such that B ∈ A2A3 and C ∈ A4A5. Build B' and C' cyclically. Then the lines AA', BB', CC' concur at X(3370). (César E. Lozada, May 14, 2019), Hyacinthos #29013).
If you have The Geometer's Sketchpad, you can view X(3370).
X(3370) lies on the Kiepert hyperbola and these lines: 3,1139 5,1140
X(3370) = isogonal conjugate of X(3368)
X(3370) = X(3380)-cross conjugate of X(1140)
X(3371) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3371) lies on this line: 3,6
X(3371) = isogonal conjugate of X(3373)
X(3371) = inverse-in-Brocard-circle of X(3386)
X(3371) = X(486)-Ceva conjugate of X(3372)
X(3371) = insimilicenter of circumcircle and 2nd Kenmotu circle
X(3371) = {X(371),X(372)}-harmonic conjugate of X(3385)
X(3372) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3372) lies on this line: 3,6
X(3372) = isogonal conjugate of X(3374)
X(3372) = inverse-in-Brocard-circle of X(3385)
X(3372) = X(486)-Ceva conjugate of X(3371)
X(3372) = exsimilicenter of circumcircle and 2nd Kenmotu circle
X(3372) = {X(371),X(372)}-harmonic conjugate of X(3386)
X(3373) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3373) lies on the Kiepert hyperbola and this line: 5,371
X(3373) = isogonal conjugate of X(3371)
X(3373) = X(372)-cross conjugate of X(3374)
X(3374) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3374) lies on the Kiepert hyperbola and this line: 5,371
X(3374) = isogonal conjugate of X(3372)
X(3374) = X(372)-cross conjugate of X(3373)
X(3375) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3375) lies on these lines: 17,1652 19,2290
X(3375) = isogonal conjugate of X(3376)
X(3375) = X(1095)-cross conjugate of X(1)
X(3375) = trilinear product of X(16) and X(17)
X(3376) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3376) lies on these lines: 1,1095 14,484 19,2166
X(3376) = isogonal conjugate of X(3375)
X(3376) = trilinear product of X(14) and X(61)
X(3377) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3377) lies on these lines: 1,1805 19,91 46,485
X(3377) = isogonal conjugate of X(3378)
X(3377) = trilinear quotient X(372)/X(371)
X(3378) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3378) lies on these lines: 1,1806 19,91 46,486
X(3378) = isogonal conjugate of X(3377)
X(3378) = trilinear quotient X(371)/X(372)
X(3379) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3379).
X(3379) lies on these lines: 3,6 5,1140
X(3379) = isogonal conjugate of X(3381)
X(3379) = inverse-in-Brocard-circle of X(3393)
X(3379) = {X(371),X(372)}-harmonic conjugate of X(3394)
X(3379) = X(1139)-Ceva conjugate of X(3380)
X(3380) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3380).
X(3380) lies on this line: 3,6
X(3380) = isogonal conjugate of X(3382)
X(3380) = inverse-in-Brocard-circle of X(3394)
X(3380) = X(1139)-Ceva conjugate of X(3379)
X(3380) = crosspoint of X(1140) and X(3370)
X(3380) = crosssum of X(3368) and X(3369)
X(3381) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
On the sides of ABC construct internal regular pentagons ABC1C2C3, BCA1A2A3, CAB1B2B3. The Aubert lines of [ABB3A1], [BCC3B1], [CAA3C1] concur in X(3381). These lines pass through A, B, C. (Aubert lines are defined at X(45010); see also A construction of X(3381) and X(5401) using Aubert lines). For the case of external pentagons, see X(5401). (Ivan Pavlov, April 15, 2022)
If you have The Geometer's Sketchpad, you can view X(3381).
X(3381) lies on the Kiepert hyperbola and these lines: {2, 3393}, {3, 5402}, {4, 3394}, {5, 3382}, {6, 5401}, {1139, 3379}, {1140, 3368}, {3369, 3370}, {3380, 3397}
X(3381) = isogonal conjugate of X(3379)
X(3381) = X(3396)-cross conjugate of X(3382)
X(3381) = X(3)-Dao conjugate of X(3379)
X(3381) = barycentric quotient X(6)/X(3379)
X(3381) = {X(5),X(3395)}-harmonic conjugate of X(3382).
X(3382) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3382).
X(3382) lies on the Kiepert hyperbola.
X(3382) = isogonal conjugate of X(3380)
X(3382) = cevapoint of X(3368) and X(3369)
X(3382) = X(3396)-cross conjugate of X(3381)
X(3383) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3383) lies on these lines: 1,1094 13,484 19,2166
X(3383) = isogonal conjugate of X(3384)
X(3383) = trilinear product of X(13) and X(62)
X(3384) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3384) lies on these lines: 18,1653 19,2290
X(3384) = isogonal conjugate of X(3383)
X(3384) = X(1094)-cross conjugate of X(1)
X(3384) = trilinear product of X(15) and X(18)
X(3385) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3385) lies on this line: 3,6
X(3385) = isogonal conjugate of X(3387)
X(3385) = inverse-in-Brocard-circle of X(3372)
X(3385) = X(485)-Ceva conjugate of X(3386)
X(3385) = insimilicenter of circumcircle and 1st Kenmotu circle
X(3385) = {X(371),X(372)}-harmonic conjugate of X(3371)
X(3386) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3386) lies on this line: 3,6
X(3386) = isogonal conjugate of X(3388)
X(3386) = inverse-in-Brocard-circle of X(3371)
X(3386) = X(485)-Ceva conjugate of X(3385)
X(3386) = exsimilicenter of circumcircle and 1st Kenmotu circle
X(3386) = {X(371),X(372)}-harmonic conjugate of X(3372)
X(3387) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3387) lies on the Kiepert hyperbola and this line: 5,372
X(3387) = isogonal conjugate of X(3385)
X(3387) = X(371)-cross conjugate of X(3388)
X(3388) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3388) lies on the Kiepert hyperbola and this line: 5,372
X(3388) = isogonal conjugate of X(3386)
X(3388) = X(371)-cross conjugate of X(3387)
X(3389) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3389) lies on these lines: 3,6 13,485 17,590 18,486 202,1124 203,2067
X(3389) = isogonal conjugate of X(3391)
X(3389) = inverse-in-Brocard-circle of X(3365)
X(3389) = X(17)-Ceva conjugate of X(3390)
X(3389) = X(3205)-cross conjugate of X(3390)
X(3389) = {X(16),X(61)}-harmonic conjugate of X(3390)
X(3389) = {X(371),X(372)}-harmonic conjugate of X(3364)
X(3390) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3390) lies on these lines: 3,6 13,486 17,615 18,485 202,1335
X(3390) = isogonal conjugate of X(3392)
X(3390) = inverse-in-Brocard-circle of X(3364)
X(3390) = X(17)-Ceva conjugate of X(3389) X(3390) = X(3205)-cross
conjugate of X(3389)
X(3390) = {X(16),X(61)}-harmonic conjugate of X(3389)
X(3390) = {X(371),X(372)}-harmonic conjugate of X(3365)
X(3391) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3391) lies on the Kiepert hyperbola and these lines: 5,16 13,371 17,590 18,372 18,485 202,1335
X(3391) = isogonal conjugate of X(3389)
X(3391) = X(61)-cross conjugate of X(3392)
X(3392) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3392) lies on the Kiepert hyperbola and these lines: 5,16 13,372 17,615 18,371 18,485 202,1335
X(3392) = isogonal conjugate of X(3390)
X(3392) = X(61)-cross conjugate of X(3391)
X(3393) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3393).
X(3393) lies on these lines: 3,6 5,1139
X(3393) = isogonal conjugate of X(5401)
X(3393) = inverse-in-Brocard-circle of X(3379)
X(3393) = X(1140)-Ceva conjugate of X(3394)
X(3394) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3394).
X(3394) lies on this line: 3,6
X(3394) = isogonal conjugate of X(5402)
X(3394) = inverse-in-Brocard-circle of X(3380)
X(3394) = X(1140)-Ceva conjugate of X(3393)
X(3394) = crosspoint of X(1139) and X(3397)
X(3394) = crosssum of X(3395) and X(3396)
X(3395) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3395).
X(3395) lies on these lines: 3,6 4,1140
X(3395) = isogonal conjugate of X(3397)
X(3395) = {X(371),X(372)}-harmonic conjugate of X(3369)
X(3395) = inverse-in-Brocard-circle of X(3368)
X(3396) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
If you have The Geometer's Sketchpad, you can view X(3396).
X(3396) lies on this line: 2,1140 3,6
X(3396) = isogonal conjugate of X(1139)
X(3396) = inverse-in-Brocard-circle of X(3369)
X(3396) = crosssum of X(3379) and X(3380)
X(3396) = crosspoint of X(3381) and X(3382)
X(3396) = {X(371),X(372)}-harmonic conjugate of X(3368)
X(3397) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
Let A' be the center of the regular pentagon AA2A3A4A5, with the same orientation than ABC and such that B ∈ A2A3 and C ∈ A4A5. Build B' and C' cyclically. Then the lines AA', BB', CC' concur at X(3397). (César E. Lozada, May 14, 2019, Hyacinthos #29013).
If you have The Geometer's Sketchpad, you can view X(3397).
X(3397) lies on the Kiepert hyperbola and this line: 2,1140
X(3397) = isogonal conjugate of X(3395)
X(3397) = X(3394)-cross conjugate of X(1139)
X(3398) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
Randy Huston, September 5, 2015, gives three constructions of X(3398):
(1) Let X be the 2nd Brocard point of pedal triangle of 1st Brocard point. Let Y be the 1st Brocard point of pedal triangle of 2nd Brocard point. Then X(3398) is the vertex conjugate of X and Y.
(2) Let U be the circle {{X(371),X(372),PU(1),PU(39)}} and V the 2nd Brocard circle. Then X(3398) is the center of the inverse-in-U of V.
(3) Let A'B'C' be the 3rd Brocard triangle. Let Oa be the circumcenter of A'BC, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3398).
X(3398) lies on these lines: 3,6 5,83 140,325 384,2782 597,1576
X(3398) = midpoint of X(1342) and X(1343)
X(3398) = isogonal conjugate of X(3399)
X(3398) = inverse-in-Brocard-circle of X(3095)
X(3398) = crosssum of X(3102) ande X(3103)
X(3398) = harmonic center of circumcircle and 1st Lemoine circle
X(3398) = harmonic center of 1st and 2nd Brocard circles
X(3398) = X(3)-of-5th-anti-Brocard-triangle
X(3398) = radical center of circumcircles of ABC, 5th Brocard triangle, and 5th anti-Brocard triangle
X(3398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,182,3), (371,372,3094), (1687,1688,39)
X(3399) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
Let A'B'C' be the 1st Brocard triangle. Let Oa be the circumcenter of A'BC, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3399). (Randy Hutson, July 20, 2016)
Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = ω. Define B' and C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3399). (Randy Hutson, July 20, 2016)
Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = 2ω. Define B' and C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(3399). (Randy Hutson, July 20, 2016)
X(3399) lies on the Kiepert hyperbola and this line: 5,1916 262,1506 597,1576
X(3399) = isogonal conjugate of X(3398)
X(3399) = cevapoint of X(3102) and X(3103)
X(3400) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3400) lies on the cubic K457.
X(3400) = isogonal conjugate of X(3401)
X(3401) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3401) lies on the cubic K457 and these lines: 1,1581 63,1934 3404,3405
X(3401) = isogonal conjugate of X(3400)
X(3402) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3402) lies on the cubic K457 and these lines: 1,1755 42,263 1923,1973
X(3402) = isogonal conjugate of X(3403)
X(3402) = trilinear product of PU(191)
X(3403) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3403) lies on the cubic K457 and these lines: 1,75 9,1921 19,1969 57,1920 63,561 76,1423 1707,1965
X(3403) = isogonal conjugate of X(3402)
X(3403) = isotomic conjugate of X(2186)
X(3404) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3404) lies on the cubic K457 and these lines: 1,163 63,561 2157,2624 3401,3405
X(3404) = isogonal conjugate of X(3405)
X(3404) = cevapoint of X(38) and X(2236)
X(3404) = crosssum of X(1755) and X(1959)
X(3404) = crosspoint of X(1821) and X(1910)
X(3405) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3405) lies on the cubic K457 and these lines: 1,82 19,1969 798,812 3401,3404
X(3405) = isogonal conjugate of X(3404)
X(3405) = cevapoint of X(1755) and X(1959)
X(3405) = crosssum of X(38) and X(2236)
Let A' be the apex of the isosceles triangle BA'C constructed outward on BC such that ∠A'BC = ∠A'CB = ω. Define B' and C' cyclically. Let Oa be the circumcenter of BA'C, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3406). (Randy Hutson, July 20, 2016)
Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = 2ω. Define B' and C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(3406). (Randy Hutson, July 20, 2016)
X(3406) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3406) lies on the Kiepert hyperbola and these lines: 3,1916 4,1691 76,182
X(3406) = isogonal conjugate of X(3095)
X(3406) = antigonal conjugate of isogonal conjugate of X(35375)
Let A'B'C' be the 1st Brocard triangle. X(3407) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, July 20, 2016)
Let A'B'C' be the 1st Brocard triangle and A"B"C" be the 1st anti-Brocard triangle. Let A* be the diagonal crosspoint of trapezoid B'C'C"B" (i.e., the intersection of lines B'C" and C'B"); define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3407). (Randy Hutson, July 20, 2016)
X(3407) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3407) lies on the Kiepert hyperbola and these lines: 1,1501 6,1916 76,384 321,2205 983,985
X(3407) = isogonal conjugate of X(3094)
X(3407) = isotomic conjugate of X(3314)
X(3407) = cevapoint of X(32) and X(182)
X(3408) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3408) lies on the cubic K457 and this line: 63,1934
X(3408) = isogonal conjugate of X(3409)
X(3409) and related points occupy Bernard Gibert's Table 38. See the note at X(3364).
X(3409) lies on the cubic K457 and this line: 1,1917
X(3409) = isogonal conjugate of X(3408)
X(3410) lies on these lines: 2,98 4,93 5,1199 23,343 68,1173 69,1369 323,427 1209,1614
Trilinears g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A + 4 sin B sin C - 33/2sin A
Barycentrics h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 16S2 - 2*33/2a2S + a2SA, where S = 2(area(ABC))
Barycentrics h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 20S2 - 2*33/2S - SBSC
X(3411) and X(3412) occur in Bernard Gibert's study of the Brocard-Kiepert quartic, which has three cusps: X(20) with Euler line as tangent; X(3411) with line X(5)X(13) as tangent, and X(3412) with line X(5)X(14) as tangent.
X(3411) lies on the Brocard-Kiepert quartic, Q073, and on these lines: 5,13 6,3412 14,382 16,20 61,631 398,548 630,3180
X(3411) = {X(6),X(3526)}-harmonic conjugate of X(3412)
Trilinears g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A + 4 sin B sin C + 33/2sin A
Barycentrics h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 16S2 + 2*33/2a2S + a2SA
Barycentrics h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 20S2 + 2*33/2S - SBSC
X(3412) lies on the Brocard-Kiepert quartic, Q073, and on these lines: 5,14 6,3411 13,382 15,20 62,631 397,548 629,3181
X(3412) = {X(6),X(3526)}-harmonic conjugate of X(3411)
As the isogonal conjugate of a point on the circumcircle, X(3413) lies on the line at infinity.
The asymptotes of the Kiepert hyperbola meet the infinity line in X(3413) and X(3414). See the note at X(2574).
X(3413) lies on the Kiepert hyperbola and these (parallel) lines: {2, 1341}, {3, 6178}, {4, 3558}, {5, 14632}, {30, 511}, {39, 14633}, {76, 13326}, {83, 14631}, {98, 1380}, {99, 1379}, {100, 36735}, {104, 36736}, {114, 2040}, {115, 2028}, {194, 3557}, {671, 6189}, {1313, 13870}, {1340, 7709}, {1349, 7694}, {1569, 2029}, {1670, 32482}, {1671, 32481}, {2037, 3029}, {2038, 34454}, {2482, 22244}, {2558, 3406}, {2559, 10131}, {5461, 22245}, {5466, 13636}, {5639, 9147}, {6040, 6054}, {6177, 13881}, {11257, 13325}, {12188, 38597}, {13188, 38596}, {14501, 14981}, {14630, 32467}, {39103, 39108}, {39105, 39107}
X(3413) = isogonal conjugate of X(1379)
X(3413) = isotomic conjugate of X(6190)
X(3413) = complementary conjugate of X(2039)
X(3413) = Thomson-isogonal conjugate of X(1380)
X(3413) = infinite point of major axis of Steiner inellipse and Steiner circumellipse
X(3413) = bicentric difference of PU(i) for these i: 116, 118
X(3413) = ideal point of PU(i) for these i: 116, 118
X(3413) = trilinear pole of line {523, 13636}
X(3413) = crossdifference of every pair of points on line X(6)X(5639)
X(3413) = X(2)-Ceva conjugate of X(39023)
X(3413) = barycentric square root of X(39023)
X(3413) = perspector of hyperbola {{A,B,C,X(2),X(6189),F1",F2"}}, where F1", F2" are the isotomic conjugates of PU(119) (the imaginary foci of the Steiner inellipse)
X(3413) = {X(115),X(14502)}-harmonic conjugate of X(2039)
As the isogonal conjugate of a point on the circumcircle, X(3414) lies on the line at infinity.
X(3414) lies on the Kiepert hyperbola and these (parallel) lines: {2, 1340}, {3, 6177}, {4, 3557}, {5, 14633}, {30, 511}, {39, 14632}, {76, 13325}, {83, 14630}, {98, 1379}, {99, 1380}, {114, 2039}, {115, 2029}, {194, 3558}, {671, 6190}, {1312, 13870}, {1341, 7709}, {1348, 7694}, {1569, 2028}, {1670, 32481}, {1671, 32482}, {2037, 34454}, {2038, 3029}, {2482, 22245}, {2558, 10131}, {2559, 3406}, {5461, 22244}, {5466, 13722}, {5638, 9147}, {6039, 6054}, {6178, 13881}, {11257, 13326}, {14502, 14981}, {14631, 32467}
X(3414) = isogonal conjugate of X(1380)
X(3414) = isotomic conjugate of X(6189)
X(3414) = complementary conjugate of X(2040)
X(3414) = Thomson-isogonal conjugate of X(1379)
X(3414) = infinite point of minor axis of Steiner inellipse and Steiner circumellipse
X(3414) = bicentric difference of PU(i) for these i: 117, 119
X(3414) = ideal point of PU(i) for these i: 117, 119
X(3414) = trilinear pole of line {523, 13722}
X(3414) = crossdifference of every pair of points on line X(6)X(5638)
X(3414) = X(2)-Ceva conjugate of X(39022)
X(3414) = barycentric square root of X(39022)
X(3414) = perspector of hyperbola {{A,B,C,X(2),X(6190),F1",F2"}}, where F1", F2" are the isotomic conjugates of PU(118) (the real foci of the Steiner inellipse)
X(3414) = {X(115),X(14501)}-harmonic conjugate of X(2040)
Vertex Conjugates
Suppose that U = u : v : w and X = x : y : z (trilinears) are points not on a sideline of ABC. Let
f(a,b,c) = a/[a2vwyz - ux(bw + cv)(bz + cy)].
The U-vertex conjugate of X is the point
f(a,b,c) : f(b,c,a) : f(c,a,b).
For a geometric interpretation, let T be the vertex triangle of the circumcevian triangles, AUBUCU and AXBXCX, of U and X; viz., the sidelines of T are AUAX, BUBX, CUCX. Then T is perspective to ABC, and the perspector is the U-vertex conjugate of X.
The definition of vertex conjugate allows X = U. To extend the geometric interpretation to the case that X = U, as X approaches U, the vertex triangle approaches a limiting triangle which we call the tangential triangle of U, a triangle perspective to ABC with perspector U-vertex conjugate of U.
The appearance of a row I, J, K in the following tables signifies that the X(i)-vertex conjugate of X(j) is X(K).
| I | J | K |
|---|---|---|
| 1 | 1 | 56 |
| 1 | 2 | 3415 |
| 1 | 3 | 84 |
| 1 | 4 | 3417 |
| 1 | 6 | 2163 |
| 1 | 7 | 3418 |
| 1 | 9 | 3420 |
| 1 | 19 | 3422 |
| 1 | 56 | 1 |
| 1 | 57 | 3423 |
| 1 | 58 | 58 |
| 2 | 2 | 25 |
| 2 | 3 | 3424 |
| 2 | 4 | 3425 |
| 2 | 6 | 1383 |
| 2 | 32 | 3407 |
| 2 | 251 | 251 |
| 2 | 523 | 23 |
| 3 | 3 | 64 |
| 3 | 4 | 4 |
| 3 | 6 | 3426 |
| 3 | 7 | 3427 |
| 3 | 20 | 3346 |
| 3 | 40 | 3345 |
| 3 | 56 | 945 |
| 3 | 64 | 3 |
| 3 | 84 | 1 |
| 3 | 1490 | 3447 |
| 3 | 1498 | 3348 |
| 3 | 2131 | 3183 |
| 3 | 3182 | 3354 |
| 4 | 4 | 3 |
| 5 | 5 | 3432 |
| 6 | 6 | 6 |
| 7 | 7 | 3433 |
| 8 | 8 | 3434 |
| 9 | 9 | 1436 |
| 10 | 10 | 3437 |
Among properties of vertex conjugation are these:
1. X(3)-vertex conjugation maps the Darboux cubic to the Darboux cubic. The appearance of (i,j) in the following list means that X(i) is on the Darboux cubic and that X(j) = X(3)-vertex conjugate of X(i):
(1,84), (3,64), (4,4), (20,3346), (40,3345), (1490,3347), (1498,3348), (2131,3183), (3182,3354)
2. The fixed point of U-vertex conjugation is the 1st Saragossa point of U. (Saragossa points are defined just before X(1166).) The appearance of (i,j) in the following list means that the 1st Saragossa point of X(i) is X(j):
(1,58), (2,251), (3,4), (4,54), (5,1166), (6,6), (7,3449), (8,3450), (9,3451), (21,961), (55,57)
If U' and X' are the isogonal conjugates of U and X, resp., then the isogonal conjugate of the U-vertex conjugate of X is the anticomplement of the midpoint of U' and X'. (Randy Hutson, July 11, 2019)
The U-vertex conjugate of U is the isogonal conjugate of the anticomplement of the isogonal conjugate of U. (Randy Hutson, January 17, 2020)
X(3415) lies on these lines: 22,55 41,386
X(3415) = isogonal conjugate of X(3416)
X(3415) = trilinear pole of line X(834)X(2483)
X(3416) lies on these lines: 1,141 2,1386 6,10 7,8 40,1503 66,72 200,223 321,1836 355,511 608,1861 612,1211 613,1737 1376,1460
X(3416) = midpoint of X(8) and X(69)
X(3416) = reflection of X(i) in X(j) for these (i,j): (1,141),
(6,10)
X(3416) = isogonal conjugate of X(3415)
X(3416) = anticomplement of X(1386)
X(3416) = crossdifference ofevery pair of points on the line
X(834)X(2483)
Let A'B'C' be the intouch triangle. Let LA be the reflection of line B'C' in the perpedicular bisector of BC, and define LB and LC cyclically. Let A" = LB∩LC, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3417). The results are the same if 'hexyl triangle' is substituted for 'intouch triangle'. (Randy Hutson, July 11, 2019)
X(3417) lies on these lines: 4,2217 24,56 36,47 48,573
X(3417) = isogonal conjugate of X(355)
X(3418) lies on this line: 36,48
X(3418) = isogonal conjugate of X(3419)
X(3419) lies on these lines: 1,442 4,8 5,78 9,80 11,997 30,63 40,1726 100,1006 150,2893 226,519 381,908 392,497 405,950 443,938 474,1210 960,1479 1060,2000 1104,1714 1376,1737 2475,2894
X(3419) = midpoint of X(8) and X(3434)
X(3419) = reflection of X(i) in X(j) for these (i,j): (1,2886),
(55,10)
X(3419) = isogonal conjugate of X(3418)
X(3419) = inverse-in-Fuhrmann circle of X(72)
X(3420) lies on these lines: 3,392 25,104 859,1444 1450,2122 1811,2932 2163,2291
X(3420) = isogonal conjugate of X(3421)
X(3421) lies on these lines: 1,2551 2,495 4,8 40,2123 57,388 100,376 144,153 145,1058 150,668 200,515 219,1249 497,519 527,1478 529,1376 631,2975
X(3421) = midpoint of X(8) and X(329)
X(3421) = reflection of X(i) in X(j) for these (i,j): (57,10),
(2096,3359)
X(3421) = isogonal conjugate of X(3420)
X(3421) = anticomplement of X(999)
X(3422) lies on these lines: 1,24 3,47 29,1479 35,78 36,77 55,1807 219,2174 573,2359 1065,1478 2163,2291 2301,2338
X(3422) = isogonal conjugate of X(1478)
X(3422) =X(1064)-cross conjugate of X(1)
X(3423) lies on these lines: 3,41 7,105 25,57 31,222 55,63 513,884 1037,1362 1790,2194 2163,2291
X(3423) = isogonal conjugate of X(2550)
X(3424) lies on these lines: 2,154 20,76 193,1916 253,1297 385,2996 671,2794 1499,2394
X(3424) = isogonal conjugate of X(1350)
X(3424) = isotomic conjugate of X(37668)
X(3424) = perspector of ABC and anticomplementary triangle of Artzt triangle
X(3425) lies on these lines: 2,2351 3,315 4,2353 22,184 32,232 98,264 378,2794 523,878
X(3425) = isogonal conjugate of X(1352)
Let PA be the parabola with focus A and directrix BC. Let LA be the polar of X(20) wrt PA. Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3426). (Randy Hutson, July 11, 2019)
X(3426) lies on these lines: 3,1495 6,1597 25,74 30,69 54,1593 64,1598 67,2777 68,382 73,1480 248,1384 265,541 381,1514 879,1499 895,1351
X(3426) = isogonal conjugate of X(376)
X(3427) lies on these lines: 9,515 104,1617 943,944 1156,2829
X(3427) = isogonal conjugate of X(3428)
X(3428) lies on these lines: 1,3 4,958 6,1064 8,411 20,2894 21,962 101,102 104,376 212,1457 221,255 347,934 405,946 516,993 602,1191 859,2328 945,1794 953,2742 1001,1006 1042,1496 1808,2716 1753,1824 2178,2256
X(3428) = midpoint of X(20) and X(3434)
X(3428) = reflection of X(i) in X(j) for these (i,j): (4,2886), (55,3),
(1012,993)
X(3428) = isogonal conjugate of X(3427)
X(3429) lies on the Kiepert hyperbola and this line: 10,1503
X(3429) = isogonal conjugate of X(3430)
X(3430) lies on these lines: 3,6 20,1330 72,1782 78,1763 101,1297 169,936 404,1730 842,2705 951,1427 1293,2842 1490,1766 2360,2915 2702,2710
X(3430) = midpoint of X(20) and X(1330)
X(3430) = reflection of X(58) in X(3))
X(3430) = isogonal conjugate of X(3429)
Let A'B'C' be the orthocentroidal triangle. X(3431) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, June 27, 2018)
In the plane of a triangle ABC, let
Oa = circle with diameter BC, and define Ob and Oc cyclically;
A' = reflection of A in Oa, and define B' and C' cyclically;
Pa = polar of A' with respect to Oa, and define Pb and Pc cyclically;
A" = Pc∩Pb, and define B" and C" cyclically.
The triangle A"B"C" is perspective to ABC, and the perspector is X(3431).
(Dasari Naga Vijay Krishna, June 8, 2021)
X(3431) lies on these lines: 2,265 3,323 4,1495 6,186 68,631 248,574
X(3431) = isogonal conjugate of X(381)
X(3431) = X(566)-cross conjugate of X(2)
X(3431) = trilinear pole of line X(526)X(647)
X(3431) = perspector of circle (X(3),R/2)
X(3431) = perspector of ABC and complement of Ehrmann vertex-triangle
X(3432) lies on these lines: 5,1601 49,52
X(3432) = isogonal conjugate of X(2888)
X(3432) = crosssum of X(195) and X(2917)
Let A'B'C' be the intouch triangle. Let A" be the crosspoint of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3433). (Randy Hutson, July 31 2018)
X(3433) lies on these lines: 3,518 7,1486 48,672 56,1876 104,378 603,1458 692,1037 1437,1780
X(3433) = isogonal conjugate of X(3434)
X(3433) = X(2175)-cross conjugate of X(6)
X(3434) lies on these lines: 1,224 2,11 4,8 10,1479 20,2894 30,956 63,516 69,674 75,1370 78,946 145,388 152,2807 200,908 226,2900 443,1058 474,496 518,1836 519,1478 595,1714 614,1738 1004,1617 1484,2932
X(3434) = midpoint of X(8) and X(3419)
X(3434) = reflection of X(i) in X(j) for these (i,j): (8,3419),
(20,3428), (55,2885), (145,2099)
X(3434) = isogonal conjugate of X(3433)
X(3434) = inverse-in-Fuhmann circle of X(3436)
X(3434) = anticomplement of X(55)
X(3434) = anticomplementary conjuate of X(144)
X(3434) = complementary conjugate of complement of X(38269)
X(3434) = inner-Conway-isogonal conjugate of X(34784)
X(3434) = homothetic center of anticomplementary triangle and cross-triangle of ABC and inner Johnson triangle
X(3434) = homothetic center of inner Johnson triangle and cross-triangle of ABC and inner Johnson triangle
Let A'B'C' be the extouch triangle. Let A" be the crosspoint of the circumcircle intercepts of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(3435). (Randy Hutson, July 31 2018)
X(3435) lies on these lines: 3,960 8,197 24,104 25,2217 48,836 56,1452 603,1193
X(3435) = isogonal conjugate of X(3436)
X(3435) = X(i)-cross conjugate of X(j) for these (i,j): (1364,513), (1397,6)
X(3436) lies on these lines: 1,908 2,12 4,8 5,956 10,46 20,100 69,313 78,515 144,1654 145,497 271,1542 315,668 341,1370 405,495 452,1621 498,993 518,1837 519,1479
X(3436) = reflection of X(i) in X(j) for these (i,j): (46,10), (56,1329), (145,2098)
X(3436) = isogonal conjugate of X(3435)
X(3436) = isotomic conjugate of X(8048)
X(3436) = anticomplement of X(56)
X(3436) = anticomplementary conjuate of X(145)
X(3436) = X(478)-cross conjugate of X(2)
X(3436) = Fuhmann-circle-inverse of X(3434)
X(3436) = complementary conjugate of complement of X(38273)
X(3436) = homothetic center of anticomplementary triangle and cross-triangle of ABC and outer Johnson triangle
X(3436) = homothetic center of outer Johnson triangle and cross-triangle of ABC and outer Johnson triangle
X(3437) lies on these lines: 10,199 35,228 184,386 2174,2200
X(3437) = isogonal conjugate of X(1330)
X(3437) = X(2206)-cross conjugate of X(6)
X(3437) = crosssum of X(1654) and X(3151)
X(3438) lies on these lines: 3,299 13,1605 2174,2200
X(3438) = isogonal conjugate of X(621)
X(3438) = X(186)-cross conjugate of X(3439)
X(3438) = crosssum of X(616) and X(628)
X(3439) lies on these lines: 3,298 14,1606 15,184
X(3439) = isogonal conjugate of X(622)
X(3439) = inverse-in-circumcircle of X(3479)
X(3439) = X(186)-cross conjugate of X(3438)
X(3439) = crosssum of X(616) and X(627)
X(3440) lies on the Neuberg cubic and these lines: 15,1495 30,298 399,1338 2132,3441
X(3440) = isogonal conjugate of X(616)
X(3440) = X(74)-cross conjugate of X(3441)
X(3441) lies on the Neuberg cubic and these lines: 16,1495 30,299 399,1337 2132,3440
X(3441) = isogonal conjugate of X(617)
X(3441) = X(74)-cross conjugate of X(3440)
X(3442) lies on these lines: 3,303 17,1607 62,184
X(3442) = isogonal conjugate of X(633)
X(3443) lies on these lines: 3,302 18,1608 61,184
X(3443) = isogonal conjugate of X(634)
X(3444) lies on these lines: 2,1029 35,37 42,2174 759,1989 1399,1400
X(3444) = isogonal conjugate of X(2895)
X(3444) = cevapoint of X(667) and X(3124)
X(3444) = X(1333)-cross conjugate of X(6)
Let A'B'C' be the tangential triangle of ABC, and let L be the line through X(1) parallel to BC. Let A'' = L∩B'C', and define B'' and C'' cyclically. Let A* = B'B''∩C'C'', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3445); also, the lines A'A*, B'B*, C'C* concur in X(3052). (Angel Montesdeoca, April 29, 2016)
X(3445) = perspector of ABC and the inner-mixtilinear tangents triangle (see X(11051).
X(3445) lies on these lines: 1,474 2,1222 3,106 6,1201 8,1120 34,1319 56,1149 58,999 87,1001 220,1015 244,2098 269,1279 937,1104 958,979 995,1126 996,1125 1220,2899 1357,2308 1388,1411 1413,1457 2176,2279 2191,2646 2256,2983
X(3445) = isogonal conjugate of X(145)
X(3445) = isotomic conjugate of complement of polar conjugate of isogonal conjugate of X(23182)
X(3445) = anticomplement of X(2885)
X(3445) = cevapoint of X(663) and X(1015)
X(3445) = X(i)-cross conjugate of X(j) for these (i,j): (55,6), (1357,513), (2347,57)
X(3445) = crosspoint of X(1) and X(2137)
X(3445) = crosssum of X(i) and X(j) for these (i,j): (1,2136), (1743,3158)
X(3445) = trilinear pole of line X(649)X(6363)
X(3445) = perspector of ABC and unary cofactor of triangle T(-2,1)
X(3445) = crossdifference of every pair of points on line X(2976)X(3667) (the radical axis of incircle and AC-incircle)
X(3446) lies on these lines: 36,518 840,1618
X(3446) = isogonal conjugate of X(149)
X(3446) = X(692)-cross conjugate of X(6)
Let E be the Euler line and TATBTC the tangential triangle of ABC. Let DA = E∩TBTC, and define DB and DC cyclically. Let A' = DBTB∩DCTC, and define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X(3447). For a sketch, click X(3447)andX(7669). (Angel Montesdeoca, April 22, 2016)
X(3447) lies on these lines: 23,325 50,232 264,2453 511,2070 2079,2485
X(3447) = isogonal conjugate of X(3448)
X(3447) = X(1576)-cross conjugate of X(6)
X(3447) = perspector of ABC and reflection of tangential triangle in Euler line
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), and X(3448) = centroid of A*B*C*. (Randy Hutson, December 10, 2016)
X(3448) lies on the anticomplementary circle and these lines: 2,98 3,2888 4,94 5,399 10,2948 20,68 23,1503 66,193 67,69 141,2930 148,690 150,2774 151,2779 152,2772 153,355 246,2782 323,858 427,1353 631,1511 1330,2842 1853,1993 2836,2895 2918,2931
X(3448) = reflection of X(i) in X(j) for these (i,j): (4,265), (20,74), (69,67), (110,125), (146,4), (193,895), (323,858),
(399,5), (2892,66), (2930,141), (2948,10)
X(3448) = isogonal conjugate of X(3447)
X(3448) = inverse-in-Fuhrmann circle of X(2475)
X(3448) = anticomplement of X(110)
X(3448) = anticomplementary conjugate of X(523)
X(3448) = X(850)-Ceva conjugate of X(2)
X(3448) = inverse-in-polar-circle of X(1112)
X(3448) = center of rectangular hyperbola through X(4), X(8), and the extraversions of X(8)
X(3448) = X(3060)-of-anti-orthocentroidal-triangle
X(3448) = orthoptic-circle-of-Steiner-circumellipse inverse of X(98)
Saragossa points are discussed just before X(1166).
X(3449) lies on these lines: 6,1602 7,2175 518,2330 572,672
X(3449) = isogonal conjugate of X(2886)
X(3449) = cevapoint of X(6) and X(2175)
X(3449) = X(92)-isoconjugate of X(22070)
X(3450) lies on these lines: 6,1603 8,1397 35,572 960,1319 1193,2003
X(3450) = isogonal conjugate of X(1329)
X(3450) = complementary conjugate of X(3452)
X(3450) = cevapoint of X(6) and X(1397)
X(3450) = X(6)-crossconjugate of X(2985)
X(3450) = X(92)-isoconjugate of X(22071)
X(3451) lies on these lines: 6,1604 9,604 55,572 284,1404 333,1412 1108,2161 1400,2316 2259,2317
X(3451) = isogonal conjugate of X(3452)
X(3451) = cevapoint of X(i) and X(j) for these (i,j): (6,604), (41,3052)
X(3451) = X(667)-cross conjugate of X(109)
X(3451) = X(92)-isoconjugate of X(22072)
X(3452) lies on these lines: 1,2551 2,7 4,936 5,10 11,210 69,1997 72,1210 78,950 118,123 120,124 200,497 281,1848 312,2321 333,645 345,2325 515,997 516,1376 631,2096 958,999 962,1706 1656,2095 1698,2093 1699,2550 1764,2183 2324,2999
X(3452) = midpoint of X(i) and X(j) for these (i,j): (1,3421), (57,329), (200,497)
X(3452) = reflection of X(999) in X(1125)
X(3452) = isogonal conjugate of X(3451)
X(3452) = complement of X(57)
X(3452) = complementary conjugate of X(3450)
X(3452) = X(668)-Ceva conugate of X(522)
X(3452) = crosspoint of X(2) and X(312)
X(3452) = crosssum of X(i) and X(j) for these (i,j): (6,604), (41,3052)
X(3452) = complement of X(57)
X(3452) = isotomic conjugate of isogonal conjugate of X(2347)
X(3452) = polar conjugate of isogonal conjugate of X(22072)
X(3452) = perspector of medial triangle and inverse-in-excircles triangle
X(3453) lies on these lines: 60,386 261,1078 404,849
X(3453) = isogonal conjugate of X(3454)
X(3453) = cevapoint of X(i) and X(j) for these (i,j): (6,2206), (2194,2220)
X(3453) = X(1324)-cross conjugate of X(759)
X(3453) = X(92)-isoconjugate of X(22073)
X(3454) lies on these lines: 2,58 5,141 10,12 34,860 44,1213 118,127 121,125 306,2901 429,1828 519,1834 1046,1698 1089,1230 1104,1125 1228,1930
X(3454) = midpoint of X(i) and X(j) for these (i,j): (4,3430), (58,1330)
X(3454) = isogonal conjugate of X(3453)
X(3454) = isotomic conjugate of isogonal conjugate of X(20966)
X(3454) = complement of X(58)
X(3454) = complementary conjugate of X(1125)
X(3454) = crosspoint of X(2) and X(313)
X(3454) = crosssum of X(i) and X(j) for these (i,j): (6,2206), (2194,2220)
X(3454) = polar conjugate of isogonal conjugate of X(22073)
Let a' be the line through A tangent to the circle {{A, X(3), X(6)}}, and define b' and c' cyclically. The lines a', b', c' concur in X(3455). (César Lozada, January 5, 2021)
X(3455) lies on these lines: 3,67 22,543 23,671 25,115 39,1576 98,186 99,1799 184,574 187,2393 228,2157 378,2794 1976,2088
X(3455) = isogonal conjugate of X(316)
X(3455) = inverse-in-circumcircle of X(67)
X(3455) = cevapoint of X(39) and X(187)
X(3455) = crosspoint of X(111) and X(1177)
X(3455) = crosssum of X(524) and X(858)
X(3455) = vertex conjugate of PU(40)
X(3455) = X(75)-isoconjugate of X(23)
X(3456) lies on these lines: 3,2916 23,1799
X(3456) = isogonal conjugate of X(7768)
X(3456) = crosssum of X(69) and X(1369)
Let A' be the free vertex of the equilateral triangle constructed outwardly on BC, and define B' and C' cyclically. X(3457) is the barycentric product A'*B'*C'. (Randy Hutson, July 31 2018)
X(3457) lies on these lines: 2,13 6,3130 15,1337 32,3124 37,1250 62,110 300,308
X(3457) = isogonal conjugate of X(298)
X(3457) = X(1495)-cross conjugate of X(3458)
X(3457) = crosspoint of X(6) and X(3440)
X(3457) = crosssum of X(2) and X(616)
X(3457) = polar conjugate of isotomic conjugate of X(36296)
X(3457) = X(15)-isoconjugate of X(75)
X(3457) = X(63)-isoconjugate of X(470)
X(3457) = barycentric product of circumcircle intercepts of inner Napoleon circle
Let A' be the free vertex of the equilateral triangle constructed inwardly on BC, and define B' and C' cyclically. X(3458) is the barycentric product A'*B'*C'. (Randy Hutson, July 31 2018)
X(3458) lies on these lines: 2,14 6,3129 16,1338 32,3124 37,1254 61,110 301,308
X(3458) = isogonal conjugate of X(299)
X(3458) = X(1495)-cross conjugate of X(3457)
X(3458) = crosspoint of X(6) and X(3441)
X(3458) = crosssum of X(2) and X(617)
X(3458) = polar conjugate of isotomic conjugate of X(36297)
X(3458) = X(16)-isoconjugate of X(75)
X(3458) = X(63)-iaoconjugate of X(471)
X(3458) = barycentric product of circumcircle intercepts of outer Napoleon circle
Let A' be the reflection of X(5) in BC, and define B' and C' cyclically. Let Oa be the circumcenter of A'BC, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(3459). (Randy Hutson, December 2, 2017)
X(3459) lies on the Napoleon cubic and these lines: 3,1263 4,1157 5,195 53,1601 137,252 288,1487 3336,3460 3462,3470
X(3459) = isogonal conjugate of X(195)
X(3459) = Cundy-Parry Phi transform of X(1263)
X(3459) = Cundy-Parry Psi transform of X(1157)
X(3459) = X(i)-cross conjugate of X(j) for these (i,j): (54,4), (2963,2)
X(3459) = Kosnita(X(1263),X(3)) point
X(3460) lies on the Napoleon cubic and these lines: 1,54 3,3469 4,484 5,2595 655,1087 1726,1745 3336,3459
X(3460) = isogonal conjugate of X(3461)
X(3460) = X(5)-Ceva conjugate of X(1)
X(3461) lies on the Napoleon cubic and these lines: 1,2120 5,2595 195,3468
X(3461) = isogonal conjugate of X(3460)
X(3461) = X(54)-cross conjugate of X(1)
X(3462) lies on the Napoleon cubic and these lines: 1,3469 2,1075 4,54 5,3463 195,2121 233,1249 499,1148 3336,3461
X(3462) = isogonal conjugate of X(3463)
X(3462) = X(5)-Ceva conjugate of X(4)
X(3463) lies on the Napoleon cubic and these lines: 3,2120 5,3462 3460,3468
X(3463) = isogonal conjugate of X(3462)
X(3463) = X(54)-cross conjugate of X(3)
Let A'B'C' be the triangle homothetic to ABC and passing through the excenters (i.e., A'B'C' is the tangential triangle of the excentral triangle.) X(3464) = X(1354)-of-A'B'C'. (Randy Hutson, July 11, 2019)
X(3464) lies on the Bevan circle, the Neuberg cubic, and on these lines: 1,74 3,3466 4,1768 13,1652 14,1653 57,1354 1054,1724
X(3464) = X(30)-Ceva conjugate of X(1)
X(3465) lies on the Neuberg cubic and these lines: 1,4 30,1807 35,228 36,1736 74,484 108,2732 912,1936 1047,1048 1157,3065 1725,1758 2250,2341 3481,3483
X(3465) = reflection of X(484) in X(2222)
X(3465) = isogonal conjugate of X(3466)
X(3465) = antigonal conjugate of X(34301)
X(3465) = X(i)-Ceva conjugate of X(j) for these (i,j): (30,484), (1807,1)
X(3465) = crosssum of X(654) and X(2638)
X(3465) = crossdifference of every pair of points on the line
X(652)X(2260)
X(3466) lies on the Neuberg cubic and these lines: 1,2816 3,3464 30,1807 78,1330 109,1794 219,1761 284,1845 399,3483 484,3484 947,2817 1718,2636 2132,3065
X(3466) = isogonal conjugate of X(3465)
X(3466) = cevapoint of X(654) and X(2638)
X(3466) = antigonal conjugate of X(34303)
X(3466) = X(i)-cross conjugate of X(j) for these (i,j): (74,3065), (1870,1)
X(3466) = trilinear pole of line X(652)X(2260)
Let J be the isogonal conjugate of the incenter I wrt the incentral triangle A'B'C', and let Ja, Jb, Jc be the reflections of J in B'C', C'A', A'B', resp. Then JaJbJc is perspective to ABC, and the perspector is X(3467). See Antreas Hatzipolakis and César Lozada, euclid 3038.
X(3467) lies on the Napoleon cubic and these lines: 1,195 4,3460 5,79 7,499 62,1251 2120,3469 3468,3470
X(3467) = isogonal conjugate of X(3336)
X(3467) = X(i)-cross conjugate of X(j) for these (i,j): (35,1), (54,3469)
X(3467) = Kosnita(X(3065),X(3)) point
X(3468) lies on the Napoleon cubic and these lines: 1,4 5,3469 36,1410 47,1758 54,3336 62,1652 62,1653 195,3461 610,2317 1047,1718 1051,2939 3460,3463 3467,3470
X(3468) = isogonal conjugate of X(3469)
X(3468) = X(5)-Ceva conjugate of X(3336)
X(3468) = SS(A → 3A) of X(1508)
X(3469) lies on the Napoleon cubic and these lines: 1,3462 3,3460 5,3468 77,1745 2120,3467
X(3469) = isogonal conjugate of X(3468)
X(3469) = X(54)-cross conjugate of X(3467)
X(3470) lies on the Napoleon cubic and these lines: 3,74 5,1117 1138,2132 3459,3462 3467,3468
X(3470) = isogonal conjugate of X(3471)
X(3470) = X(2914)-cross conjugate of X(1157)
X(3470) = antigonal conjugate of X(33567)
X(3471) lies on the Napoleon cubic and these lines: 3,1138 5,1117 140,523 477,550
X(3471) = isogonal conjugate of X(3470)
X(3471) = X(1511)-cross conjugate of X(30)
X(3472) lies on the Darboux cubic and these lines: 1,3348 3,3354 4,3347 20,3473 40,2131 64,3345 84,3346
X(3472) = reflection of X(3354) in X(3)
X(3472) = isogonal conjugate of X(3473)
X(3472) = X(20)-Ceva conjugate of X(3345)
X(3473) lies on the Darboux cubic and these lines: 20,3472 40,3355 1490,2130 1498,3353 3182,3183
X(3473) = isogonal conjugate of X(3472)
X(3473) = X(64)-cross conjugate of X(1490)
X(3474) lies on these lines: 1,376 2,1155 4,46 7,55 8,529 20,65 25,1633 33,1721 40,388 42,3000 43,2635 56,962 57,497 63,2550 73,1044 79,498 109,278 144,210 165,226 196,243 200,527 212,948 329,1376 354,390 412,1118 484,1478 515,2093 528,2094 580,1777 653,1857 658,2898 1040,2263 1707,1738 1761,2345
X(3474) = reflection of X(i) in X(j) for these (i,j): (329,1376), (497,57)
X(3474) = cevapoint of X(46) and X(1721)
X(3474) = crosssum of X(652) and X(3022)
X(3474) = {X(7),X(55)}-harmonic conjugate of X(3475)
X(3475) lies on these lines: 1,4 2,210 7,55 12,938 142,200 165,553 329,1001 942,1788 954,1617
X(3475) = isogonal conjugate of X(3477)
X(3475) = {X(7),X(55)}-harmonic conjugate of X(3474)
X(3476) lies on these lines: 1,4 2,1319 7,528 8,56 10,1420 12,1388 43,1450 57,519 100,1470 329,529 381,1387 604,2345 952,999 956,1617 993,2078
X(3476) = reflection of X(i) in X(j) for these (i,j): (8,1376),
(497,1)
X(3476) = isogonal conjugate of X(3478)
X(3477) lies on these lines: 1,1827 3,1471 77,354 78,1001 219,2280
X(3477) = isogonal conjugate of X(3475)
X(3478) lies on these lines: 1,1828 3,902 77,1122 219,2347 1037,1457 1473,1795
X(3478) = isogonal conjugate of X(3476)
X(3479) is the tangential of X(1276) on the Neuberg cubic.
X(3479) lies on the Neuberg cubic and these lines: 3,298 30,1337 3480,3484
X(3479) = isogonal conjugate of X(1337)
X(3479) = antigonal conjugate of X(39134)
X(3479) = circumcircle-inverse of X(3439)
X(3480) is the tangential of X(1277) on the Neuberg cubic.
X(3480) lies on the Neuberg cubic and these lines: 3,299 30,1338 3479,3484
X(3480) = isogonal conjugate of X(1338)
X(3480) = antigonal conjugate of X(39135)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3481) lies on the Neuberg cubic and these lines: 3,2120 4,2121 30,3482 3465,3483
X(3481) = isogonal conjugate of X(3482)
X(3482) lies on the Neuberg cubic and these lines: 3,2121 4,1157 30,3481
X(3482) = isogonal conjugate of X(3481)
X(3482) = antigonal conjugate of X(34304)
Barycentrics (a (a^6+a^5 b-a^4 b^2-2 a^3 b^3-a^2 b^4+a b^5+b^6+a^5 c+a^4 b c-a b^4 c-b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^3 c^3+2 b^3 c^3-a^2 c^4-a b c^4-b^2 c^4+a c^5-b c^5+c^6))/(-a^3-a^2 b+a b^2+b^3-a^2 c-a b c-b^2 c+a c^2-b c^2+c^3) : :
X(3483) lies on the Neuberg cubic and these lines: 1,1157 4,484 399,3466 3465,3481
X(3483) = isogonal conjugate of X(7165)
X(3483) = antigonal conjugate of X(34300)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3484) is the tangential of X(4) on the Neuberg cubic.
X(3484) lies on the Neuberg cubic, then circle {{X(4),X(15),X(16),X(186)}}, and on these lines: 3,2120 4,54 74,1157 484,3466 3479,3480
X(3484) = isogonal conjugate of X(8439)
X(3484) = X(30)-Ceva conjugate of X(1157)
X(3485) lies on these lines: 1,4 2,65 7,21 8,12 11,938 20,1836 29,1118 46,631 55,411 57,1125 78,2550 85,350 196,1940 221,940 238,1451 277,2140 329,958 354,1858 376,1770 443,997 459,1482 551,1420 608,2303 908,2551 945,1065 986,1393 1038,2263 1159,1656 1468,1935 1854,2883 2171,2345
X(3485) = X(968)-cross conjugate of X(966)
X(3486) lies on these lines: 1,4 2,1837 3,1788 8,21 11,2476 20,65 25,1610 29,1857 46,376 78,2551 80,498 90,1000 144,145 281,284 452,960 519,1697 529,2098 1159,1657 1468,1936 1503,1854 2268,2345
X(3486) = reflection of X(i) in X(j) for these (i,j): (8,958), (388,1)
X(3486) = extangents-to-intangents similarity image of X(8)
Let A' be the midpoint of X(1) and the A-intouch point. Define B' and C' cyclically. The triangle A'B'C' is homothetic to the 2nd extouch triangle, and the center of homothety is X(3487). (Randy Hutson, September 14, 2016)
X(3487) lies on these lines: 1,4 2,72 3,7 5,938 8,442 9,1125 57,631 78,443 142,936 281,1148 329,405 386,2140 474,1260
X(3488) lies on these lines: 1,4 3,938 8,405 9,519 20,942 35,1788 57,376 387,1104 952,954
X(3488) = reflection of X(1056) in X(1)
X(3489) lies on the Napoleon cubic and these lines: 3,1337 5,302 51,61
X(3489) = isogonal conjugate of X(627)
X(3489) = X(54)-cross conjugate of X(3490)
X(3490) lies on the Napoleon cubic and these lines: 3,1338 5,303 51,62
X(3490) = isogonal conjugate of X(628)
X(3490) = X(54)-cross conjugate of X(3489)
X(3491) lies on the cubic Z(X(384)) and these lines: 1,295 4,69 32,1613 39,695 147,185 211,754 626,2387 3494,3497
X(3491) = X(384)-Ceva conjugate of X(39)
X(3492) lies on the cubic Z(X(384)) and these lines: 1,3497 4,83 32,695 39,1915 184,194 3500,3502
X(3492) = X(384)-Ceva conjugate of X(32)
X(3494) lies on the cubic Z(X(384)) and these lines: 1,87 3,2053 32,2319 384,3502 3491,3501
X(3494) = isogonal conjugate of X(3502)
X(3495) lies on the cubic Z(X(384)) and these lines: 1,3499 32,983 39,256 384,3503 3498,3501
X(3495) = isogonal conjugate of X(3503)
X(3496) lies on the cubic Z(X(384)) and these lines: 1,32 4,9 6,986 39,893 57,348 63,194 191,2795 220,2943 257,384 517,2329 910,960 2170,2975 2288,2323
X(3496) = isogonal conjugate of X(3497)
X(3496) = complement of X(33867)
X(3496) = crossdifference of every pair of points on the line X(1459)X(1491)
X(3497) lies on the cubic Z(X(384)) and these lines: 1,3492 3,984 63,2896 257,384 607,2221 3491,3494
X(3497) = isogonal conjugate of X(3496)
X(3497) = X(172)-cross conjugate of X(1)
X(3497) = trilinear pole of line X(1459)X(1491)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3498) lies on the cubic Z(X(384)) and these lines: 3,83 3495,3501
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3499) lies on the cubic Z(X(384)) and these lines: 3,695 6,76 2076,2916
X(3499) = X(i)-Ceva conjugate of X(j) for these (i,j): (384,3), (3051,6)
X(3499) = vertex conjugate of PU(141)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(3500) lies on the cubic Z(X(384)) and these lines: 1,295 3,238 63,194 222,1424 384,3501 3492,3502
X(3500) = isogonal conjugate of X(3501)
X(3500) = trilinear pole of line X(659)X(1459)
X(3500) = cevapoint of X(649) and X(3123)
X(3500) = X(2275)-cross conjugate of X(1)
X(3501) lies on the cubic Z(X(384)) and these lines: 1,39 2,1334 ,3,2329 4,9 7,979 8,672 32,3494 37,986 41,100 43,213 57,345 63,2896 76,1423 145,1475 220,1376 321,2198 346,1400 384,3500 404,644 484,1759 579,2321 1575,2176
X(3501) = isogonal conjugate of X(3500)
X(3501) = crosssum of X(649) and X(3123)
X(3501) = crossdifference of every pair of points on the line X(659)X(1459)
X(3501) = {X(1),X(1018)}-harmonic conjugate of X(3208)
X(3502) lies on the cubic Z(X(384)) and these lines: 1,2896 39,3496 384,3494 3492,3500 3493,3503
X(3502) = isogonal conjugate of X(3494)
X(3503) lies on the cubic Z(X(384)) and these lines: 1,3 76,3501 83,3496 109,1923 384,3495 3493,3502
X(3503) = isogonal conjugate of X(3495)
X(3504) lies on the cubic Z(X(385)) and these lines: 25,385 32,3229 1423,1716
X(3504) = isogonal cojugate of X(3186)
X(3504) = X(69)-Ceva conjugate of X(3)
X(3504) = X(2998)-cross conjugate of X(3224)
X(3505) lies on the cubic Z(X(385)) and these lines: 2,3186 32,695 98,783 3510,3512
X(3506) lies on the cubic Z(X(385)) and these lines: 1,3512 2,98 32,694 648,1974 1196,1915
X(3506) = X(385)-Ceva conjugate of X(32)
X(3507) lies on the cubic Z(X(385)) and these lines: 1,2 32,2319 511,3512 694,3508 3225,3510
X(3507) = X(385)-Ceva conjugate of X(3508)
X(3508) lies on the cubic Z(X(385)) and these lines: 1,6 32,983 76,1423 98,813 190,1966 256,1500 385,3512 694,3510 1018,1756 1334,1655
X(3508) = X(385)-Ceva conjugate of X(3507)
X(3509) lies on the cubic Z(X(385)) and these lines: 1,32 2,7 6,982 19,2319 65,2329 98,813 101,758 171,846 199,228 291,1757 295,511 335,385 484,1018 522,649 529,1146 594,2160 666,2311 902,968 910,1282 1054,1575
X(3509) = isogonal cojugate of X(3512)
X(3509) = X(i)-Ceva conjugate of X(j) for these (i,j): (335,1),
(385,3511)
X(3509) = crosssum of X(i) and X(j) for these (i,j): (659,2170),
(2238,2292)
X(3509) = crossdifference of every pair of points on the line
X(663)X(1193)
X(3510) lies on the cubic Z(X(385)) and these lines: 1,76 6,43 42,894 291,511 292,2238 1402,1423 1492,1933 3225,3507 3505,3512
X(3510) = X(i)-Ceva conjugate of X(j) for these (i,j): (385,3509), (1911,1)
X(3511) is the point P on line X(3)X(76) for which the P-Brocard triangle is perspective to ABC. (Randy Hutson, August 19, 2019)
X(3511) lies on the cubic Z(X(385)) and these lines: 3,76 6,694 147,446 237,385
X(3511) = isogonal conjugate of isotomic conjugate of X(25332)
X(3511) = isogonal conjugate of anticomplement of X(39092)
X(3511) = X(i)-Ceva conjugate of X(j) for these (i,j): (237,3), (385,6)
X(3511) = crosssum of X(523) and X(2679)
X(3511) = crossdifference of every pair of points on the line
X(804)X(2023)
X(3512) lies on the cubic Z(X(385)) and these lines: 1,3506 6,2114 19,1423 55,846 335,385 511,3507 1086,2160 1654,2893 3505,3510
X(3512) = isogonal cojugate of X(3509)
X(3512) = cevapoint of X(i) and X(j) for these (i,j): (659,2170), (2238,2292)
X(3512) = X(1914)-cross conjugate of X(1)
X(3512)= crosssum of X(846) and X(1282)
X(3512) = trilinear pole of line X(663)X(1193)
For any two points P, Q in the set {A, B, C} and arbitrary point X, let M(X) = (|PX| + |QX|)/|PQ|. The point X that minimizes M(X) is X(3513). See
David Eppstein, The dilation center of a triangle
for a discussion of X(3513) as a point of intersection of three similar ellipses, each having two of the points A, B, C as foci. (Contributed by David Eppstein, 7/10/08)
X(3513) is the point P such that the incenter of the circumcevian triangle of P is also the incenter of triangle ABC. (Randy Hutson, 9/23/2011)
X(3513) and X(3514) are the limiting points (point-circles) of the coaxal system that includes the circumcircle and incircle. Their midpoint, X(3660), is the radical trace of the incircle and circumcircle. (Peter Moses, November 15, 2011)
In the plane of a triangle ABC, let P be a point on the line X(1)X(3), and let DEF = intouch triangle. Let RA = reflection of AP in EF, and define RB and RC cyclically. The lines RA, RB, RC concur in a point, P', that lies on a rectangular hyperbola that circumscribes DEF. Let P'' = DEF-isogonal conjugate of P'. The mapping P → P'' is a projective transformation of the line X(1)X(3), and its fixed points are X(3513) and X(3514). (Angel Montesdeoca, April 17, 2021)
If you have The Geometer's Sketchpad, you can view X(3513).
X(3513) lies on this line: 1,3
X(3513) = reflection of X(3514) in X(3660)
X(3513) = circumcircle-inverse of X(3514)
X(3513) = incircle-inverse of X(3514)
X(3513) = one of two harmonic traces of the Soddy circles; the other is X(3514)
X(3513) = {X(i),X(j)}-harmonic conjugate of X(3514) for these (i,j): (1,57), (3,1617), (36,2078), (55,56), (65,354), (165,1420)
Like X(3513), the point X(3514) is the point of intersection of three similar ellipses, each having two of the points A, B, C as foci. (Contributed by Peter Moses, 7/24/08)
If you have The Geometer's Sketchpad, you can view X(3514).
X(3514) lies on this line: 1,3
X(3514) = reflection of X(3513) in X(3660)
X(3514) = circumcircle-inverse of X(3513)
X(3514) = incircle-inverse of X(3513)
X(3514) = one of two harmonic traces of the Soddy circles; the other is X(3513)
X(3514) = {X(i),X(j)}-harmonic conjugate of X(3513) for these (i,j): (1,57), (3,1617), (36,2078), (55,56), (65,354), (165,1420)
As a point on the Euler line, X(3515) has Shinagawa coefficients (3F, -E - 3F).
X(3515) lies on these lines: 2,3 36,1398 64,1620 154,185 159,2929 165,1902 187,2207 232,3053 1033,1609 1204,1495 1350,1974 1452,2646
X(3515) = isogonal conjugate of X(15077)
X(3515) = crosspoint of X(250) and X(1301)
X(3515) = polar conjugate of isotomic conjugate of X(37672)
X(3515) = homothetic center of Kosnita triangle and mid-triangle of orthic and circumorthic triangles
As a point on the Euler line, X(3516) has Shinagawa coefficients (3F,E - 3F).
X(3516) lies on these lines: 2,3 6,1204 393174 51,1192 55,1398 64,184 165,1829 574,2207 1181,3357
X(3516) = homothetic center of Trinh triangle and mid-triangle of orthic and circumorthic triangles
As a point on the Euler line, X(3517) has Shinagawa coefficients (3F, -2E - 3F).
X(3517) lies on these lines: 2,3 52,3167 154,389 231,1609 1147,1351 1181,1495 1384,2207 3053,3199 3357,3426
X(3517) = circumcircle-inverse of X(37942)
X(3517) = homothetic center of tangential triangle and mid-triangle of orthic and circumorthic triangles
As a point on the Euler line, X(3518) has Shinagawa coefficients (4F, -3E - 4F).
X(3518) lies on these lines: 2,3 49,143 51,54 52,110 53,1601 93,324 98,3456 107,1141 112,3199 156,568 184,1199 389,1495 575,1843 576,1974 1112,3043 1147,3060 1304,3470 1866,1870
X(3518) = isogonal conjugate of X(3519)
X(3518) = anticomplement of X(37452)
X(3518) = circumcircle-inverse of X(37943)
X(3518) = X(1179)-Ceva conjugate of X(4)
X(3518) = X(2965)-cross conjugate of X(1994)
X(3518) = crosspoint of X(3442) and X(3443)
X(3518) = crosssum of X(633) and X(634)
X(3518) = X(3)-isoconjugate of X(2962)
X(3518) = {X(4),X(24)}-harmonic conjugate of X(186)
X(3518) = perspector of ABC and orthocentroidal-of-orthic triangle
X(3518) = crosssum of X(14813) and X(14814)
Let A'B'C' be the reflection triangle. Let Oa be the circle centered at A' and passing through A, and define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(3519). Let Na be the reflection of X(5) in the perpendicular bisector of BC, and define Nb and Nc cyclically. The lines ANa, BNb, CNc concur in X(3519). Let A''B''C'' be the Trinh triangle. Let A* be the orthopole of line B'C', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3519). (Randy Hutson, October 13, 2015)
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26850.
X(3519) lies on the the Jerabek hyperbola, the cubic K618, and these lines: {2, 1493}, {3, 539}, {4, 93}, {5, 1173}, {6, 17}, {20, 13452}, {49, 343}, {54, 140}, {64, 1657}, {65, 2962}, {68, 12606}, {69, 12363}, {70, 14791}, {74, 550}, {185, 14861}, {265, 5562}, {290, 7768}, {340, 8795}, {394, 15317}, {399, 14862}, {511, 13433}, {524, 5576}, {542, 13564}, {567, 12899}, {895, 11585}, {1176, 3564}, {1503, 9935}, {1899, 11577}, {2889, 3448}, {2914, 14643}, {2918, 5898}, {2979, 12291}, {3426, 5073}, {3431, 3523}, {3521, 13754}, {3522, 11270}, {3527, 3574}, {3850, 14483}, {4857, 6286}, {5056, 13565}, {5059, 11738}, {5068, 14491}, {5449, 15002}, {5504, 12359}, {6101, 12226}, {6145, 7574}, {6515, 9827}, {7517, 15069}, {10018, 11597}, {10625, 14864}, {10628, 11744}, {11138, 11600}, {11139, 11601}, {11225, 15047}, {11412, 12280}, {12936, 15232}
X(3519) = midpoint of X(i) and X(j) for these {i,j}: {2888, 12325}, {11412, 12280}
X(3519) = reflection of X(i) in X(j) for these {i,j}: {195, 1209}, {6243, 6152}, {6288, 2888}, {11271, 1493}, {12226, 6101}, {12316, 3574}, {13431, 12242}, {13432, 13431}
X(3519) = isotomic conjugate of X(32002)
X(3519) = complement of X(11271)
X(3519) = anticomplement of X(1493)
X(3519) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1487, 8}, {2962, 2889}
X(3519) = cevapoint of X(i) and X(j) for these (i,j): {633, 634}, {14813, 14814}
X(3519) = X(11140)-Ceva conjugate of X(2963)
X(3519) = X(1216)-cross conjugate of X(3)
X(3519) = barycentric product X(i)X(j) for these {i,j}: {3, 11140}, {63, 2962}, {69, 2963}, {93, 394}, {252, 343}, {525, 930}
X(3519) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1994}, {5, 14129}, {6, 3518}, {17, 472}, {18, 473}, {48, 2964}, {51, 14577}, {69, 7769}, {93, 2052}, {184, 2965}, {216, 143}, {252, 275}, {562, 14165}, {570, 6152}, {577, 49}, {647, 1510}, {930, 648}, {2962, 92}, {2963, 4}, {8603, 10633}, {8604, 10632}, {11140, 264}, {14111, 11547}
X(3519) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3518}, {4, 2964}, {19, 1994}, {49, 158}, {92, 2965}, {143, 2190}, {162, 1510}, {1973, 7769}, {2148, 14129}, {2167, 14577}, {2216, 6152}
X(3519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11271, 1493), (17, 18, 2963), (93, 562, 14111), (195, 1656, 12242), (195, 13432, 13431), (1209, 12242, 1656), (1209, 13431, 12242), (1656, 13432, 195), (12242, 13431, 195), (19778, 19779, 11140)
As a point on the Euler line, X(3520) has Shinagawa coefficients (4F, E - 4F).
X(3520) lies on these lines: 2,3 35,1870 39,112 49,3043 54,74 64,3431 93,930 99,1235 184,3357 567,1986 574,1968 578,1199 1970,3269
X(3520) = reflection of X(4) in X(1594)
X(3520) = homothetic center of circumorthic triangle and Trinh triangle
X(3520) = X(35)-of-Trinh-triangle if ABC is acute
X(3520) = Trinh-isogonal conjugate of X(7691)
X(3520) = exsimilicenter of circumcircle and Trinh circle; the insimilicenter is X(2071))
X(3520) = {X(3),X(4)}-harmonic conjugate of X(186)
Let A'B'C' be the orthocentroidal triangle. Let A" be the isogonal conjugate, wrt the A-altimedial triangle, of A'. Define B" and C" cyclically. The lines AA", BB", CC" concur in X(3521). (Randy Hutson, November 2, 2017)
Let Pa be the isotomic conjugate of X(1105) with respect to the triangle AX(3)X(4), and define Pb and Pc cyclically. Then PaPbPc and ABC are perspective at X(3521). (Angel Montesdeoca, September 23, 2018)
X(3521) lies on these lines: 3,1568 5,74 6,382 20,3431 30,54 64,381 185,265 597,1177
X(3521) = X(1531)-cross conjugate of X(265)
As a point on the Euler line, X(3522) has Shinagawa coefficients (3, -4).
X(3522) is one of many points on the Euler line that have a certain convenient representation, received from Peter Moses, February 8, 2010. Specifically, the appearance of (t,k) in the following list means that
X(k) = f(A,B,C) : f(B,C,A): f(C,A,B), where f(A,B,C) = t cos A + cos B cos C:
(-4,3528), (-3,3522), (-5/2, 548), (-2,376), (-3/2,550), (-5/4,3534), (-1,20), (-3/4,1657), (-2/3,3529), (-1/3,3146), (-1/4,382), (-1/5,3543), (0,4), (1/6,546), (1/4,381), (1/3,3091), (2/5, 3545), (4/9,3544), (1/2,5), (2/3,3090), (7/10,547), (3/4,1656), (1,2), (7/6,632), (6/5,3533), (5/4,3526), (4/3,3525), (3/2,140), (2,631), (5/2,549), (3,3523), (7/2,3530), (4,3525), (infinity,3)
Other recent results on representations of points on the Euler line are given in
Peter J. C. Moses and Clark Kimberling, "Perspective isoconjugate triangle pairs, Hofstadter pairs, and crosssums on the nine-point circle," Forum Geometricorum 11 (2011) 83-93. Click here to download a pdf.
X(3522) lies on these lines: 2,3 8,165 40,145 56,390 78,144 84,3219 97,3346 185,2979 193,1350 323,1181 489,1270 490,1271 590,1131 615,1132 1038,3100 1155,3486 1193,1742 1204,3098 2646,3474
X(3522) = midpoint of X(20) and X(3091)
X(3522) = reflection of X(i) in X(j) for these (i,j): (4,1656), (631,3), (3091,631)
X(3522) = complement of X(17578)
X(3522) = anticomplement of X(3091)
X(3522) = crosspoint of X(22235) and X(22237)
X(3522) = crosssum of X(22236) and X(22238)
X(3522) = circumcircle-inverse of X(37944)
X(3522) = Thomson-isogonal conjugate of X(5644)
X(3522) = homothetic center of 2nd Conway triangle and cross-triangle of excentral and 2nd circumperp triangles
X(3522) = {X(2),X(3)}-harmonic conjugate of X(15717)
As a point on the Euler line, X(3523) has Shinagawa coefficients (3, -2).
X(3523) lies on these lines: 1,5265 2,3 35,390 36,3085 40,3306 84,3305 95,253 145,1385 147,620 153,2551 165,962 182,193 389,2979 391,572 487,1270 488,1271 991,3216 1151,3069 1152,3068 1155,3485 1788,2646 1189,2888
X(3523) = midpoint of X(i) in X(j) for these (i,j): (3,3526), (3090,3528)
X(3523) = reflection of X(i) in X(j) for these (i,j): (3090,3526), (3528,3)
X(3523) = anticomplement of X(3090)
X(3523) = circumcircle-inverse of X(37945)
X(3523) = homothetic center of dual of orthic triangle and (mid-triangle of orthic and circumorthic triangles)
X(3523) = Euler line intercept, other than X(2), of conic {{X(2),X(15),X(16),X(17),X(18)}}
As a point on the Euler line, X(3524) has Shinagawa coefficients (4, -3).
X(3524) lies on these lines: 2,3 35,1058 36,1056 40,551 69,3431 98,2482 182,1992 187,1285 519,3158 553,3487 597,1350 943,1466 1000,1319 1385,3241 3058,3086 3070,3316 3071,3317
X(3524) = midpoint of X(376) in X(3545)
X(3524) = reflection of X(i) in X(j) for these (i,j): (4,3545), (3545,2)
X(3524) = isogonal conjugate of X(3531)
X(3524) = crosssum of X(11485) and X(11486)
X(3524) = circumcircle-inverse of X(37946)
X(3524) = trisector nearest X(3) of segment X(2)X(3)
X(3524) = X(2)-of-Thomson-triangle
X(3524) = {X(376),X(631)}-harmonic conjugate of X(2)
As a point on the Euler line, X(3525) has Shinagawa coefficients (4, -1).
X(3525) lies on these lines: 2,3 69,575 394,1199 498,1056 499,1058 944,1698 1007,1078 1285,2548 1587,3316 1588,3317 3053,3055 3085,3304 3086,3303
X(3525) = crosspoint of X(3316) and X(3317)
X(3525) = crosssum of X(3311) and X(3312)
As a point on the Euler line, X(3526) has Shinagawa coefficients (5, -1).
Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). Let A'B'C' be the medial-of-medial triangle. X(3526) is the radical center of the circumcircles of triangles A'AeAi, B'BeBi, C'CeCi. (Randy Hutson, June 7, 2019)
X(3526) lies on these lines: 2,3 6,3411 49,182 143,2979 195,394 302,628 303,627 498,999 499,3295 567,1092 568,1216 575,599 590,3312 615,3311 1125,1482 1159,1788 1384,2548 1385,1698 1506,3053
X(3526) = midpoint of X(3090) and X(3523)
X(3526) = reflection of X(3) in X(3523)
X(3526) = complement of X(3090)
X(3526) = circumcircle-inverse of X(37947)
X(3526) = orthocentroidal-circle-inverse of X(3628)
X(3526) = homothetic center of medial triangle and mid-triangle of Euler and anti-Euler triangles
X(3526) = homothetic center of X(5)-altimedial and X(2)-anti-altimedial triangles
X(3526) = {X(3411),X(3412)}-harmonic conjugate of X(6)
X(3526) = {X(2),X(4)}-harmonic conjugate of X(3628)
Let HA be the foot of the A-altitude. Let BA and CA be the feet of perpendiculars from HA to CA and AB, resp. Let GA be the centroid of HABACA. Define GB and GC cyclically. The lines AGA, BGB, CGC concur in X(3527). (Randy Hutson, June 7, 2019)
X(3527) lies on these lines: 3,51 5,69 6,1598 24,3431 25,54 64,389 68,381 72,1482 73,999 74,1112 185,3426 1092,3066
X(3527) = isogonal conjugate of X(631)
X(3527) = isotomic conjugate of polar conjugate of X(34818)
X(3527) = X(1181)-cross conjugate of X(3)
X(3527) = perspector of 2nd Lemoine circle
As a point on the Euler line, X(3528) has Shinagawa coefficients (4, -5).
X(3528) lies on these lines: 2,3 35,1056 36,1058 39,1285 40,3244 165,944 519,3161
X(3528) = reflection of X(i) in X(j) for these (i,j): (4,3090), (3090,3523), (3523,3)
X(3528) = reflection of X(2) in X(15700)
X(3528) = anticomplement of X(3851)
X(3528) = orthocentroidal-circle-inverse of X(3544)
X(3528) = pole of van Aubel line wrt conic {{X(13),X(14),X(17),X(18),X(20)}}
X(3528) = {X(2),X(3)}-harmonic conjugate of X(10299)
X(3528) = {X(2),X(4)}-harmonic conjugate of X(3544)
X(3528) = {X(2),X(20)}-harmonic conjugate of X(382)
X(3528) = {X(2),X(3529)}-harmonic conjugate of X(4)
X(3528) = {X(376),X(631)}-harmonic conjugate of X(20)
As a point on the Euler line, X(3529) has Shinagawa coefficients (2, -5).
X(3529) lies on these lines: 2,3 516,944 1056,3303 1058,3304 1249,3284 1770,3486
X(3529) = reflection of X(i) in X(j) for these (i,j): (4,20), (20,1657), (382,550), (3146,3), (3543,3534)
X(3529) = anticomplement of X(382)
X(3529) = circumcircle-inverse of X(37948)
X(3529) = orthocentroidal-circle-inverse of X(3855)
X(3529) = Ehrmann-mid-to-Johnson similarity image of X(20)
X(3529) = homothetic center of cevian triangle of X(3) and mid-triangle of medial and anticomplementary triangles
X(3529) = {X(2),X(4)}-harmonic conjugate of X(3855)
X(3529) = {X(2),X(20)}-harmonic conjugate of X(550)
X(3529) = {X(3),X(4)}-harmonic conjugate of X(3090)
X(3529) = {X(4),X(3528)}-harmonic conjugate of X(2)
X(3529) = {X(382),X(550)}-harmonic conjugate of X(2)
As a point on the Euler line, X(3530) has Shinagawa coefficients (7, -5).
Let P7' and U7' be the bicentric pair constructed as with PU(7), but with circles half the size. The midpoint of P7' and U7' is X(3530). (Randy Hutson, December 26, 2015)
X(3530) lies on these lines: 2,3 15,3411 16,3412 1385,3244
X(3530) = midpoint of X(i) in X(j) for these (i,j): (3,140), (5,548), (546,550)
X(3530) = complement of X(546)
X(3530) = anticomplement of X(35018)
X(3530) = circumcircle-inverse of X(37949)
X(3530) = orthocentroidal-circle-inverse of X(5079)
X(3530) = {X(2),X(3)}-harmonic conjugate of X(550)
X(3530) = {X(2),X(4)}-harmonic conjugate of X(5079)
X(3530) = {X(2),X(20)}-harmonic conjugate of X(3855)
X(3530) = {X(3),X(5)}-harmonic conjugate of X(548)
X(3531) lies on the Jerabek hyperbola and these lines: 3,373 25,3431 51,3426 54,1598 69,381 74,1597 399,895
X(3531) = isogonal conjugate of X(3524)
X(3531) = cevapoint of X(11485) and X(11486)
X(3531) = perspector of circle (X(3),2R)
Let DEF be the cevian triangle of X(69) and A' the center of the circle that passes through D and through points where the line EF cuts to the circumcircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3532). (Angel Montesdeoca, October 30, 2019)
X(3532) lies on the Jerabek hyperbola and these lines: 4,1192 6,1204 64,1620 68,550 71,3207 72,165 74,1498 265,1657 378,1173 1181,3431 3357,3426\
X(3532) = isogonal conjugate of X(3146)
X(3532) = cevapoint of X(1151) and X(1152)
X(3532) = X(154)-cross conjugate of X(6)
X(3532) = polar conjugate of isotomic conjugate of X(36609)
X(3532) = X(63)-isoconjugate of X(33630)
As a point on the Euler line, X(3533) has Shinagawa coefficients (6, -1).
X(3533) lies on these lines: 2,3 371,3317 372,3316
X(3533) = pole of Napoleon axis wrt conic {{X(2),X(15),X(16),X(17),X(18)}}As a point on the Euler line, X(3534) has Shinagawa coefficients (5, -9).
Let A'B'C' be the medial triangle. Let A" be the pole of line B'C' wrt the A-power circle, and define B", C" cyclically. X(3534) = X(2)-of-A"B"C". (Randy Hutson, March 14, 2018)
Let OA be the circle centered at the A-vertex of the Gemini triangle 107 and passing through A; define OB and OC cyclically. X(3534) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(3534) lies on these lines: 2,3 154,2777 394,399 542,1350 590,1327 599,3098 615,1328 999,3058 1159,3474 1384,2549
X(3534) = midpoint of X(i) and X(j) for these (i,j): (20,376), (381,1657), (3529, 3543)
X(3534) = reflection of X(i) in X(j) for these (i,j): (3,376), (4,549), (376,550), (381,3), (382, 381), (549,548),
(599,3098), (3543,5)
X(3534) = circumcircle-inverse of X(37950)
X(3534) = Stammler isogonal conjugate of X(6)
X(3534) = {X(2),X(3)}-harmonic conjugate of X(15693)
X(3534) = trisector nearest X(20) of segment X(3)X(20)
As a point on the Euler line, X(3535) has Shinagawa coefficients (2F, S).
X(3535) lies on these lines: 2,3 275,3317 281,1659 393,590 587,1826 615,3087 1249,3068 1870,3083 2052,3316
X(3535) = inverse-in-orthocentroidal-circle of X(3536)
X(3535) = X(1151)-cross conjugate of X(1270)
X(3535) = polar conjugate of X(1131)
As a point on the Euler line, X(3536) has Shinagawa coefficients (2F, -S).
X(3536) lies on these lines: 2,3 19,587 275,3316 393,615 590,3087 1249,3069 1870,3084 2052,3317
X(3536) = inverse-in-orthocentroidal-circle of X(3535)
X(3536) = X(1152)-cross conjugate of X(1271)
X(3536) = polar conjugate of X(1132)
As a point on the Euler line, X(3537) has Shinagawa coefficients (2E, -2E + F).
X(3537) lies on these lines: 2,3 577,1285 1038,1058 1040,1056
As a point on the Euler line, X(3538) has Shinagawa coefficients (2E, -2E - F).
X(3538) lies on these lines: 2,3 1038,1056 1040,1058
X(3538) = {X(376),X(631)}-harmonic conjugate of X(24)
As a point on the Euler line, X(3539) has Shinagawa coefficients (2E, S).
X(3539) lies on these lines: 2,3 1056,3084 1058,3083
X(3539) = inverse-in-orthocentroidal-circle of X(3540)
As a point on the Euler line, X(3540) has Shinagawa coefficients (2E, -S).
X(3540) lies on these lines: 2,3 1056,3083 1058,3084
X(3540) = inverse-in-orthocentroidal-circle of X(3539)
As a point on the Euler line, X(3541) has Shinagawa coefficients (2F, E).
X(3541) lies on these lines: 2,3 33,499 34,498 54,66 254,264 317,1078 393,570 495,1398 571,3087 578,1899 590,3093 615,3092 847,1217 1092,1352 1199,2904 1870,3085 1968,2548
X(3541) = inverse-in-orthocentroidal-circle of X(3542)
X(3541) = anticomplement of X(3549)
As a point on the Euler line, X(3542) has Shinagawa coefficients (2F, -E).
X(3542) lies on these lines: 2,3 33,498 34,499 70,1177 107,1299 158,1068 193,2904 206,1614 230,2207 254,1093 393,847 590,3092 615,3093 1300,1301 1352,1974 1870,3086 3186,3462
X(3542) = inverse-in-orthocentroidal-circle of X(3541)
X(3542) = anticomplement of X(3548)
X(3542) = circumcircle-inverse of X(37951)
X(3542) = X(1093)-Ceva conjugate of X(4)
X(3542) = cevapoint of X(155) and X(454)
X(3542) = X(155)-cross conjugate of X(4)
X(3542) = polar conjugate of X(6504)
As a point on the Euler line, X(3543) has Shinagawa coefficients (1, -6).
Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Then X(3543) = X(193)-of-A'B'C'. (Randy Hutson, September 5, 2015)
X(3543) lies on these lines: 2,3 144,3419 146,148 147,543 152,544 153,528 253,317 371,1131 372,1132 388,3058 390,1478 393,3163 515,3241 519,962 541,3448 551,1699 553,938 671,2794 1503,1992
X(3543) = midpoint of X(2) and X(3146)
X(3543) = reflection of X(i) in X(j) for these (i,j): (2,4), (20,2), (376,381), (3529,3534), (3534,5)
X(3543) = anticomplement of X(376)
X(3543) = circumcircle-inverse of X(37952)
X(3543) = harmonic center of polar and de Longchamps circles
X(3543) = intersection of tangents to conic {{X(4),X(13),X(14),X(15),X(16)}} at X(13) and X(14)
X(3543) = pole of Fermat axis wrt conic {{X(4),X(13),X(14),X(15),X(16)}}
X(3543) = Euler line intercept, other than X(20), of conic {{X(13),X(14),X(15),X(16),X(20)}}
X(3543) = {X(2),X(3)}-harmonic conjugate of X(15708)
X(3543) = {X(381),X(382)}-harmonic conjugate of X(15684)
X(3544) lies on these lines: {2, 3}, {6, 14842}, {83, 39143}, {113, 15025}, {146, 15088}, {147, 15092}, {148, 38628}, {149, 38629}, {150, 38630}, {153, 38631}, {355, 20057}, {373, 6241}, {576, 11008}, {590, 23275}, {615, 23269}, {944, 7988}, {1056, 7741}, {1058, 7951}, {1131, 13951}, {1132, 8976}, {1285, 7746}, {1352, 22330}, {3070, 3317}, {3071, 3316}, {3244, 5587}, {3303, 3614}, {3304, 7173}, {3431, 18296}, {3448, 38632}, {3567, 27355}, {3590, 31487}, {3592, 23273}, {3594, 23267}, {3616, 38140}, {3617, 38034}, {3618, 33412}, {3620, 38136}, {3623, 38138}, {3626, 3817}, {3629, 10516}, {3631, 11477}, {3632, 5603}, {3636, 7967}, {3746, 10588}, {3982, 5714}, {4031, 9612}, {5158, 33630}, {5261, 10593}, {5274, 10592}, {5343, 16644}, {5344, 16645}, {5365, 16772}, {5366, 16773}, {5418, 12819}, {5420, 12818}, {5563, 10589}, {5881, 38076}, {5886, 32900}, {5889, 14845}, {5892, 11439}, {5943, 15058}, {6000, 11465}, {6154, 20400}, {6225, 32767}, {6329, 14912}, {6337, 15031}, {6361, 12571}, {6419, 13886}, {6420, 13939}, {6425, 23259}, {6426, 23249}, {6427, 18538}, {6428, 18762}, {6453, 32785}, {6454, 32786}, {6488, 9541}, {7317, 30323}, {7603, 7738}, {7612, 18843}, {7617, 14023}, {7687, 15034}, {7758, 8176}, {7772, 31415}, {7816, 39142}, {7991, 10175}, {8252, 10148}, {8253, 10147}, {8797, 32001}, {9624, 34627}, {9729, 16261}, {9779, 9956}, {9862, 38740}, {9955, 12245}, {10171, 18492}, {10222, 20050}, {10514, 26339}, {10515, 26340}, {11002, 11591}, {11017, 34783}, {11362, 30308}, {11451, 12162}, {11459, 16625}, {11482, 18358}, {11522, 34631}, {11695, 12290}, {12112, 37514}, {12244, 38729}, {12317, 15027}, {12383, 15044}, {12900, 15020}, {13172, 38751}, {13199, 38763}, {13452, 18418}, {13464, 34747}, {13472, 15077}, {14061, 35021}, {14094, 15081}, {14561, 22234}, {14639, 20399}, {14644, 20125}, {14651, 38745}, {14654, 38807}, {14683, 15046}, {15012, 15024}, {15018, 32139}, {15019, 15083}, {15029, 36253}, {15052, 36753}, {15054, 23515}, {15069, 20583}, {15808, 19925}, {16982, 23039}, {17852, 23251}, {18383, 35260}, {18424, 31652}, {18436, 18874}, {18439, 32205}, {18489, 18504}, {18493, 20054}, {18584, 31404}, {19872, 28150}, {19877, 22793}, {22681, 32522}, {23235, 36519}, {23513, 38669}, {23514, 38664}, {23516, 38683}, {28194, 30315}, {32815, 32887}, {32816, 32886}, {32823, 32868}, {32826, 37647}, {34641, 38021}, {36520, 38686}
As a point on the Euler line, X(3545) has Shinagawa coefficients (2, 3).
X(3545) lies on these lines: 2,3 11,1056 12,1058 69,1568 114,671 230,1285 262,538 355,3241 371,1328 372,1327 551,944 1131,3312 1132,3311 1352,1992 3058,3085
X(3545) = midpoint of X(4) and X(3524)
X(3545) = reflection of X(i) in X(j) for these (i,j): (376,3524), (3524,2)
X(3545) = complement of X(10304)
X(3545) = anticomplement of X(5054)
X(3545) = circumcircle-inverse of X(37953)
X(3545) = orthocentroidal-circle inverse of X(376)
X(3545) = circle-O(PU(5))-inverse of X(382)
X(3545) = trisector nearest X(2) of segment X(2)X(4)
X(3545) = trisector nearest X(381) of segment X(2)X(381)
X(3545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,15702), (2,4,376), (2,5,5071), (2,20,549), (2,381,4), (2,382,15715), (2,384,33224), (2,11737,3544), (3,4,33703), (3,5,5056), (3,381,3845), (4,5,3090), (5,381,2), (5,5068,3544)
As a point on the Euler line, X(3546) has Shinagawa coefficients (E - 2F, -E).
X(3546) lies on these lines: 2,3 498,1038 499,1040 590,1578 615,1579 1060,3085 1062,3086 1092,1899
X(3546) = complement of X(3089)
As a point on the Euler line, X(3547) has Shinagawa coefficients (E + 2F, -E).
X(3547) lies on these lines: 2,3 68,1176 69,155 343,1181 498,1040 499,1038 577,2548 590,1579 615,1578 1060,3086 1062,3085
As a point on the Euler line, X(3548) has Shinagawa coefficients (E - 4F, -E).
Let La be the polar of X(4) wrt the circle centered at A and passing through X(3), and define Lb and Lc cyclically. (Note that X(4) is the perspector of any circle centered at a vertex of ABC.) Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is homothetic to the medial triangle, and the center of homothety is X(3548). (Randy Hutson, December 11, 2015)
X(3548) lies on these lines: 2,3 68,125 254,2970 498,1060 499,1062 1147,1899 1204,1568
X(3548) = complement of X(3542)
As a point on the Euler line, X(3549) has Shinagawa coefficients (E + 4F, -E).
X(3549) lies on these lines: 2,3 68,184 155,343 193,195 206,1209 216,2165 498,1062 499,1060 577,1506
X(3549) = complement of X(3541)
X(3549) = homothetic center of orthocevian triangle of X(2) and Johnson triangle
X(3550) lies on these lines: 1,3 2,902 31,34 32,2319 42,3097 63,3099 81,2177 87,2209 172,3208 200,1707 238,1376 595,978 609,1018 612,846 614,1054 750,1621 968,1961 983,1423 1120,2163 1740,1918 1955,2947 2308,3240 2329,3053
X(3550) = isogonal conjugate of X(3551)
X(3550) = X(983)-Ceva conjugate of X(1)
X(3550) = crosssum of X(513) and X(3123)
X(3551) lies on the Feuerbach hyperbola and these lines: 1,1463 8,726 9,1575 87,3123 983,1423 1149,2320
X(3551) = isogonal conjugate of X(3550)
X(3551) = X(982)-cross conjugate of X(1)
As a point on the Euler line, X(3552) has Shinagawa coefficients ((E + F)2 - 3S2, 4S2).
X(3552) lies on these lines: 2,3 32,99 69,2076 76,187 83,574 172,192 287,1204 330,1914 385,1975 1613,3360
X(3552) = reflection of X(2) in X(33246)
X(3552) = reflection of X(5025) in X(7807)
X(3552) = complement of X(33019)
X(3552) = cevapoint of X(3) and X(3360)
X(3552) = anticomplement of X(5025)
X(3552) = isogonal conjugate of perspector of 2nd Brocard circle
X(3552) = {X(2),X(3)}-harmonic conjugate of X(33004)
X(3552) = {X(2),X(4)}-harmonic conjugate of X(32966)
X(3552) = {X(2),X(20)}-harmonic conjugate of X(6655)
X(3552) = orthocentroidal circle inverse of X(32966)
X(3553) lies on these lines: 1,6 19,41 33,42 34,3087 35,1609 48,2285 57,2178 65,198 71,1182 78,2345 165,1030 172,577 200,594 205,1474 216,1040 284,1766 326,894 380,584 571,609 610,2174 612,2318 800,1500 1490,1901 1834,2910 1953,2082 2099,2262 2266,2294 2270,3340 2303,2327 3068,3084 3069,3083
X(3553) = {X(1),X(6)}-harmonic conjugate of X(3554)
X(3554) lies on these lines: 1,6 19,604 33,3087 34,393 36,1609 48,2082 56,2262 57,2164 198,319 216,1038 239,326 269,1086 282,1146 374,1696 380,2278 577,1040 800,1015 836,3086 1033,1398 1182,2260 1249,1870 1404,2261 1407,1422 1420,2178 1953,2285 3068,3083 3069,3084
X(3554) = crosspoint of X(1) and X(1422)
X(3554) = crosssum of X(i) and X(j) for these (i,j): (1,2324), (3083, 3084)
X(3554) = {X(1),X(6)}-harmonic conjugate of X(3553)
See Tibor Dosa, "Some triangle centers associated with excircles," Forum Geometricorum 7 151-158.
X(3555) lies on these lines: 1,6 8,443 10,354 20,145 28,1280 63,3295 65,519 78,999 200,474 210,1125 244,3214 319,2891 329,1058 379,3187 388,3419 496,908 528,1770 758,3057 912,1482 938,3421 962,971 997,3304 1009,2350 1320,2771 1376,3338 1858,2098 1872,1897 1998,3149 2093,2136
X(3555) = reflection of X(i) in X(j) for these (i,j): (8,942), (72,1), (3057,3244)
X(3555) = anticomplement of X(34790)
X(3555) = X(16655)-of-excentral-triangle
X(3555) = orthologic center of these triangles: Hutson-extouch to Hutson-intouch; the reciprocal orthologic center is X(5920)
X(3556) lies on these lines: 1,159 3,960 6,1245 20,1610 25,65 31,56 40,197 55,976 63,1619 64,71 73,2187 100,1265 161,2099 222,1660 859,1780 958,1503 1012,2217 1398,1456 1498,3428 1593,2182 2176,2178
X(3556) = isogonal conjugate of X(7219)
X(3556) = X(63)-Ceva conjugate of X(6)
X(3556) = crosssum of X(i) and X(j) for these (i,j): (123,522), (513,2968)
X(3556) = pole wrt circumcircle of line X(521)X(656)
X(3556) = X(188)-of-the-tangential-triangle
X(3556) = polar conjugate of isotomic conjugate of X(22119)
X(3556) = crosspoint of X(109) and X(15385)
X(3556) = X(65)-of-the-excentral-triangle-of-the-tangential-triangle
X(3557) was submitted and called the Pappus point by Roland Bacher (March 11, 2009); coordinates and the related point X(3558) were found by Peter Moses (March 12, 2009).
X(3557) is the crosspoint (and crosssum) of the real foci of the Steiner inellipse. (Bernard Gibert, January 4, 2015)
X(3557) is the Brocard axis intercept, other than X(1340), of the circle {{X(1340),PU(1)}}. Also, X(3557) is the insimilicenter of the 2nd Lemoine circle and the circle {{X(371),X(372),PU(1),PU(39)}}. (Randy Hutson, January 5, 2015)
X(3557) lies on the cubics K028, K289, K704 and these lines: 3,6 4,3414 194,3413 262,6178
X(3557) = reflection of X(i) in X(j) for these (i,j): (1379,2029), (3558,3095)
X(3557) = isogonal conjugate of X(6177)
X(3557) = inverse-in-2nd-Brocard-circle of X(1341)
X(3557) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,576,3558), (6,32,3558), (6,1379,1341), (6,2029,1340), (61,62,3558), (371,372,1380), (1670,1671,1341), (1687,1688,2558), (1689,1690,3558), (3104,3105,3558)
X(3557) = crosssum of PU(118)
X(3557) = crosspoint of PU(118)
X(3558) is the crosspoint (and crosssum) of the imaginary foci of the Steiner inellipse. (Bernard Gibert, January 4, 2015)
X(3558) is the Brocard axis intercept, other than X(1341), of the circle {{X(1341),PU(1)}}. Also, X(3558) is the exsimilicenter of the 2nd Lemoine circle and the circle {{X(371),X(372),PU(1),PU(39)}}. (Randy Hutson, January 5, 2015)X(3558) lies on the cubics K028, K289, K704 and these lines: 3,6 4,3413 194,3414 262,6177
X(3558) = reflection of X(i) in X(j) for these (i,j): (1380,2028), (3557,3095)
X(3558) = isogonal conjugate of X(6178)
X(3558) = crosssum of PU(119)
X(3558) = crosspoint of PU(119)
X(3558) = inverse-in-2nd-Brocard-circle of X(1340)
X(3558) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,576,3557), (6,32,3557), (6,1380,1340), (6,2028,1341), (61,62,3557), (371,372,1379), (1670,1671,1340), (1687,1688,2559), (1689,1690,3557), (3104,3105,3557)
As a point on the Euler line, X(3559) has Shinagawa coefficients ($aSA$F, $aSBSC$ + ($aSA$ - abc)F - $a$S2).
X(3559) lies on these lines: 2,3 58,1785 90,1896 158,920 162,1780 225,283 243,1858 318,333 393,1778 908,1819 1068,3157 1838,2328
X(3559) = X(1896)-Ceva conjugate of X(29)
X(3559) = cevapoint of X(i) and X(j) for these (i,j): (4,920), (46,1068), (453,3193)
X(3559) = X(i)-cross conjugate of X(j) for these (i,j): (46,3193), (3193,29)
As a point on the Euler line, X(3560) has Shinagawa coefficients ($aSA$, 2abc - $aSA$), and also Shinagawa coefficients (R + r, R - r).
X(3560) lies on these lines: {1, 90}, {2, 3}, {35, 5587}, {40, 5251}, {55, 355}, {56, 5886}, {58, 5707}, {65, 920}, {84, 5436}, {100, 5818}, {104, 3616}, {119, 498}, {255, 2654}, {284, 5778}, {329, 5761}, {499, 1470}, {515, 5248}, {517, 958}, {573, 4877}, {581, 4653}, {938, 5770}, {943, 4313}, {944, 1621}, {946, 993}, {952, 3295}, {954, 5779}, {956, 1482}, {971, 1001}, {999, 3485}, {1125, 5450}, {1259, 3419}, {1698, 2077}, {1780, 5398}, {1898, 2646}, {2975, 5603}, {3576, 5259}, {3601, 5720}, {3746, 5881}, {3817, 5267}, {3929, 5258}, {4267, 5788}, {4654, 5563}, {5086, 5687}, {5260, 5657}, {5446, 5752}, {5698, 5762}, {5703, 5811}, {5744, 5804}, {5841, 6585}
X(3560) = complement of X(6850)
X(3560) = anticomplement of X(37438)
X(3561) lies on these lines: 1,21 3,1425 20,109 225,412 377,1771 1040,3215 1092,1813 1259,1331
X(3561) = crosssum of X(2310) and X(2501)
X(3562) lies on these lines: 1,21 4,651 6,938 8,394 20,222 40,77 65,775 73,411 100,1771 110,270 145,280 221,962 404,3075 416,820 950,2003 1066,3072 1071,3100 1406,3474 1895,3194 1935,2654
X(3562) = crosssum of X(647) and X(2310)
The antipodal pair X(3563) to X(3565) were described by Peter Moses, Dec. 13, 2004.
Let A' be the reflection in line BC of the A-vertex of the antipedal triangle of X(6), and define B' and C' cyclically. Let OA be the circumcenter of AB'C', and define OB and OC cyclically. Let U be the circumcenter of A'BC, and define V and W cyclically. Let O' be the circumcenter of OAOBOC. Let O'' be the circumcenter of UVW. Then X(3563) = Λ(O',O''). Also, O'O''∩(infinity line) = X(3564). (Randy Hutson, June 19, 2015)
X(3563) lies on the circumcircle and these lines: 2,136 3,2971 4,99 24,112 25,110 100,1824 101,2333 107,459 186,691 232,1692 378,1296 403,935 427,930 468,476 934,1426
X(3563) = reflection of X(3565) in X(3)
X(3563) = isogonal conjugate of X(3564)
X(3563) = anticomplement of X(31842)
X(3563) = cevapoint of X(25) and X(232)
X(3563) = Λ(X(5), X(6))
X(3563) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(2),X(24)}
X(3563) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(25)}
X(3563) = inverse-in-polar-circle of X(114)
X(3563) = inverse-in-{circumcircle, nine-point circle}-inverter of X(136)
X(3563) = trilinear pole of line X(6)X(924)
X(3563) = trilinear pole, wrt Thomson triangle, of line X(182)X(3167)
X(3563) = Ψ(X(6), X(924))
X(3563) = Cundy-Parry Phi transform of X(14248)
X(3563) = Cundy-Parry Psi transform of X(6337)
X(3563) = perspector, wrt dual of orthic triangle, of polar circle
X(3564) and X(3566) are on the line at infinity. They may be regarded as directions in the plane of ABC, and as such are perpendicular. Continuing from X(3563), the infinite point of the line O'O'' is X(3564). (Randy Hutson, June 19, 2015)
X(3564) lies on these (parallel) lines: 2,3167 3,69 4,193 5,6 26,159 30,511 52,1843 98,325 110,468 114,230 115,1570 125,3292 140,141 147,385 156,206 184,343 265,895 287,441 323,858 381,1992 383,3181 394,1368 427,1993 428,3060 429,3193 495,611 496,613 546,576 547,597 548,3098 549,599 550,1350 1080,3180 1483,3242 1625,2211 1994,3410 2023,2025 2930,2931
X(3564) = isogonal conjugate of X(3563)
X(3564) = X(2)-Ceva conjugate of X(35067)
X(3564) = crossdifference of every pair of points on line X(6)X(924)
X(3564) = Cundy-Parry Phi transform of X(6337)
X(3564) = Cundy-Parry Psi transform of X(12248)
X(3564) = X(92)-isoconjugate of X(32654)
Let La be the reflection of the line X(5)X(6) in line BC, and define Lb and Lc cyclically. Let A' = Lc∩Lc, B' = Lc∩La, C' = La∩Lb. The lines AA', BB', CC' concur in X(3565). (Randy Hutson, January 29, 2018)
X(3565) lies on the circumcircle and these lines: 2,2374 3,2971 20,98 22,111 376,1300 378,1299 842,2071 858,2770 1370,2373 &nbnbsp; 2456,2698
X(3565) = reflection of X(3563) in X(3)
X(3565) = isogonal conjugate of X(3566)
X(3565) = crosssum of X(512) and X(2519)
X(3565) = X(2489)-cross conjugate of X(2)
X(3565) = anticomplement of X(5139)
X(3565) = trilinear pole of line X(6)X(1196)
X(3565) = Ψ(X(6), X(1196))
X(3565) = Λ(trilinear polar of X(459))
X(3566) lies on these (parallel) lines: 30,511 64,879 669,3265 1640,1853 2088,3143 2489,2506 2491,2524 2450,3569
X(3566) = isogonal conjugate of X(3565)
X(3566) = isotomic conjugate of X(35136)
X(3566) = X(2)-Ceva conjugate of X(15525)
X(3566) = crossdifference of every pair of points on line X(6)X(1196)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2[2S4 + SBSC(2S2 - SBSC - (1/2)(2 + b2 + c2)SASBSC
This point is the first of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see
The other two Spanish Points are the Sancho Panza Point X(3060) and the Miguel de Cervantes Point X(143).
X(3567) = X(1698)-of-the-orthic-triangle, if ABC is acute. (See also X(3060).)
X(3567) lies on these lines: 2,52 3,143 4,51 5,568 6,24 25,1614 74,1112 140,2979 155,1995 184,1199 186,578 511,631 567,1658 576,1092 1147,1994 1154,1656
X(3567) = crosssum of X(3) and X(1656)
X(3567) = homothetic center of orthocentroidal triangle and X(4)-adjunct anti-altimedial triangle
X(3567) = homothetic center of X(5)-altimedial and X(2)-adjunct anti-altimedial triangles
X(3567) = insimilicenter of the circumcircle and the nine-point circle of the orthic triangle. (The exsimilicenter is X(3060).)
As a point on the Euler line, X(3568) has Shinagawa coefficients ((E + F)3F - (E + F)(E + 10F)S2 + 5S4, (E + F)4 - 2(E + F)(2E - 7F)S2 - 3S4).
Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X(3568) is the point of intersection of the lines P(2)P(40) and U(2)U(40). The name "Beltrami-Euler Point" signifies the fact that X(3568) lies on the Euler line. See also X(2966) and X(3569). Contributed by Chris van Tienhoven, June 7, 2009.
X(3568) lies on this line: 2,3
Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X(3569) is the point of intersection of the lines P(2)U(2) and P(40)U(40). The name "Beltrami-Parry Point" signifies the fact that X(3569) lies on the line X(74)X(111); where X(111) is the Parry point.and X(74) is the isogonal conjugate of the Euler infinity point. See also X(3568). Contributed by Chris van Tienhoven, June 7, 2009..
X(3569) lies on the Walsmith rectangular hyperbola and these lines: 6,526 74,111 110,112 113,1560 115,125 187,237 248,2422 297,525 520,2451 684,2491 688,2514 694,804 879,1987 924,2485 1499,1513 1510,3050 1769,2294 2450,3566
X(3569) = isogonal conjugate of X(2966)
X(3569) = reflection of X(i) in X(j) for these (i,j): (6,2492), (1640,1637), (2451,2489), (3049,2485), (3288,647), (32120,468)
X(3569) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,38987), (297,868), (648,2967), (694,3124), (878,2451), (1987,3269), (2421,511), (2715,6)
X(3569) = complement of anticomplementary conjugate of X(39359)
X(3569) = perspector of hyperbola {{A,B,C,X(6),X(232)}}
X(3569) = crosspoint of X(i) and X(j) for these (i,j): (6,2515), (98,648), (110,2987)
X(3569) = crosssum of X(i) and X(j) for these (i,j): (2,2799), (98,2395), (112,2409), (230,523), (248,879), (441,525), (511,647)
X(3569) = crossdifference of every pair of points on the line X(2)X(98)
X(3569) = Parry-circle-inverse of X(647)
X(3569) = midpoint of Jerabek hyperbola intercepts of Lemoine axis
X(3569) = Parry-isodynamic-circle-inverse of X(1495); see X(2)
X(3569) = X(850)-of-1st-Brocard-triangle
X(3569) = X(6)-of-2nd-Parry-triangle
X(3569) = X(6)-of-3rd-Parry-triangle
X(3569) = 1st-Brocard-isotomic conjugate of X(1316)
X(3569) = intersection of trilinear polar of X(6) and polar wrt polar circle of X(6)
X(3569) = centroid of (degenerate) cross-triangle of 2nd and 3rd Parry triangles
X(3569) = center of the (degenerate) perspeconic of the 2nd and 3rd Parry triangles
X(3569) = antipode of X(32120) in Walsmith rectangular hyperbola
X(3569) = orthocenter of X(6)X(74)X(110)
X(3569) = orthocenter of X(6)X(113)X(125)
X(3569) = orthocenter of X(125)X(1495)X(3580)
The points given by trilinears P(6) = b : c : a and U(6) = c : a : b are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) = 1/b : 1/c : 1/a and U(8) = 1/c : 1/a : 1/b. The bicentric pair of lines P(6)U(8) and P(8)U(6) concur in X(3570). Contributed by Peter Moses, July 7, 2009..
If you have The Geometer's Sketchpad, you can view X(3570).
X(3570) lies on these lines: 2,6 100,190 101,668 106,3227 662,799 666,1026 1016,1023 1914,3253
X(3570) = isogonal conjugate of X(3572)
X(3570) = isotomic conjugate of X(4444)
X(3570) = X(666)-Ceva conjugate of X(190)
X(3570) = cevapoint of X(659) and X(2238)
X(3570) = crossdifference of every pair of points on the line X(512)X(1015)
X(3570) = trilinear pole of line X(238)X(239)
The points given by trilinears P(6) = b : c : a and U(6) = c : a : b are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) = 1/b : 1/c : 1/a and U(8) = 1/c : 1/a : 1/b. The bicentric pair of lines P(6)P(8) and U(6)U(8) concur in X(3571). Contributed by Peter Moses, July 7, 2009.
Continuing, the line P(6)P(8) is the tangent to the 1st bicentric of the Kiepert hyperbola at P(8), and line U(6)U(8) is the tangent to the 2nd bicentric of the Kiepert hyperbola at U(8). (Randy Hutson, March 25, 2016)
X(3571) lies on these lines: 1,512 9,43 1621,1964
X(3571) = crossdifference of PU(90)
X(3571) = crossdifference of every pair of points on line X(2238)X(4367)
The points given by trilinears P(6) = b : c : a and U(6) = c : a : b are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) = 1/b : 1/c : 1/a and U(8) = 1/c : 1/a : 1/b. The pair of central lines P(6)U(6) and P(8)U(8) concur in X(3572). Contributed by Peter Moses, July 7, 2009.
X(3572) lies on these lines: 2,661 6,798 37,513 42,649 111,741 291,1635 335,812 660,1026 813,901 1019,2084
X(3572) = isogonal conjugate of X(3570)
X(3572) = isotomic conjugate of X(27853)
X(3572) = X(813)-Ceva conjugate of X(292)
X(3572) = X(665)-cross conjugate of X(649)
X(3572) = crosspoint of X(i) and X(j) for these (i,j): (292,813), (1438,2702)
X(3572) = crosssum of X(i) and X(j) for these (i,j): (239,812), (659,2238)
X(3572) = crossdifference of every pair of poins on the line X(238)X(239)
X(3572) = trilinear pole of line X(512)X(1015)
X(3573) lies on these lines: 1,21 2,1083 99,110 100,101 109,932 192,2175 660,2284 666,885 692,1492 785,835 898,901 1332,1633 1618,2397
X(3573) = isogonal conjugate of X(876)
X(3573) = cevapoint of X(238) and X(659)
X(3573) = X(659)-cross conjugate of X(238)
X(3573) = crosspoint of X(99) and X(666)
X(3573) = crosssum of X(512) and X(665)
X(3573) = crossdifference of every pair of points on the line X(244)X(661)
X(3573) = intersection of tangents to Steiner circumellipse at X(99) and X(666)
X(3574) = X(21)-of-the-orthic-triangle if ABC is acute. Let A'B'C' be the orthic triangle, which is the pedal (and cevian) triangle of the orthocenter, H = X(4) . The Euler lines of the triangles HB'C', HC'A', HA'B' concur in X(3574). (Michel Garitte, July 7, 2009)
Let A'B'C' be the cevian triangle of X(5). Let La be the reflection of line B'C' in line BC, and define Lb, Lc cyclically. Let A" = Lb ∩ Lc, and define B", C" cyclically. The lines AA", BB", CC" concur in the isogonal conjugate of X(3574). (Randy Hutson, January 29, 2018)
Let A'B'C' be the orthic triangle. X(3574) is the radical center of the circles O(3,4) of triangles AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)
X(3574) lies on these lines: 4,54 5,51 25,2917 113,137 125,389 130,132 155,195 185,427 193,576 235,1843 1204,3541 1839,2182 1907,2883 3519,3527
X(3574) = midpoint of X(4) and X(54)
X(3574) = reflection of X(1209) in X(5)
X(3574) = anticomplement of X(32348)
X(3574) = X(4)-Ceva conjugate of X(3575)
X(3574) = crosspoint of X(4) and X(5)
X(3574) = crosssum of X(3) and X(54)
As a point on the Euler line, X(3575) has Shinagawa coefficients (F, -E - 3F).
Let KH denote the hyperbola discussed at X(1112). As noted in the paper cited at X(1112), KH is the X(4)-Ceva conjugate of the Euler line. Inversely, the Euler line is the X(4)-Ceva conjugate of KH. Since X(3574) lies on KH, its X(4)-Ceva conjugate, which is X(3575), lies on the Euler line. (Peter Moses, July 7, 2009)
X(3575) lies on these lines: 1,1892 2,3 53,571 64,66 128,135 185,1503 225,1852 317,1975 515,1829 516,1902 1179,1300 1192,1853 1452,1837 1828,2829 1862,1872
X(3575) = reflection of X(1885) in X(4)
X(3575) = circumcircle-inverse of X(37954)
X(3575) = X(4)-Ceva conjugate of X(3574)
X(3575) = X(65)-of-orthic-triangle if ABC is acute
X(3575) = Kosnita-to-orthic similarity image of X(5)
X(3575) = {X(4),X(24)}-harmonic conjugate of X(5)
X(3575) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(52)
X(3575) = crosspoint, wrt orthic triangle, of X(4) and X(52)
X(3575) = Ehrmann-vertex-to-orthic similarity image of X(5)
X(3575) = excentral-to-ABC functional image of X(65)
X(3575) = X(4)-of-anti-Ara-triangle
X(3576) occurs in Hyacinthos #18095 (Quang Tuan Bui, August 2, 2009). Following a note by Seiichi Kirikami [see X(165)], Peter Moses, October 19, 2010, found that if DEF is the pedal triangle of X(3576), then |FB| + |CE| = |DC| + |AF| = |EA| + |BD|.
Let Pa be the parabola with focus A and directrix BC, and define Pb and Pc cyclically. X(3576) is the centroid of the six points of tangency of lines from X(1) to Pa, Pb, and Pc. (Randy Hutson, December 2, 2017)
Let A'B'C' be the hexyl triangle. Let Ab = BC∩C'A', Ac = BC∩A'B', and define Bc, Ba, Ca, Cb cyclically. Then X(3576) is the centroid of AbAcBcBaCaCb. (Randy Hutson, December 2, 2017)
Let I be the incenter, X(1), and Ia, Ib, Ic be the excenters. Let Oa be the de Longchamps circle of triangle IbIcI, and define Ob and Oc cyclically. Then X(3576) is the radical center of Oa, Ob, Oc. (Randy Hutson, December 2, 2017)
If you have The Geometer's Sketchpad, you can view X(3576).
X(3576) lies on these lines: 1,3 2,515 4,1125 8,3523 9,48 10,631 11,3523 20,946 21,84 30,1699 58,602 77,102 78,947 106,1292 140,355 154,392 187,1572 198,374 200,956 223,1455 284,2257 376,516 380,1108 405,1490 519,3158 549,952 573,1449 580,1468 581,1193 595,601 936,958 950,3086 953,1308 960,1071 962,3522 963,2910 970,1051 991,995 1001,1012 1151,1702 1152,1703 1210,3486 1212,3207 1350,1386 1435,1870 1511,2948 1732,2364 1766,3247 1829,3515 1902,3516 2202,3100 2320,3306 2718,2742
X(3576) = midpoint of X(1) and X(165)
X(3576) = reflection of X(i) in X(j) for these (i,j): (40,165),(165,3)
X(3576) = isogonal conjugate of X(3577)
X(3576) = X(2320)-Ceva conjugate of X(1)
X(3576) = crosssum of X(2) and X(2093)
X(3576) = crossdifference of every pair points on the line X(650)X(1769)
X(3576) = X(2)-of-hexyl-triangle
X(3576) = anticomplement of X(10175)
X(3576) = X(2)-of-2nd-circumperp-triangle
X(3576) = X(381)-of-excentral-triangle
X(3576) = {X(1),X(3)}-harmonic conjugate of X(40)
X(3576) = homothetic center of hexyl triangle and medial triangle of 2nd circumperp triangle
X(3576) = homothetic center of Johnson triangle and cross-triangle of Aquila and anti-Aquila triangles
X(3576) = Thomson-isogonal conjugate of X(9)
X(3576) = orthocenter of cross-triangle of ABC and anti-Aquila triangle
X(3576) = insimilicenter of circumcircles of ABC and anti-Aquila triangle; the exsimilicenter is X(1)
X(3576) = trisector nearest X(3) of segment X(1)X(3)
X(3576) = endo-homothetic center of Ehrmann side-triangle and orthic triangle; the homothetic center is X(381)
In the plane of a triangle ABC, let
A'B'C' = excentral triangle,
Oa = circle with diameter BC, and define Ob and Oc cyclically;
Pa = polar of A' with respect to Oa, and define Pb and Pc cyclically;
A" = Pc∩Pb, and define B" and C" cyclically.
The triangle A"B"C" is perspective to ABC, and the perspector is X(3577). (Dasari Naga Vijay Krishna, July 15, 2021)
X(3577) lies on these lines: 1,227 4,3340 7,515 8,908 9,374 21,40 30,3255 80,1537 314,322 943,1697 971,1159 1000,1512 1156,2800 1172,2331 1476,3333 1709,3065 2320,3306 3243,3254
X(3577) = isogonal conjugate of X(3576)
X(3577) = cevapoint of X(1) and X(2093)
X(3577) = X(2099)-cross conjugate of X(1)
X(3577) = trilinear pole of line X(650)X(1769)
X(3578) lies on these lines: 2,6 8,30 319,3219 340,445 519,2292 540,1046 542,1281
X(3578) = anticomplement of X(37631)
X(3578) = X(75)-Ceva conjugate of X(1125)
Let OA be the circle centered at the A-vertex of the Aquila triangle and passing through A; define OB and OC cyclically. X(3579) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(3579) lies on these lines: 1,3 4,2355 5,516 8,376 10,30 12,1770 20,355 24,1902 31,582 37,2160 42,500 44,573 45,1766 71,2173 72,74 109,227 140,946 191,210 267,2941 378,1829 381,1698 392,404 498,1836 515,550 518,3098 548,952 549,1125 631,962 759,1293 896,3214 901,2687 944,3522 971,1158 1191,1480 1216,2807 1250,2306 1656,1699 1702,3312 1703,3311 1827,1872 3085,3474 3218,3555
X(3579) = midpoint of X(i) and X(j) for these (i,j): (3,40), (20,355)
X(3579) = reflection of X(i) in X(j) for these (i,j): (946,140), (1385,3)
X(3579) = X(3210)-Ceva conjugate of X(37)
X(3579) = X(5)-of-1st-circumperp-triangle
X(3579) = X(140)-of-excentral-triangle
X(3579) = centroid of excenters and X(40)
Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the orthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the orthic axis. The triangle A"B"C" is homothetic to ABC, and its centroid is X(3580); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)
X(3580) lies on the Walsmith rectangular hyperbola and these lines: 2,6 5,568 22,1899 23,1503 24,68 30,74 52,1594 110,468 113,403 125,511 140,567 146,1514 186,2931 278,2994 281,445 287,2373 297,525 340,687 451,3193 467,2052 542,1495 1154,2072 1352,1995 1368,2979
X(3580) = midpoint of X(i) and X(j) for these (i,j): (23,3448), (265,3581)
X(3580) = reflection of X(i) in X(j) for these (i,j): (110,468), (146,1514), (858,125)
X(3580) = isotomic conjugate of X(2986)
X(3580) = complement of X(323)
X(3580) = X(340)-Ceva conjugate of X(30)
X(3580) = cevapoint of X(6) and X(2931)
X(3580) = X(3003)-cross conjugate of X(403)
X(3580) = crosspoint of X(i) and X(j) for these (i,j): (2,94), (76,1494)
X(3580) = crosssum of X(i) and X(j) for these (i,j): (6,50), (32, 1495), (647,2088)
X(3580) = crossdifference of every pair of points on the line X(184)X(512)
X(3580) = crosspoint of X(6) and X(2931) wrt both the excentral and tangential triangles
X(3580) = pole wrt polar circle of trilinear polar of X(1300)
X(3580) = X(48)-isoconjugate (polar conjugate) of X(1300)
X(3580) = antipode of X(110) in Walsmith rectangular hyperbola
X(3580) = orthocenter of X(125)X(1495)X(3569)
Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A* be the crosssum of A1 and A2, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(3581).
X(3581) lies on these lines: 3,6 30,74 49,1658 185,2937 186,323 381,1531 399,1495 541,1533 1204,1657
X(3581) = reflection of X(i) in X(j) for these (i,j): (265,3580), (323,1511), (399,1495)
X(3581) = perspector of ABC and cross-triangle of ABC and 2nd isogonal triangle of X(4)
X(3582) lies on these lines: 1,2 11,30 12,547 35,496 56,381 57,1727 58,3615 80,1319 244,1725 376,1479 497,3524 542,1428 553,1776 599,613 946,3336 1478,3545 1519,1768 1656,3304 1749,3218 1837,3656 2964,3075 3303,3526
X(3583) lies on these lines: 1,4 5,35 11,30 12,546 20,499 55,381 56,382 79,942 80,517 115,1914 149,519 316,350 484,516 495,3058 1203,1834 1539,3028 1749,1776 1770,3336 1781,1839 2308,3017 2964,3073 3065,3218 3070,3299 3071,3301 3100,3153
X(3583) = reflection of X(i) in X(j) for these (i,j): (36,11), (484,1737)
X(3583) = crosspoint of X(79) and X(80)
X(3583) = crosssum of X(35) and X(36)
X(3583) = homothetic center of 2nd isogonal triangle of X(1) and the reflection of the Johnson triangle in X(4); see X(36)
X(3583) = homothetic center of Mandart-incircle triangle and (cross-triangle of ABC and 2nd isogonal triangle of X(1))
X(3583) = homothetic center of Ehrmann vertex-triangle and intangents triangle
X(3583) = homothetic center of Ehrmann mid-triangle and Mandart-incircle triangle
X(3583) = {X(1),X(4)}-harmonic conjugate of X(3585)
X(3584) lies on these lines: 1,2 5,3058 11,547 12,30 13,1250 36,495 37,1989 55,381 79,3579 226,484 376,1478 388,3524 542,2330 553,3336 599,611 756,1725 1479,3545 1656,3303 1749,3219 2964,3074 3304,3526
X(3585) lies on these lines: 1,4 5,36 10,191 11,546 12,30 13,2307 20,498 55,382 56,381 65,79 115,172 149,3244 316,1909 355,1836 377,1698 535,2975 993,2476 1254,1725 1539,3024 1781,1826 2964,3072 3070,3301 3071,3299
X(3585) = reflection of X(35) in X(12)
X(3585) = homothetic center of 2nd isogonal triangle of X(1) and Johnson triangle; see X(36)
X(3585) = {X(1),X(4)}-harmonic conjugate of X(3583)
X(3585) = homothetic center of Ehrmann vertex-triangle and anti-tangential midarc triangle
X(3586) lies on these lines: 1,4 7,3543 9,80 10,452 11,3576 20,1210 30,57 40,1728 46,2955 90,191 165,1737 329,519 355,1697 380,1826 405,1376 496,1420 516,2093 517,1864 936,2478 938,3146 993,1005 1449,1901 1453,1834
X(3586) = reflection of X(i) in X(j) for these (i,j): (1,497), (1750,4)
X(3586) = insimilicenter of hexyl and 2nd Johnson-Yff circles; the exsimilicenter is X(9614)
X(3586) = {X(1), X(1479)}-harmonic conjugate of X(9614)
X(3587) lies on these lines: 1,3 4,3305 9,30 20,3219 63,376 84,550 515,3358 582,1453 1445,3488 3306,3524
In 2008, Alexei Myakishev gave the following construction of a conic. Let CA and CB be points on the line AB satisfying |BCA| = |CB| and |ACB| = |CA| and arranged in this order: CA, B, A, and CB. Define points AB, CB, BC, AC cyclically. The six points CA, BA, AB, CB, BC, AC lie on a conic. Myakishev's proof is by Carnot's theorem, since
[c/(c+a)][(a+b)/b][a/(a+b)][(b+c)/c][b/(b+c)][(c+a)/a] = 1.
For details, in Russian, visit Item 4 at Geometry.ru,
Peter Moses found that X(3588) is the point of concurrence of three lines constructed from points numbered 1,3,4,6,8,9,10,11, as follows:
X(3588) = X(8)X(573)∩X(2269,3057), where
X(573) = X(3)X(6)∩X(4)X(9)
X(2269) = X(8)X(9)∩X(1)X(573)
X(3057) = X(1)X(3)∩X(10)X(11)
The six points Ca, Ba, Ab, Cb, Bc, Ac can be constructed from the extangents triangle as follows. Let A'B'C' be the extangents triangle. Then Ab = BC∩C'A', Ac = BC∩A'B', and Bc, Ba, Ca, Cb are defined cyclically. (Randy Hutson, January 29, 2018)
X(3588) lies on these lines: 8,573 37,1953 42,181 71,594 213,2347 1824,2354 2225,2264 3059,3198
In 2003, Peter Moses gave a general construction as follows: Suppose that P = u : v : w (barycentrics). The centroids of the triangles BCP, CAP, ABP form a triangle homothetic to ABC, with ratio -1/3 and center
P' = 2u + v + w : u + 2v + w : u + v +2w.
In 2010, Seiichi Kirikami gave another construction for P': let
D = AP∩BC, E = BP∩CA, F = CP∩AB.
Then the Newton lines of the quadrilaterals PEAF, PFBD, PDCE concur in P'. (The Newton line of a quadrilateral is the line of the midpoints of the two diagonals of the quadrilateral.)
X(3589) is the Moses-Kirikami image of the symmedian point. In the following list (from P. Moses, Aug. 23, 2010), the appearance of I, J means that X(j) is the Moses-Kirikami image of X(i).
1,1125 2,6 3,140 4,5 6,3589 7,142 8,10 20,3 23,468 69,141 99,620 100,3035 144,9 145,1 146,113 147,114 148,115 149,11 150,116 151,117 152,118 153,119 192,37 193,6 194,39 239,3008 315,626 316,625 329,3452 376,549 381,547 382,546 385,230 390,1001 410,441 487,642 488,641 550,3530 616,618 617,619 621,623 622,624 627,629 628,630 631,632 633,635 634,636 637,639 638,640 944,1385 962,946 1278,75 1320,1387 1330,3454 1370,1368 1654,1213 1657,548 1916,2023 1992,597 2475,442 2888,1209 2895,1211 3091,1656 3146,4 3151,440 3153,2072 3164,216 3177,1212 3180,396 3181,395 3241,551 3434,2886 3436,1329 3448,125 3522,631 3523,3526 3529,550 3543,381
Early in 2012, Seiichi Kirikami found a simple relation between the tetrahedron and the Moses-Kirikami image. Using 3-dimensional cartesian coordinates, suppose that the vertices of triangle ABC are placed in the xy-plane:
A = (x1, y1, 0), B = (x2, y2, 0), C = (x3, y3, 0).
Let P = (x4, y4, z4) be an arbitrary point not in the plane of ABC. Then the centroid of the tetrahedron ABCP is the point
Q = ( (x1 + x2 + x3 + x4)/4, (y1 + y2 + y3 + y4)/4, z4/4 ).
The orthogonal projections of P and Q onto the xy plane coincide with the Moses-Kirikami images of X(6) and X(3589). That is, X(6) has coordinates (x4, y4, 0), and X(3589) has coordinates ( (x1 + x2 + x3 + x4)/4, (y1 + y2 + y3 + y4)/4, 0 ).
Let A' be the midpoint of segment AX(6), and define B' and C' cyclically. Then A'B'C' is homothetic to the medial triangle, and the homothetic center is X(3589). (Randy Hutson, December 26, 2015)
Let E be the bicevian conic of X(2) and X(6); i.e., the ellipse that passes through the vertices of the medial and symmedial triangles. Then X(3589) is the center of E. This ellipse is also the locus of centers of circumconics passing through X(6). Let L be a line through X(2), and let P and P' be the points of intersection of L and the circumcircle. Let V be the locus of the crosssum of P and P'. The locus of V generated by L is E. (Randy Hutson, December 26, 2015)
Let A' be the reflection of X(6) in line BC. Let Oa be the circle with center A' and tangent to BC. Define Ob and Oc cyclically. The radical center of Oa, Ob, Oc is X(3589). (Randy Hutson, December 26, 2015)
If you have The Geometer's Sketchpad, you can view X(3589).
X(3589) lies on these lines:
2,6 5,182 10,1386 11,2330 12,1428 23,2916 39,698 51,3313 53,458 83,316 140,143 216,441 239,594 264,1990 373,468 397,622 398,621 427,1974 498,613 499,611 518,1125 542,547 549,3098 575,3564 576,632 625,2030 631,1350 742,3008 894,1086 1351,3526 1352,1656 1506,1692 1698,3416 2450,3566
X(3589) = midpoint of X(i) and X(j) for these (i,j): (2,597), (5,182), (6,141), (10, 1386), (625, 2030)
X(3589) = isogonal conjugate of X(3108)
X(3589) = isotomic conjugate of X(10159)
X(3589) = complement of X(141)
X(3589) = complement of complement of X(6)
X(3589) = anticomplement of X(34573)
X(3589) = polar conjugate of isogonal conjugate of X(22352)
X(3589) = polar conjugate of isotomic conjugate of X(7767)
X(3589) = complementary conjugate of X(21248)
X(3589) = crosssum of X(6) and X(39)
X(3589) = {X(2),X(6)}-harmonic conjugate of X(141)
X(3589) = centroid of {A,B,C,X(6)}
X(3589) = centroid of PU(11)PU(45)
X(3589) = Kosnita(X(6),X(2)) point
X(3589) = crosspoint of X(2) and X(83)
X(3589) = X(620) of 1st Brocard triangle
X(3589) = antipode of X(141) in conic {{X(13),X(14),X(15),X(16),X(141)}}
X(3589) = complement of isotomic conjugate of cevapoint of X(2) and X(6)
X(3589) = polar conjugate of isogonal conjugate of X(22352)
X(3589) = perspector of the medial triangle and the tangential triangle, wrt the symmedial triangle, of the bicevian conic of X(2) and X(6)
X(3589) = crossdifference of every pair of points on line X(523)X(2076)
X(3589) = X(9)-of-submedial-triangle if ABC is acute
On Dec. 22, 2000, Atul Dixit constructed this point in Hyacinthos message #2183, as follows. Let ABC be a triangle with medians AD, BE, FC and centroid G. Construct semicircles with diameters BD, DC, BC outwardly. Let TA be the circle tangent to the three semicircles BD, DC, BC, and let A' be the center of TA. Define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(3590). Barycentric coordinates were found by Paul Yiu (2000), and further properties and related points, by Peter Moses (2010).
The radius of circle TA is |BC|/6, and the Kiepert angle (e.g., B-to-C-to-X(3890)) is arctan(2/3).
If you have The Geometer's Sketchpad, you can view X(3590) and Dixit Chains of Circles.
X(3590) lies on the Kiepert hyperbola and these lines: {2,3594}, {4,6221}, {6,3591}, {20,1327}, {76,3595}, {140,3316}, {485,3523}, {486,5056}, {590,1131}, {1132,3068}, {1271,5490}, {1328,3091}, {1656,3317}, {3543,6482}
This point is obtained in the manner of X(3590) using the inward semicircles instead of outward.
If you have The Geometer's Sketchpad, you can view X(3591).
X(3591) lies on the Kiepert hyperbola and these lines: 2,3592 6,3590 20,1328 76,3593 140,3317 486,3523 615,1132 1131,3069 1327,3091 1656,3316
X(3592) lies on these lines: 2,3591 3,6 485,546 487,3589 590,1588 615,3525 1587,3529 2066,3298 2067,3297 3068,3071 3070,3146
X(3592) = radical center of the Lucas(3) circles
X(3592) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,3594), (6,371,1151), (6,1151,1152), (61,62,3311), (371,3311,6)
X(3593) lies on these lines: 2,6 4,1165 76,3591 488,3091 637,3523 641,1588
X(3593) = {X(2),X(69)}-harmonic conjugate of X(3595)
X(3594) lies on these lines: 2,3590 3,6 486,546 488,3589 590,3525 615,1587 1588,3529 3069,3070 3071,3146 3297,3303 3298,3304
X(3594) = radical center of the Lucas(-3) circles
X(3594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,3592), (6,372,1152), (6,1152,1151), (61,62,3312), (372,3312,6)
X(3595) lies on these lines: 2,6 4,1164 76,3590 487,3091 638,3523 642,1587
X(3595) = {X(2),X(69)}-harmonic conjugate of X(3593)
Let A', B', C' denote the respective excircles of a triangle ABC. Let A'' be the circle tangent to A', B', C' whose interior includes A', and likewise for circles B'' and C''. Let Kaa be the point of tangency of circles A' and A'', and likewise for Let Kbb and Let Kcc. The lines A-to-Kaa, B-to-Let Kbb, C-to-Let Kcc concur in X(3596). For more, see
Boris Odehnal, "Some Triangle Centers Associated with the Circles Tangent to the Excircles," http://forumgeom.fau.edu/FG2010volume10/FG201006.pdf
Barycentrics for the center of the circle A'' are given by Peter Moses (August 26, 2014):
-2 a^4 (b+c)-(b-c)^2 (b+c)^3+a (b^2-c^2)^2+a^2 (b+c) (b^2+c^2)-a^3
(3 b^2+4 b c+3 c^2)
: (b-c) c^2 (b+c)^2+a (b-c) c (b+c) (2 b+c)+a^2 (b+c) (2 b^2+c^2)+a^3
(2 b^2+2 b c+c^2)
: (a-b) b^2 (a+b)^2+(a-b) b (a+b) (2 a+b) c+(a+b) (2 a^2+b^2) c^2+(2
a^2+2 a b+b^2) c^3
If you have The Geometer's Sketchpad, you can view X(3596) and X(3597).
X(3596) lies on these lines: 2,1240 8,314 10,75 43,350 69,150 86,996 92,2064 190,573 192,2092 219,645 261,958 264,1969 281,345 304,309 305,561 312,2321 346,646 730,1740
X(3596) = isogonal conjugate of X(1397)
X(3596) = isotomic conjugate of X(56)
X(3596) = isotomic conjugate of isogonal conjugate of X(8)
X(3596) = isotomic conjugate of isogonal conjugate of anticomplement of X(1)
X(3596) = anticomplement of X(17053)
X(3596) = complement of polar conjugate of isogonal conjugate of X(23159)
X(3596) = polar conjugate of X(608)
X(3596) = trilinear product of extraversions of X(7)
X(3596) = trilinear product of vertices of Gemini triangle 39
Let MA, MB, MC be the centers of the circles A'', B'', C'' used to construct X(3596), respectively. Then the lines A-to-MA, B-to-MB, C-to-MC concur in X(3597). For more, see the reference and sketch at X(3596).
Let P and Q be the intersections of line BC and the excircles radical circle. Let X = X(10). Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3597). (cf. X(592), where the circle is the 1st Lemoine circle and X = X(182)) (Randy Hutson, January 15, 2019)
X(3597) lies on these lines: 2,970 4,2092 226,986 429,2052
Let OA be the circle passing through B and C, internally tangent to the incircle, and likewise for OB and OC. Let PA be the point where OA meets the incircle, and likewise for PB and PC. Let QA be the point of intersection of the tangents to the incircle at PB and PC, and likewise for PB and PC. Then X(3598) is the perspector of the triangles PAPBPC and QAQBQC; this point is also X(7)-of-QAQBQC. This point and X(3599) were discovered by Kang-Ying Liu of St. Andrew's Priory School, Honolulu, Hawaii, during 2010.
X(3598) lies on the circumconic {{A, B, C, X(9), X(105), X(269), X(390), X(3598), X(7218), X(9309)}} and these lines: {1, 7218}, {2, 7}, {20, 3673}, {22, 347}, {25, 1119}, {55, 3672}, {56, 105}, {75, 7172}, {77, 7191}, {85, 3600}, {145, 3212}, {165, 3663}, {193, 33891}, {241, 26242}, {244, 7204}, {269, 479}, {273, 6995}, {320, 37668}, {348, 5265}, {354, 3056}, {388, 7198}, {612, 4328}, {910, 4000}, {942, 7390}, {1002, 1469}, {1014, 4228}, {1086, 7735}, {1111, 4293}, {1122, 7248}, {1155, 4346}, {1323, 13462}, {1418, 3290}, {1420, 3160}, {1429, 2280}, {1442, 17024}, {1467, 4350}, {1477, 9086}, {1788, 30617}, {2266, 18162}, {2348, 37681}, {3091, 4911}, {3340, 5543}, {3361, 10481}, {3616, 3674}, {3664, 10980}, {3665, 7288}, {3705, 10513}, {3785, 33940}, {3920, 7190}, {4056, 10591}, {4225, 18600}, {4294, 7264}, {4308, 9312}, {4352, 37575}, {4441, 10030}, {4644, 37665}, {4848, 32003}, {4872, 5274}, {4888, 24239}, {5228, 5276}, {5232, 25631}, {5261, 7247}, {5268, 7274}, {5272, 7240}, {5542, 9746}, {5838, 24600}, {6049, 25716}, {6180, 33854}, {6356, 7494}, {6904, 20880}, {7081, 31995}, {7146, 29624}, {7185, 17081}, {7223, 7268}, {7238, 7778}, {7249, 30712}, {7263, 8667}, {7269, 29815}, {7272, 10590}, {7282, 7378}, {7465, 37541}, {7485, 15804}, {7736, 17365}, {9058, 15728}, {10106, 31994}, {10404, 24805}, {11038, 14828}, {12632, 17158}, {14986, 17170}, {16823, 32086}, {22464, 26228}, {24796, 32636}, {26236, 26245}, {34253, 37657}
X(3598) = X(7290)-cross conjugate of X (5222)
X(3598) = X(i)-isoconjugate of X(j) for these (i,j): {220, 21446}, {663, 37223}
X(3598) = crosssum of X(4513) and X (37658)
X(3598) = barycentric product X(i)*X(j) for these {i,j}: {7, 5222}, {85, 7290}, {269, 30854}, {279, 390}, {651, 30804}, {1434, 3755}, {4626, 14330}, {24002, 35280}
X(3598) = barycentric quotient X (i)/X(j) for these {i,j}: {269, 21446}, {390, 346}, {651, 37223}, {3755, 2321}, {4989, 3686}, {5222, 8}, {5575, 10322}, {7290, 9}, {14330, 4163}, {28017, 21450}, {30804, 4391}, {30854, 341}, {35280, 644}
X(3598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 1447, 2}, {7, 5435, 9436}, {56, 7195, 279}, {57, 1423, 672}, {75, 15589, 7172}, {3705, 21296, 10513}
Let PAPBPC be as at X(3598). Let TA be the point of intersection of the lines BPC and CPB. X(3599) is the perspector of triangles PAPBPC and TATBTC.
X(3599) lies on these lines: {1, 8916}, {2, 7}, {55, 479}, {165, 279}, {354, 5543}, {390, 31527}, {497, 36620}, {651, 1190}, {658, 10580}, {1088, 9778}, {1323, 31508}, {1615, 6180}, {1996, 9812}, {2898, 30332}, {3748, 31721}, {4350, 10857}, {5218, 30623}, {7056, 10578}, {8236, 31526}, {10004, 13405}, {14100, 15913}, {32624, 37578}
X(3599) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {55, 479, 3160}
X(3600) was found by Peter Moses in connection with the 1st Liu point.
X(3600) lies on these lines: 1,7 2,12 3,1056 4,496 8,57 10,3361 21,1617 30,1058 36,3085 55,3522 65,145 100,1466 144,960 171,1106 193,330 226,452 346,2285 354,3486 376,3295 391,1400 443,956 474,3421 495,631 497,3146 515,938 519,3339 553,3241 942,944 948,1104 982,1254 1010,1014 1043,1434 1159,1483 1319,3485 1385,3487 1435,1891 1446,3424 1478,3086 1479,3543 2646,3475 3057,3474
X(3600) = reflection of X(8) in X(1706)
X(3600) = reflection of X(938) in X(3333)
X(3600) = {X(1),X(20)}-harmonic conjugate of X(390)
X(3601), the {X(1), X(3611)}-harmonic conjugate of X(3576), was found by Peter Moses in connection with the 1st Liu point.
Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the intouch triangle at X(3601). (Randy Hutson, July 31 2018)
X(3601) lies on these lines: 1,3 2,950 5,3586 7,3522 8,3158 9,21 10,3486 20,226 28,33 37,610 58,212 73,991 77,738 84,943 100,1706 142,390 200,958 376,3487 386,1453 405,936 443,497 452,3452 515,3085 516,3485 579,1449 631,1210 728,2329 938,3523 1012,1490 1055,3100 1104,2999 1193,2293 1253,1468 1439,3532 1682,3056 1698,1837 1876,3516
X(3601) = {X(1),X(3)}-harmonic conjugate of X(57)
X(3601) = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of Aquila and anti-Aquila triangles
Points Associated with Morley Cubics
Points X(3602)-X(3609) are associated with the three Morley Cubics indexed as K29, K30, K31 at Bernard Gibert's Cubics in the Triangle Plane. The points are related to those in the section just before X(3272), under the heading "Points Associated with Equilateral Triangles." See also
Jean-Pierre Ehrmann and Bernard Gibert, "A Morley Configuration," Forum Geometricorum 1 (2001) 51-58. (Click here to download a PDF of this article.)
Let A'B'C' be the 1st Morley triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3602). (Randy Hutson, July 11, 2019)
Let A'B'C' be the 1st Morley triangle. Let A* be the foot of the perpendicular from A to line B'C', and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(3602). (Randy Hutson, July 11, 2019)
Let A'B'C' be the 2nd Morley triangle. Let Ba and Ca be points on BC such that A'BcCa is an equilateral triangle; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'', here named the 2nd MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC, with perspector X(3602). (Thanh Oai Dao and Peter Moses, March 30, 2018)
If you have GeoGebra, you can view X(3602).
X(3602) lies on these lines: 356,357 1134,1137 1135,1136 3274, 3604 3275,3603
X(3602) = trilinear pole of perspectrix of ABC and 1st Morley triangle
X(3602) = {X(357), X(358)}-harmonic conjugate of X(356). (Bernard Gibert, November 3, 2010).
X(3603) is {X(1136), X(1137)}-harmonic conjugate of X(3276). (Bernard Gibert, November 3, 2010).
Let A'B'C' be the 3rd Morley triangle. Let Ba and Ca be points on BC such that A'BcCa is an equilateral triangle; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'', here named the 3rd MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC, with perspector X(3603). (Thanh Oai Dao and Peter Moses, March 30, 2018)
Let A'B'C' be the 2nd Morley triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3603). (Randy Hutson, July 11, 2019)
Let A'B'C' be the 2nd Morley triangle. Let A* be the foot of the perpendicular from A to line B'C', and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(3603). (Randy Hutson, July 11, 2019)
X(3603) lies on these lines: 357,1135 358,1134 1136,1137 3273, 3604 3275,3602
X(3603) = trilinear pole of perspectrix of ABC and 2nd Morley triangle
X(3603) = {X(1135),X(3272)}-harmonic conjugate of X(357)
X(3604) is {X(1134), X(1135)}-harmonic conjugate of X(3277). (Bernard Gibert, November 3, 2010)
Let A'B'C' be the 1st Morley triangle. Let Ba and Ca be points on BC such that A'BcCa is an equilateral triangle; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'', here named the 1st MORLEY-MIDPOINT TRIANGLE, is an equilateral triangle that is homothetic to A'B'C' and perspective to ABC, with perspector X(3604). (Thanh Oai Dao and Peter Moses, March 30, 2018)
Let A'B'C' be the 3rd Morley triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3604). (Randy Hutson, July 11, 2019)
Let A'B'C' be the 3rd Morley triangle. Let A* be the foot of the perpendicular from A to line B'C', and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(3604). (Randy Hutson, July 11, 2019)
X(3604) lies on these lines: 356,1134 357,1137 358,1136 3273,3603 3274,3602
X(3604) = trilinear pole of perspectrix of ABC and 3rd Morley triangle
X(3605) lies on these lines: 358,3279 1507,1508
X(3605) = orthology center of 1st Morley triangle and 1st Morley adjunct triangle
X(3606) lies on this line: 1137,3281
X(3607) lies on this line: 1135,3279
X(3608) is the perspector of the Morley triangle and the triangle of the cusps of the Steiner deltoid. (Bernard Gibert, November 3, 2010). See
X(3608) lies on these lines: 5,356 20,3278
631,3279 3273,3609
X(3609) is the perspector of the triangle of the cusps of the Steiner deltoid and the equilateral triangle J defined just before X(3272). (Bernard Gibert, November 3, 2010)
X(3609) lies on these lines: 5,3272 20,3334 631,3335
In a Hyacinthos message dated January 10, 2011, Jean-Louis Ayme introduced a triangle as follows. Let RA be the radical axis of the circumcircle and the A-excircle, and define RB and RC cyclically. Let TA = RB∩RC, and define TB and TC cyclically. (TA is also the radical center of the circumcircle and the B- and C- excircles.) The Ayme triangle TATBTC is perspective to triangle ABC and also perspective to many other triangles. Peter Moses found that its perspector with the cevian triangle of X(346) is X(3610). He also found that the A-vertex of the Ayme triangle has first barycentric as follows:
- (b + c)(a2 + b2 + c2 + 2bc) : b(a2 + b2 - c2) : c(a2 - b2 + c2),
from which the other two vertices are easily obtained. The Ayme triangle is perspective to ABC with perspector X(19).
Moses found that the locus of X such that the cevian triangle of X is perspective to the Ayme triangle is a cubic which passes through the points X(i) for i = 1, 2, 19, 75, 279, 304, 346, 2184. A barycentric equation for this Ayme-Moses cubic follows:
(Cyclic sum of ayz[by(a2 + b2 - c2) - cz(a2 - b2 + c2] ) = 0.
The Ayme triangle is homothetic to the incentral triangle, and the center of homothety is X(612). (Randy Hutson, September 14, 2016)
X(3610) lies on these lines: 10,37 19,346 612,2345
X(3610) = perspector of ABC and cross-triangle of ABC and Ayme triangle
In a Hyacinthos message dated January 7, 2011, Jean-Louis Ayme noted that the orthic triangle of ABC is perspective to the medial triangle of the extangents triangle of ABC. Peter Moses found coordinates for the perspector, X(3611).
X(3611) lies on these lines: 19,51 25,3197 40,185 42,1409 55,184 65,225 71,228 209,3198 511,3101 1899,2550
Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the reflection triangle of X(1) at X(3612). (Randy Hutson, July 31 2018)
X(3612) lies on these lines: 1,3 9,2174 21,90 78,993 80,1698 140,1837 186,1452 284,1723 376,1770 377,1125 442,3586 498,515 499,950 550,1836 631,1737 1006,1728 1047,3362 1449,2245 1788,3524 1905,3515 3474,3528
X(3612) = {X(1),X(3)}-harmonic conjugate of X(46)
Let A' be the point of intersection of the tangents to the nine-point circle at the points where the circle meets line BC, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(3613). Also, A'B'C' is the side-triangle of the tangential triangle of the medial triangle and the tangential triangle of the orthic triangle. (Randy Hutson, August 30, 2011.)
X(3613) is the pole of the Lemoine axis with respect to the nine-point circle. (Luis González, Hyacinthos #20253, October 5, 2011)
X(3613) is the perspector of the nine-point circle and lies on the hyperbola that passes through the points A, B, C, X(4), X(5). (Randy Hutson, December 30, 2012.)
Let LA be the radical axis of the nine-point circle and the circle having AN as diameter, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABC, and the perspector is X(3613). See Vijay Krishna, X(3613).pdf , (November 13, 2019)
X(3613) lies on these lines: 4,160 5,141 53,232 66,2548 157,3425 184,2980 311,325 1316,3447 1485,3148
X(3613) = isogonal conjugate of X(5012)
X(3613) = isotomic conjugate of X(1078)
X(3613) = polar conjugate of X(36794)
X(3613) = X(8053)-of-orthic-triangle if ABC is acute
Let A'B'C' be the Feuerbach triangle, L the line through A and X(5), and A'' = L∩B'C'; define B'' and C'' cyclically. Then the lines A'A'', B'B'', C'C'' concur in X(3614). X(3614) = {X(5),X(12)}-harmonic conjugate of X(11). Further, X(3614) is the trilinear pole (with respect to the Feuerbach triangle) of the perspectrix of ABC and the Feuerbach triangle. (Randy Hutson, August 30, 2011.)
See also X(3615) and X(5954).
Let A'B'C' be the Feuerbach triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC; define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(3614). (Randy Hutson, July 11, 2019)
X(3614) lies on these lines: 1,5 35,546 55,3091 56,3090 140,3585 381,498 442,1155 899,3136 1329,2476 1478,1656 1698,1836 2475,3035 2635,3142 3058,3085
Let A'B'C' be the Feuerbach triangle, L the line through A' and X(5), and A'' = L∩BC; define B'' and C'' cyclically. Then the lines AA'', BB'', CC'' concur in X(3615). (Randy Hutson, August 30, 2011.)
Let A'B'C' be the Feuerbach triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. (The lines AA", BB", CC" concur in X(1).) Let A* be the trilinear pole of line B"C", and define B*and C* cyclically. The lines AA*, BB*, CC* concur in X(3615). (Randy Hutson, July 11, 2019)
X(3615) lies on these lines: 1,564 2,582 5,49 11,60 12,59 21,36 29,1870 58,3582 86,1443 476,953 501,3583 655,2595 946,1325 2287,2323
X(3615) = isogonal conjugate of X(2594)
X(3615) = cevapoint of X(1) and X(5)
X(3615) = intersection of tangents at X(1) and X(5) to hyperbola passing through X(1), X(5) and the excenters
X(3615) = crosspoint of X(1) and X(5) wrt the excentral triangle
X(3615) = trilinear pole of line X(654)X(1021)
Homothetic Centers: X(3616)-X(3637)
Suppose that X is a triangle center, that M is the medial triangle, and that t is a real number. The t-dilation of M from X, denoted by H(X; M, t), is a triangle center. If X = x : y : z (trilinears), then
H(X; M, t) = bc[(1 + t)ax + (1 - t)by + (1 - t)cz] : ca[(1 - t)ax + (1 + t)by + (1 - t)cz] : ab[(1 - t)ax + (1 - t)by + (1 + t)cz],
with inverse given by
H-1(X; M, t) = bc[(1 - t)(by + cz) - 2ax] : ca[(1 - t)(cz + ax) - 2by : ab[(1 - t)(ax + by) - 2cz].
César E. Lozada contributed several such triangle centers (March 21, 2011), as summarized here:
| X | t=1/2 | t=2 | t=-1/2 | t=-2 |
|---|---|---|---|---|
| X(1) | X(3616) | X(145) | X(667) | X(3617) |
| X(3) | X(631) | X(20) | X(3090) | X(3091) |
| X(4) | X(3091) | X(3146) | X(3523) | X(3522) |
| X(5) | X(1656) | X(4) | X(3526) | X(631) |
| X(6) | X(3618) | X(193) | X(3619) | X(3620) |
| X(8) | X(3617) | X(3621) | X(3622) | X(3623) |
| X(10) | X(1698) | X(8) | X(3624) | X(3616) |
| X | t=1/2 | t=2 | t=-1/2 | t=-2 |
|---|---|---|---|---|
| X(1) | X(3244) | X(1125) | X(3625) | X(3626) |
| X(3) | X(550) | X(140) | X(3627) | X(546) |
| X(4) | X(382) | X(5) | X(1657) | X(550) |
| X(5) | X(546) | X(3628) | X(548) | X(3530) |
| X(6) | X(3630) | X(3589) | X(3631) | X(3632) |
| X(8) | X(3633) | X(10) | X(3634) | X(3244) |
| X(10) | X(3626) | X(3635) | X(3636) | X(3637) |
Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define OB and OC cyclically. Let LA be the external tangent to circles OB and OC that is the nearer of the two to OA. Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Then A'B'C' is homothetic to ABC, and the center of homothety is X(3616). See the reference at X(1001).
See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.
X(3616) lies on these lines: 1,2 3,962 4,1385 5,944 7,21 9,1475 11,2476 12,1388 20,946 29,278 37,2272 40,3306 55,404 63,3333 69,1386 85,3160 100,474 104,3560 105,1036 106,835 140,1482 142,390 149,214 226,452 238,1468 244,986 320,3246 329,405 330,1655 346,3247 354,960 355,3090 377,497 388,1319 391,1449 392,942 406,1870 442,496 443,1058 459,1829 515,3091 516,3522 517,631 940,1191 941,2277 952,1656 958,3304 966,1100 968,988 982,2292 1000,1392 1056,3436 1220,2899 1320,3035 1376,3303 1479,2475 1699,3146 1788,2099 1962,3210 3218,3338 3242,3589 3524,3579
X(3616) = isogonal conjugate of X(2334)
X(3616) = {X(1),X(2)}-harmonic conjugate of X(8)
X(3616) = {X(1),X(10)}-harmonic conjugate of X(145)
X(3616) = {X(2),X(145)}-harmonic conjugate of X(10)
X(3616) = X(10)-of-cross-triangle-of-Aquila-and-anti-Aquila-triangles
X(3617) lies on these lines: 1,2 20,355 40,3146 44,391 45,346 63,1706 88,1219 100,958 144,1654 149,1145 193,3416 321,341 377,3421 390,1837 404,956 442,1159 452,3419 496,1000 515,3522 517,3091 631,952 944,3523 984,1278 1376,2975 1482,3090 1483,3526 1697,3305 1788,3600 2551,3434
X(3617) = isotomic conjugate of X(30712)
X(3617) = insimilicenter of Spieker circle and AC-incircle
X(3617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,8,145), (8,10,2)
X(3618) lies on these lines: 1,344 2,6 4,83 8,1386 20,5085 140,1351 159,1995 239,2345 264,1249 297,3087 316,2030 348,1445 388,1428 393,458 459,1843 487,3311 488,3312 497,2330 511,631 575,1352 576,3525 611,3086 613,3085 625,1692 1350,3523 1503,3091 1656,3564 2297,2999 3060,3313 3098,3524
X(3618) = reflection of X(69) in X(3620)
X(3618) = antipode of X(69) in conic {{X(13),X(14),X(15),X(16),X(69)}}
X(3618) = trilinear pole of line X(3800)X(3804) (the polar wrt the polar circle of X(8801))
X(3618) = pole wrt polar circle of trilinear polar of X(8801) (line X(826)X(2501))
X(3618) = polar conjugate of X(8801)
X(3618) = {X(2),X(6)}-harmonic conjugate of X(69)
X(3618) = {X(2),X(69)}-harmonic conjugate of X(3619)
X(3618) = antipode of X(69) in conic {{X(13),X(14),X(15),X(16),X(69)}}
X(3619) lies on these lines: 2,6 4,3096 182,3525 511,3090 631,1352 1350,3091 1503,3523 3526,3564
X(3619) = {X(2),X(69)}-harmonic conjugate of X(3618)
X(3619) = {X(6),X(141)}-harmonic conjugate of X(3620)
X(3620) lies on these lines: 2,6 8,1738 20,1352 66,3410 76,2996 145,3416 253,3164 320,2345 340,3087 511,3091 631,3564 1350,3146 1351,3090 1353,3526 1503,3522 1843,2979
X(3620) = midpoint of X(69) and X(3618)
X(3620) = center of conic {{X(13),X(14),X(15),X(16),X(69)}}
X(3620) = {X(6),X(141)}-harmonic conjugate of X(3619)
X(3621) lies on these lines: 1,2 20,952 44,346 45,391 63,2136 89,1219 149,3436 377,1159 517,3146 518,1278 631,1483 944,3522 1482,3091 1697,3219
X(3621) = isotomic conjugate of X(36606)
X(3622) lies on these lines: 1,2 7,1420 20,1385 21,999 56,1621 79,2320 81,1191 86,3445 100,3303 144,1001 149,377 193,1386 320,1279 388,1388 390,2646 391,1100 404,3295 452,3487 4496,2476 497,2475 517,3523 631,1482 940,1616 944,3091 946,3146 952,3090 962,3522 1056,2478 1219,1255 1319,3485 1483,1656 1697,3306 2325,3247 3218,3333
X(3622) = {X(1),X(2)}-harmonic conjugate of X(145)
X(3623) lies on these lines: 1,2 4,1483 20,1483 20,1482 79,1392 100,3304 144,3243 149,388 193,3242 346,1100 390,2098 517,3522 944,3146 952,3091 1056,2475 1120,2334 1320,3296 1449,2325 1697,3218 2099,3600 2136,3306 2975,3303
Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically. A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. A'B'C' is homothetic to the 3rd Euler triangle at X(3624). (Randy Hutson, July 31 2018)
X(3624) lies on these lines: 1,2 3,1699 5,3576 9,583 11,3601 12,1420 34,451 35,474 36,405 40,140 57,191 58,748 63,3337 90,3255 165,631 226,3361 377,3583 442,3586 443,1479 515,3090 516,3523 517,3526 595,750 940,1203 1213,1449 1385,1656 1706,3035 2478,3585 3306,3336 3339,3485
X(3624) = complement of X(9780)
X(3624) = {X(1),X(2)}-harmonic conjugate of X(1698)
X(3624) = homothetic center of ABC and cross-triangle of Aquila and anti-Aquila triangles
X(3625) lies on these lines: 1,2 44,2321 72,2802 515,1657 536,1358 548,952
X(3625) = {X(1),X(8)}-harmonic conjugate of X(3626)
X(3625) = {X(8),X(10)}-harmonic conjugate of X(4669)
Let OA be the circle centered at the A-vertex of the 6th mixtilinear triangle and passing through A; define OB, OC cyclically. X(3626) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(3626) lies on these lines: 1,2 40,3529 44,594 45,2321 106,1339 355,382 515,550 517,546 952,3530 960,2802
X(3626) = midpoint of X(8) and X(10)
X(3626) = {X(1),X(2)}-harmonic conjugate of X(15808)
X(3626) = {X(1),X(8)}-harmonic conjugate of X(3625)
X(3626) = {X(1),X(10)}-harmonic conjugate of X(3634)
As a point on the Euler line, X(3627) has Shinagawa coefficients (1, -7).
Let ABC be a triangle and Γa a variable circle passing through A and the midpoint, Ma, of side BC. Let Ab and Ac be the points, other than A and Ma, in which the circle Γa meets AC and AB, respectively. Let A' be the vertex of parallelogram with vertices A, Ab, A', Ac. When Γa varies, the locus of A' is a line ℓa. Define ℓb and ℓc cyclically. The triangle formed by the lines ℓa, ℓb, ℓc is orthologic to ABC with orthology center X(3627). The reciprocal orthologic center of these triangles is X(3). See X(3627). (Angel Montesdeoca, July 13, 2022)
X(3627) lies on these lines: 2,3 53,3284 143,185 156,1514 495,3585 496,3583 515,1483 576,1353 1173,3521 1478,3303 1479,3304 1484,2829 1493,2883 1533,3574
X(3627) = circumcircle-inverse of X(37955)
X(3627) = {X(3),X(4)}-harmonic conjugate of X(547)
X(3627) = X(1482)-of-orthic-triangle if ABC is acute
X(3627) = {X(381),X(382)}-harmonic conjugate of X(5073)
X(3627) = X(20)-of-Ehrmann-mid-triangle
X(3627) = Johnson-to-Ehrmann-mid similarity image of X(382)
As a point on the Euler line, X(3628) has Shinagawa coefficients (5, 1).
X(3628) is the centroid of the set {A', B', C', X(5)}, where A'B'C' is the medial triangle; more generally, H-1(X; M, 2) is the centroid of the set {A', B', C', X}. (Angel Montesdeoca, 12/20/2011)
For another construction see Antreas Hatzipolakis and Peter Moses, Euclid 2576 .
X(3628) lies on these lines: 2,3 17,395 18,396 32,3054 39,3055 52,373 141,576 143,1216 156,182 230,1506 233,3284 485,3594 486,3592 495,499 496,498 575,3564 623,630 624,629 952,1125 1209,1493
X(3628) = midpoint of X(5) and X(140)
X(3628) = isogonal conjugate of X(34567)
X(3628) = isotomic conjugate of isogonal conjugate of X(34565)
X(3628) = complement of X(140)
X(3628) = centroid of {A,B,C,X(5)}
X(3628) = circumcircle-inverse of X(37956)
X(3628) = center of the Vu pedal-centroidal circle of X(140)
X(3628) = Kosnita(X(5),X(2)) point
X(3628) = center of conic which is locus of centers of circumconics passing through X(5)
X(3628) = center of bicevian conic of X(2) and X(5)
X(3628) = {X(2),X(5)}-harmonic conjugate of X(140)
X(3628) = {X(3),X(5)}-harmonic conjugate of X(546)
X(3628) = homothetic center of X(2)-altimedial and X(140)-anti-altimedial triangles
X(3628) = inverse-in-orthocentroidal-circle of X(3526)
X(3628) = {X(2),X(4)}-harmonic conjugate of X(3526)
X(3629) lies on these lines: 2,6 53,648 182,3530 317,1990 382,1351 397,621 398,622 487,3594 488,3592 511,550 518,3244 542,1539 546,576 1112, 1843 1350,3528
X(3630) lies on these lines: 2,6 53, 340 548,3098 1205,2854 1503,1657
X(3630) = anticomplement of X(32455)
X(3631) lies on these lines: 2,6 319,1086 320,594 338,1232 382,1352 511,546 545,2321 550,1503 1350,3529 3530,3564
X(3631) = midpoint of X(69) and X(141)
X(3632) lies on these lines: {1,2}, {3,5288}, {4,4900}, {5,16200}, {6,4007}, {9,3943}, {35,956}, {36,5687}, {37,4034}, {40,550}, {44,4873}, {45,4727}, {46,6762}, {55,5258}, {56,17573}, {57,10944}, {58,4720}, {63,11010}, {65,3894}, {69,1266}, {72,3586}, {75,4888}, {80,3680}, {90,12641}, {100,7280}
X(3632) = {X(8),X(10)}-harmonic conjugate of X(4668)
X(3632) = homothetic center of outer Garcia triangle and mid-triangle of medial and anticomplementary triangles
X(3633) lies on these lines: 1,2 40,548 46,2136 191,1697 210,1357 517,1657 1317,1420 1482,1699 1483,3576 1743,2325 2093,3189 3339,3476
X(3634) lies on these lines: 1,2 5,516 37,1574 40,3090 44,1213 46,3305 88,1224 140,515 165,3091 355,3526 442,1155 451,1861 474,993 632,1385 750,1724 944,3533 946,1656 1739,2292 3039,3161 3219,3336 3525,3576
X(3634) = {X(1),X(10)}-harmonic conjugate of X(3626)
X(3634) = {X(2),X(10)}-harmonic conjugate of X(1125)
X(3634) = centroid of ABCX(10)
X(3634) = complement of X(1125)
X(3634) = Kosnita(X(10),X(2)) point
X(3634) = homothetic center of medial triangle and midpoint triangle of X(10)
X(3634) = center of conic which is locus of centers of circumconics passing through X(10)
X(3634) = center of bicevian conic of X(2) and X(10)
X(3634) = {X(1),X(2)}-harmonic conjugate of X(19862)
X(3634) = {X(8),X(10)}-harmonic conjugate of X(4745)
Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(3635) and X(15519) are equal.
X(3635) lies on these lines: {1, 2}, {6, 4029}, {20, 11224}, {21, 13602}, {37, 4700}, {44, 4982}, {65, 3892}, {72, 3898}, {75, 4909}, {86, 4464}, {149, 5270}, {214, 13996}, {355, 5072}, {392, 4005}, {405, 8162}, {726, 4718}, {2796, 4409}, {2802, 9945}, {3555, 3962}, {3630, 5847}, {4072, 4898}
X(3635) = midpoint of X(i) and X(j) for these {i,j}: {1, 3244}, {10, 145}, {944, 4301}, {1482, 5882}, {2650, 4065}, {3057, 3874}, {3555, 3878}, {3625, 3633}, {3924, 5262}, {4084, 5697}, {4297, 7982}, {4314, 12559}, {5493, 11531}, {10912, 12437}, {12675, 13600}
X(3635) = reflection of X(i) in X(j) for these (i,j): (8, 3634), (10, 3636), (1125, 1), (3625, 4691), (3626, 1125), (3632, 4746), (3754, 5045), (4665, 4758), (4701, 10), (4745, 551), (6684, 15178), (7317, 15223), (13753, 5570)
X(3635) = complement of X(3625)
X(3635) = anticomplement of X(4691)
X(3635) = {X(1),X(10)}-harmonic conjugate of X(3636)
X(3636) lies on these lines: 1,2 86,1266 226,1388 382,946 515,546 516,550 517,3530 993,3304 3528,3576
X(3636) = {X(1),X(2)}-harmonic conjugate of X(3244)
X(3636) = {X(1),X(10)}-harmonic conjugate of X(3635)
X(3636) = midpoint of X(1) and X(1125) (the anticomplement and complement of X(10))
X(3637) = isogonal conjugate of X(3355)
X(3637) lies on the Darboux cubic and these lines: 4,3356 20,3355 1490,3472 1498,2130 3182,3353
In the plane of triangle ABC, let X,Y,Z be the points of contact of the incircle with sidelines BC,CA,AB, and let X',Y',Z' be the harmonic conjugates of X,Y,Z with respect to {B,C}, {C,A}, {A,B}, respectively. It is well known that AX, BY, CZ concur in the Gergonne point, X(7), and that X',Y',Z' lie on the Gergonne line. The circles with centers X',Y',Z' passing respectively through X,Y,Z are orthogonal to the incircle and to the Gergonne line, forming a coaxal system with the Gergonne line as line of centers. The radical axis is the Soddy line, which passes through X(1) and X(7) and is the line of centers of the orthogonal coaxal system, which includes the Soddy circles. Among the points on this line are the two limiting points (point-circles of the system): X(3638) and X(3639). The former lies between X(7) and the inner Soddy point, X(175). (Richard Guy, September 2, 2011)
Coordinates and properties for the points P=X(3638) and Q=X(3639) were found by Peter Moses (September 3, 2011):
midpoint(P,Q) = X(1323)
crossdifference(P,Q) = X(657)
crosssum(P,Q) = X(36)
cevapoint(P,Q) = X(80)
idealpoint(P,Q) = X(516)
Q = {X(1),X(7)}-harmonic conjugate of P
Q = inverse-in-incircle of P
Write I = X(1), U = X(7), r = inradius, R = circumradius, and d = 4*sqrt(3)*area(ABC). Then distance ratios are given by |IP|/|PU| = 2r(r + 4R)/d and |IQ|/|QU| = -2r(r + 4R)/d
X(3638) lies on these lines: 1,7 14,226 57,1277 553,1082
X(3638) = X(15)-of-intouch-triangle
X(3638) = isogonal conjugate of X(36737)
X(3638) = crossdifference of every pair of points on line X(657)X(7127)
See X(3638) for a description of the inner and outer points.
X(3639) lies on these lines: 1,7 13,226 57,1276 553,559
X(3639) = isogonal conjugate of X(36738)
X(3639) = X(16)-of-intouch-triangle
Seiichi Kirikami (November 1, 2010), defined a point P having actual trilinear distances (p,q,r) by the equations a + p = b + q = c + r and a point P' with distances (p',q',r') satisfying p - a = q - b = r - c. Here, P = X(3640) and P' = X(3641). Kirikami also discovered a bicentric pair of points satisfying p + b = q + c = r + a and another bicentric pair satisfying p - b = q - c = r - a. The six point are given by homogeneous trilinears t(a,b,c) : t(b,c,a) : t(c,a,b) as follows:
X(3640): t(a,b,c) = b2 + c2 - ab - ac +
2*area(ABC)
X(3641): t(a,b,c) = b2 + c2 - ab - ac -
2*area(ABC)
K1: t(a,b,c) = ac - b2 + 2*area(ABC)
K2: t(a,b,c) = ab - c2 + 2*area(ABC)
K3: t(a,b,c) = ac - b2 + 2*area(ABC)
K4: t(a,b,c) = ab -cb2 + 2*area(ABC)
Peter Moses (November 1, 2010) found that the six points lie on an ellipse, E, here called the Kirikami-Moses ellipse. The center of E is the incenter of ABC, and an equation for E is as follows:
e(a,b,c,x,y,z) + e(b,c,a,y,z,x) + e(c,a,b,z,x,y) = 0, where
e(a,b,c,x,y,z) = h(a,b,c)x2 - j(a,b,c)yz, where
h(a,b,c) = (5a4 + b4 + c4 - 4a3b - 4a3c - 4a2bc + 2b2c2 + 2c2a2 + 2a2b2)
and
j(a,b,c) = a4 + b4 + c4 - 8b3c - 8bc3 + 6b2c2 - 2a2b2 - 2a2c2 + 8abc(b+c-a).
Moses found six more points on E:
M1 = (b - c) (a + b + c) + 2*sqrt(3)*area(ABC) : :
M2 = (b - c) (a + b + c) - 2*sqrt(3)*area(ABC) : :
M3 = 2ab + ac - b2 - 2c2 +
2*sqrt(3)*area(ABC) : :
M4 = 2ac + ab - c2 - 2b2 +
2*sqrt(3)*area(ABC) : :
M5 = 2ab + ac - b2 - 2c2 -
2*sqrt(3)*area(ABC) : :
M6 = 2ac + ab - c2 - 2b2 -
2*sqrt(3)*area(ABC) : :
'
These six points comprise three bicentric pairs: {M1, M2}, {M3, M4}, {M5, M6}.
Further properties found by Moses:
(1) The incenter is the midpoint of the segments X(3640)X(3641), K1K2, and K3K4.
(2) X(3640) and X(3641) lie on the line X(1)X(6).
(3) The lines K1K4 and K2K3 are parallel to the line X(1)X(6).
(4) The lines X(3640)K4, K1K3, and X(3641)K2 are parallel; indeed, the reflection of the first in the second is the third.\
(5) The lines X(3640)K3, K2K3, and X(3641)K1 are parallel, and the reflection of the first in the second is the third.
(6) The lines tangent to E at X(3640) and X(3641) are perpendicular to the line X(1)X(3), as are the lines K1K2 and K3K4.
(7) The following four lines are parallel: the tangents to E at K2 and K4, and X(3640)K1 and X(3641)K3.
(8) The following four lines are parallel: the tangents to E at K1 and K3, and X(3640)K2 and X(3641)K4.
X(3640) lies on these lines: {1, 6}, {3, 11498}, {8, 175}, {10, 5590}, {30, 16131}, {40, 11825}, {55, 8199}, {65, 10976}, {142, 30342}, {144, 30334}, {145, 30333}, {200, 13388}, {355, 6214}, {481, 2550}, {515, 5870}, {517, 1160}, {519, 5860}, {537, 24832}, {730, 6272}, {944, 10784}, {946, 6201}, {952, 5874}, {976, 36567}, {982, 8945}, {1374, 5880}, {1385, 26348}, {1482, 11917}, {1703, 8416}, {1829, 11389}, {1837, 10926}, {2800, 12754}, {2802, 13270}, {2809, 6213}, {3057, 10928}, {3083, 3873}, {3084, 3681}, {3244, 26340}, {3576, 45553}, {3579, 35247}, {4847, 13390}, {5252, 10924}, {5393, 25568}, {5405, 24477}, {5587, 10515}, {5594, 9798}, {5657, 10518}, {5689, 10513}, {5698, 31567}, {5881, 6278}, {6001, 6257}, {6203, 30320}, {8218, 12440}, {8219, 12441}, {8975, 8983}, {9941, 9995}, {10793, 12194}, {11902, 12438}, {12697, 40268}, {13950, 13971}, {16232, 39959}, {18480, 18511}, {18525, 26346}, {26338, 45717}, {26344, 45711}, {26345, 45712}, {26347, 45718}, {35641, 35794}, {35642, 35793}, {36480, 36536}, {45551, 45716}
X(3640) = reflection of X(i) in X(j) for these {i,j}: {1, 45713}, {3641, 1}, {12698, 1160}, {45720, 3244}
X(3640) = X(1)-of-outer-Grebe-triangle
X(3640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 11371}, {1, 3751, 18991}, {1, 5223, 30557}, {1, 5588, 6}, {1, 5589, 7969}, {1, 18992, 11370}, {1, 19003, 1386}, {1, 19004, 45398}, {1, 45426, 18992}, {6, 5604, 1}, {6, 10930, 10041}, {6, 10932, 10049}, {8, 1270, 5688}, {3242, 7968, 1}, {4663, 45398, 19004}, {5588, 5604, 11371}, {5605, 44636, 1}, {6762, 7174, 3641}, {7969, 45476, 1}, {8199, 8206, 55}, {10920, 10922, 6214}, {11498, 22757, 3}, {12594, 19049, 10040}, {12595, 19047, 10048}, {26349, 26350, 6}, {45422, 45729, 3301}, {45424, 45728, 3299}
See X(3640) for a discussion of X(3641).
X(3641) lies on these lines: {{1, 6}, {3, 11497}, {8, 176}, {10, 5591}, {30, 16130}, {40, 11824}, {55, 8198}, {65, 10975}, {142, 30341}, {144, 30333}, {145, 30334}, {200, 13389}, {355, 6215}, {482, 2550}, {515, 5871}, {517, 1161}, {519, 5861}, {537, 24831}, {730, 6273}, {944, 10783}, {946, 6202}, {952, 5875}, {976, 36566}, {982, 8941}, {1373, 5880}, {1385, 26341}, {1482, 11916}, {1659, 4847}, {1702, 8396}, {1829, 11388}, {1837, 10925}, {2362, 39959}, {2800, 12753}, {2802, 13269}, {2809, 6212}, {3057, 10927}, {3083, 3681}, {3084, 3873}, {3244, 26339}, {3576, 45552}, {3579, 35246}, {5252, 10923}, {5393, 24477}, {5405, 25568}, {5587, 10514}, {5595, 9798}, {5657, 10517}, {5688, 10513}, {5698, 31568}, {5881, 6281}, {6001, 6258}, {6204, 30319}, {8216, 12440}, {8217, 12441}, {8974, 8983}, {9615, 45531}, {9941, 9994}, {10792, 12194}, {11901, 12438}, {12698, 40268}, {13949, 13971}, {18480, 18509}, {18525, 26336}, {26334, 45711}, {26335, 45712}, {26337, 45718}, {35641, 35792}, {35642, 35795}, {36480, 36535}, {45550, 45715}, {45594, 45717}
X(3641) = reflection of X(i) in X(j) for these {i,j}: {1, 45714}, {3640, 1}, {12697, 1161}, {45719, 3244}
X(3641) = X(1)-of-inner-Grebe-triangle
X(3641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 11370}, {1, 3751, 18992}, {1, 5223, 30556}, {1, 5588, 7968}, {1, 5589, 6}, {1, 18991, 11371}, {1, 19003, 45399}, {1, 19004, 1386}, {1, 45427, 18991}, {6, 5605, 1}, {6, 10929, 10040}, {6, 10931, 10048}, {8, 1271, 5689}, {3242, 7969, 1}, {4663, 45399, 19003}, {5589, 5605, 11370}, {5604, 44635, 1}, {6762, 7174, 3640}, {7968, 45477, 1}, {8198, 8205, 55}, {10919, 10921, 6215}, {11497, 22756, 3}, {12594, 19050, 10041}, {12595, 19048, 10049}, {26342, 26343, 6}, {45423, 45729, 3299}, {45425, 45728, 3301}
X(3642) and X(3644) lie on the line through X(141) parallel to the Euler line.
X(3642) lies on these lines: 2,14 3,618 4,636 13,76 16,298 17,628 30,141 32,395 62,633 69,532 381,624 530,599 616,2896 629,631 630,1656 2926,3131
X(3642) = 1st-Brocard-isogonal conjugate of X(22687)
X(3643) lies on these lines: 2,13 3,619 4,635 6,532 14,76 15,299 18,627 30,141 32,396 61,634 69,533 381,623 531,599 617,2896 629,1656 630,631 2925,3132
X(3643) = 1st-Brocard-isogonal conjugate of X(22689)
For any point P, let L(A) be the line through P parallel to BC. Let U = AB∩L(A) and V = AC∩L(A), and let a' = |UV|. Define b' and c' cyclically. Let L be the line consisting of points P such that a + ta' = b + tb' = c + tc'. Then X(3644) is given by t=1. The points corresponding to t = -1, -1/2, and 0, are X(75), X(37), and X(192), respectively. (Seiichi Kirikami, November 7, 2010)
X(3644) = reflection of X(i) in X(j) for these (i,j): (75,192), (1278,37)
X(3644) lies on these lines: 2,37 190,1743 726,3244
X(3644) = anticomplement of X(4686)
Denote the incenter and excenters by I, IA, IB, IC. Let JA be the incenter of triangle BCI, and define JB and JC cyclically. The lines IAJA, IBJB, ICJC concur in X(3645).
X(3645) lies on the Euler line of the BCI triangle. (Randy Hutson, January 29, 2018)
X(3645) lies on these lines: 1,168 40,483 258,1127
Denote the incenter and excenters by I, IA, IB, IC. Let KA be the centroid of triangle BCI, and define KB and KC cyclically. The lines IAKA, IBKB, ICKC concur in X(3646).
Of the 2 intersections of the Bevan circle and line BC, let Ab be the one closer to B, and define Bc and Ca cyclically. Let Ac be the one closer to C, and define Ba and Cb cyclically. Let Ab' = {B,C}-harmonic conjugate of Ab, and define Bc' and Ca' cyclically. Let Ac' = {B,C}-harmonic conjugate of Ac, and define Ba' and Cb' cyclically. The points Ab', Ac', Bc', Ba', Ca', Cb' lie on an ellipse centered at X(3646). (Randy Hutson, December 10, 2016)
X(3646) lies on these lines: 1,210 2,40 3,2951 9,1125 10,1058 11,1697 57,191 405,1490 748,1453 936,1001 978,1045
Kirikami-Schiffler Points: X(3647)-X(3652)
Suppose that X is a point and A'B'C' is a central triangle. Let LA be the line through A' parallel to the Euler line of triangle BCX, let LB be the line through B' parallel to the Euler line of CXA, and let LC be the line through C' parallel to the Euler line of AXB.
It is well known that if X=X(1), the incenter, then the three aforementioned Euler lines concur in the Schiffler point, X(21). If their parallels, the lines LA, LB, LC concur, the point of concurrence is the Kirikami-Schiffler point of the triangle A'B'C', denoted by KS(A'B'C'). Seiichi Kirikami (February 1, 2011) found that those lines concur if A'B'C' is the reference triangle ABC and also concur if A'B'C' is the medial triangle. Peter Moses found additional cases and properties. A summary follows:
| triangle A'B'C' | LA∩LB∩LC |
|---|---|
| ABC | X(79) |
| medial | X(3647) |
| excentral | X(191) |
| anticomplementary | X(3648) |
| intouch | X(3649) |
| extouch | X(3650) |
| Feuerbach | X(442) |
| Fuhrmann | X(191) |
| 1st circumperp | X(3651) |
| 2nd circumperp | X(21) |
| Carnot | X(3652) |
Suppose that A'B'C' is the cevian triangle of a point P. Then LA, LB, LC concur if and only if P lies on the cubic K455.
Suppose that A'B'C' is the anticevian triangle of a point P. Then LA, LB, LC concur if and only if P lies on the cubic given by the barycentric equation
a(2a + b + c)yz(y - z) + b(2b + c + a)zx(z - x) + c(2c + a + b)xy(x - y) = 0.
Suppose that A'B'C' is the pedal triangle of a point P on the line X(1)X(3). Then LA, LB, LC concur.
Suppose that A'B'C' is the antipedal triangle of a point P on the line X(1)X(4). Then LA, LB, LC concur.
Let (OA) be the reflection in A of the A-excircle, and define (OB) and (OC) cyclically. X(3647) is the radical center of (OA), (OB), (OC). (Randy Hutson, June 7, 2019)
X(3647) lies on these lines: 1,21 2,79 9,2173 10,30 35,3219 44,2092 45,2305 100,3065 124,128 214,960 442,1155 540,3178 553,1125
X(3647) = complement of X(79)
Let OA be the circle centered at the A-vertex of the 2nd Fuhrmann triangle and passing through A; define OB and OC cyclically. X(3648) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
Let OA be the circle centered at the A-vertex of the K798e triangle and passing through A; define OB and OC cyclically. X(3648) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(3648) lies on these lines: 2,79 7,21 8,30 10,191 40,153 63,2894 145,758 149,3065
X(3648) = anticomplement of X(79)
Two proofs of Dao's theorem about cyclic hexagons were published in 2014: Nikolaos Dergiades's Dao's Theorem on Six Circumcenters Associated with a Cyclic Hexagon and Telv Cohl's A Purely Synthetic Proof of Dao's Theorem on Six Circumcenters Associated with a Cyclic Hexagon. In the degenerate case that the hexagon is the intouch triangle A'B'C', the triangle of the circumcenters of AB'C', A'BC', A'B'C is perspective to A'B'C' with perspector X(3649). (Dao Thanh Oai, Francisco Javier Garcia Capitan, ADGEOM #1718, September 16, 2014)
Let A' be the midpoint of A and X(1), and define B' and C' cyclically. Triangle A'B'C' is perspective to the intouch triangle at X(3649). (Randy Hutson, June 7, 2019)
X(3649) lies on these lines: 1,30 7,21 10,12 11,113 55,3487 57,191 140,3336 145,388 354,946 396,2306 429,1835 553,1125 940,1406 962,3303 1056,2098 1086,1193 1100,1839 1317,1365 1358,2795 1360,1367 1411,2647 1749,3337 1834,2650 1852,1870 1901,2294 3318,3324
X(3649) = X(54)-of-intouch-triangle
X(3649) = X(3651)-of-Mandart-incircle-triangle
X(3650) lies on these lines: 8,30 9,46 21,999 553,1125 758,3057
As a point on the Euler line, X(3651) has Shinagawa coefficients (2$a$E + 2abc, - 3$a$E - 2abc).
For a construction of X(3651) as the orthocenter of a certain triangle, see: Antreas Hatzipolakis and César Lozada, Hyacinthos 29288.
X(3651) lies on these lines: {2,3}, {4,442}, {9,2173}, {35,79}, {36,950}, {40,758}, {55,3487}, {56,3488}, {72,74}, {78,3587}, {81,500}, {98,1292}, {104,4297}, {108,1294}, {165,191}, {201,3465}, {212,1745}, {329,3648}, {477,1290}, {515,5258}, {581,1754}, {601,1742}, {842,2691}, {944,3428}, {1030,1901}, {1058,1617}, {1260,3650}, {1330,1792}, {1444,2893}, {1612,3772}, {1768,2949}, {1936,4303}, {2635,3074}, {2693,2766}, {2975,3419}, {3072,4300}, {3652,5777}, {4333,5010}, {5204,5427}, {5217,5714}, {5584,5657}, {5752,5890}
X(3651) = midpoint of X(2) and X(2475)
X(3651) = reflection of X(i) in X(j) for these (i,j): (4,442), (21,3)
X(3651) = X(78)-gimel conjugate of X(3430)
X(3651) = reflection of X(21) in X(3)
X(3651) = X(195)-of-2nd-extouch-triangle
X(3651) = X(i)-zayin conjugate of X(j) for these (i,j): (9,71), (522,6003), (3465,6000)
Quim Castellsaguer defines the Carnot triangle as the triangle formed by the circumcenters of the triangles BCH, CAH, ABH, where H denotes the orthocenter. The Carnot triangle is also known as the Johnson triangle.
X(3652) lies on these lines: 5,79 12,1727 21,104 ,30,40 140,1768 500,846 758,1482 942,1776 952,3065 3219,3579
X(3653) lies on these lines: 1,549 2,355 3,551 30,1699 381,1125 517,3524 631,3241 946,3534 3058,3612
X(3653) = complement of X(38074)
X(3653) = anticomplement of X(38083)
X(3654) lies on these lines: 1,549 2,392 3,519 8,376 10,381 30,40 63,1145 165,952 495,2093 515,3534 542,3416 547,1698 551,1482 962,3545 1385,3241 1836,3245
X(3655) lies on these lines: 1,30 2,355 3,519 8,3524 40,1483 145,3579 376,517 381,515 549,952 1387,3586 1482,3534 1837,3582
X(3656) lies on these lines: 1,30 2,392 3,551 4,1392 8,3545 40,549 57,1387 119,3577 355,381 376,962 496,3340 516,3534 553,999 940,1480 944,3543 952,1699 1388,1770 3524,3579
Let A'B'C' be the medial triangle of the reference triangle ABC. Let P be the point of intersection of the lines X(1)X(3) and BC, and let P' be the point where the line through P perpendicular to line AX(3) meets that line. Let LA be the line PP', and define LB and LC cyclically. Let A''=LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(3657). (Jean-Louis Ayme, Hyacinthos #16676, August 21, 2008)
The Ayme triangle A''B''C'' is also perspective to these triangles: tangential, orthic, intangents, extangents, and the circumorthic. (Peter Moses, November 7, 2011)
X(3657) lies on these lines: 3,513 65,924 68,521 69,693 71,661 72,523 74,915 895,2990
X(3657) = isogonal conjugate of X(3658)
As a point on the Euler line, X(3658) has Shinagawa coefficients ($aSA3$(E+F) -$aSA2$[(E+F)2-2S2] +3$aSA$FS2-$a$(E+F)FS2, 2$aSBSC$S2 -2$aSA2$S2).
X(3658) lies on the Euler line.
X(3658) lies on these lines: 2,3 100,110 108,925 109,2617 476,1290 1292,1302
X(3658) = isogonal conjugate of X(3657)
Let I be the incenter and IA the A-excenter of the triangle ABC. Let LA be the line joining the circumcenter of triangle BCI and the incenter of triangle BCIA, and define LB and LC cyclically. The lines LA, LB, LC concur in X(3659). (Seiichi Kirikami, April 12, 2010)
Let Ea be the ellipse with B and C as foci and passing through X(1), and define Eb and Ec cyclically. Let La be the line tangent to Ea at X(1), and define Lb and Lc cyclically. Let A' = La∩BC, B' = Lb∩CA, C' = Lc∩AB. Then A', B', C' are collinear, and the line A'B'C' meets the line at infinity at the isogonal conjugate of X(3659). (Randy Hutson, April 9, 2016)
Let IAIBIC be the excentral triangle. The lines X(1)X(3) of triangles BCIA, CAIB, ABIC concur in X(3659). (Randy Hutson, June 7, 2019)
X(3659) lies on the circumcircle and these lines: 3,164 55,258 106,1130 759,1128
X(3659) = X(11)-of-excentral-triangle
X(3659) = X(119)-of-hexyl-triangle
X(3659) = center of hyperbola {{X(1),X(164),X(166),X(167),X(168),excenters}}
X(3659) = X(100) of 1st circumperp triangle
X(3659) = Λ(X(6728), X(10492))
X(3659) = isogonal conjugate of infinite point of antiorthic axis of intouch triangle (or excentral triangle)
Let CA be the circumcircle analog of the Conway circle; that is, the circle with center X(3) and radius (R2 + s2)1/2. Then X(3660) is the radical trace of CA and the Conway circle. (Randy Hutson, October 13, 2015)
If you have The Geometer's Sketchpad, you can view X(3660) as the radical trace of the incircle and circumcircle.
X(3660) lies on these lines: 1,3 11,971 105,2720 108,840 109,1279 222,614 244,1458 513,676 518,3035 1071,3086 1357,3322 1360,3025 1404,2246 1421,1456 1426,1878 1477,2222 1788,3555
X(3660) = isogonal conjugate of X(34894)
X(3660) = inverse-in-incircle of X(57)
X(3660) = X(468)-of-intouch-triangle
X(3660) = inverse-in-{circumcircle, nine-point circle}-inverter of X(56)
X(3660) = radical trace of the incircle and circumcircle; see X(3513)
X(3660) = radical trace of circumcircle and Moses-Longuet-Higgins circle
X(3660) = radical trace of incircle and Moses-Longuet-Higgins circle
X(3660) = inverse-in-circumcircle of X(1617)
X(3661) and X(3662) were considered by Seiichi Kirikami and Peter Moses in December, 2011, in connection with combos.
X(3661) lies on these lines: 1,2 6,319 7,3620 9,1654 63,2896 69,894 75,141 76,321 100,761 192,2321 226,3212 257,312 297,318 320,599 344,966 469,1829 1726,3219 2082,3305
X(3661) = isotomic conjugate of X(14621)
X(3661) = complement of X(4393)
X(3661) = anticomplement of X(17023)
X(3661) = {X(75),X(141)}-harmonic conjugate of X(3662)
X(3661) = {X(594),X(17228)}-harmonic conjugate of X(3662)
X(3661) = {X(1086),X(17227)}-harmonic conjugate of X(3662)
X(3661) = {X(17389),X(17397)}-harmonic conjugate of X(1)
X(3662) = 3(a2 + b2 + c2)*X(2) +
(a2 + b2 + c2 -2bc - 2ca -
2ab)X(7)
X(3662) = (a2 + b2 + c2 - bc - ca -
ab)X(7) + (a2 + b2 + c2)*X(9
X(3662) = (bc + ca + ab)*X(75) - 2(a2 + b2 +
c2)X(141)
X(3662) lies on these lines: 1,2896 2,7 6,320 8,1738 69,239 75,141 85,257 86,1333 273,297 306,3210 319,599 614,1716 982,2887 1266,1278 2345,3619
X(3662) = anticomplement of X(17353)
X(3662) = isotomic conjugate of X(17743)
X(3662) = complement of X(17350)
X(3662) = {X(75),X(141)}-harmonic conjugate of X(3661)
X(3662) = {X(594),X(17228)}-harmonic conjugate of X(3661)
X(3662) = {X(1086),X(17227)}-harmonic conjugate of X(3661)
Combos Using Central Triangles
The discussion of combos near the beginning of ETC is continued here. Suppose that T is a central triangle, and let nT its normalization, so that the triangle nT is essentially a 3x3 matrix with row sums equal to 1, and the rows of nT are normalized barycentrics for the A-, B-, C- vertices of T.
Let X be a triangle center, given by barycentrics x : y : z, not necessarily normalized. The point whose rows are the matrix product X*(nT) is then a triangle center, denoted by Xcom(T).
Among central triangles T are cevian and anticevian triangles and others described at MathWorld. A brief list follows, with A-vertices given in barycentrics (not normalized):
Intouch triangle = cevian triangle of X(7)
A-vertex = 0 : 1/(c + a - b) : 1/(a + b - c)
Extouch triangle = cevian triangle of X(8)
A-vertex = 0 : c + a - b : a + b - c
Incentral triangle = cevian triangle of X(1)
A-vertex = 0 : b : c
Excentral triangle = anticevian triangle of X(1)
A-vertex = -a : b : c
Hexyl triangle
A-vertex = a(1 + a1 + b1 + c1) : b(-1
+ a1 + b1 - c1) : c(1 + a1
- b1 + c1), where a1 = cos A,
b1 = cos B, c1 = cos C
Tangential triangle = anticevian triangle of X(6)
A-vertex = -a2 : b2 : c2
1st Brocard triangle
A-vertex = a2 : c2 : b2
Let A'B'C' be the inverse-in-excircles triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3663). (Randy Hutson, July 20, 2016)
X(3663) lies on these lines: 1,7 2,2415 6,527 9,3008 10,75 37,142 57,1766 63,1723 69,519 86,99 141,536 144,1743 165,3598 226,1465 256,2481 273,1785 307,1210 319,3625 320,3244 329,2999 545,3589 553,940 572,1429 573,1423 988,1125 1074,1111 1122,3057 1439,2823
X(3663) = isotomic conjugate of X(1222)
X(3663) = anticomplement of X(17355)
X(3663) = trilinear product of vertices of inverse-in-excircles triangle
X(3663) = perspector of ABC and cross-triangle of ABC and inverse-in-excircles triangle
X(3663) = complement of X(3729)
X(3663) = complement, wrt intouch triangle, of X(12723)
X(3664) lies on these lines: 1,7 2,1743 6,142 10,69 11,1366 36,1014 37,527 57,573 58,86 75,519 222,226 255,307 319,3626 354,1122 511,942 948,1419 1086,1100 1099,1111 1266,3636 1394,3485 1396,1848 1565,2792
X(3664) = complement of X(4416)
X(3665) lies on these lines: 1,1565 5,1111 7,21 12,85 57,1759 65,760 226,241 269,1038 279,388 1086,2275 1355,1367 1441,3264 3160,3476
Let A' be the trilinear pole of the tangent to the Apollonius circle where it meets the A-excircle, and define B' and C' cyclically. Triangle A'B'C' is homothetic to the medial triangle at X(3666). (Randy Hutson, January 15, 2019)
X(3666) lies on these lines: 1,3 2,37 6,63 7,941 9,2999, 11,114 21,1104 27,1841 31,1386 38,42 39,712 43,210 44,3219 45,3305 72,386 77,1407 81,593 88,1255 141,306 191,1203 226,1465 227,388 238,846 239,257 240,1859 244,1962 392,995 474,975 581,1071 612,1376 614,968 650,824 726,1215 756,899 896,2308 918,3310 960,1193 1015,2482 1036,3556 1054,1961 1150,3187 1211,2092 1279,1621 1355,1364 1396,1870 1762,2264 1817,2303 2236,2309
X(3666) = isogonal conjugate of X(2298)
X(3666) = isotomic conjugate of X(30710)
X(3666) = complement of X(321)
X(3666) = complementary conjugate of X(21245)
X(3666) = polar conjugate of isogonal conjugate of X(22345)
X(3666) = X(1215)com(Incentral) = X(1064)com[n(Inverse Hexyl)]
X(3666) = {X(1),X(3)}-harmonic conjugate of X(37539)
X(3667) lies on these lines: 4,2457 30,511 74,2758 98,2712 99,2705 100,2743 101,2737 102,2757 103,2726 104,2718 109,2731 110,2692 145,2403 572,1919 649,3239 885,3062 1027,1721 1292,2748 1294,2755 1295,2756 1296,2759 1297,2760 1519,1769 1768,2957 2394,3429 2487,2496
X(3667) = isogonal conjugate of X(1293)
X(3667) = isotomic conjugate of isogonal conjugate of X(8643)
X(3667) = excentral-isogonal conjugate of X(1054)
X(3667) = Thomson-isogonal conjugate of X(106)
X(3667) = Lucas-isogonal conjugate of X(106)
Let A'B'C' be the 3rd extouch triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(3668). (Randy Hutson, September 14, 2016)
Let A13B13C13 be Gemini triangle 13. Let A' be the center of conic {{A,B,C,B13,C13}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(3668). (Randy Hutson, January 15, 2019)
X(3668) lies on these lines: {1, 7}, {9, 948}, {10, 307}, {19, 57}, {37, 226}, {56, 2218}, {65, 1439}, {75, 1088}, {142, 241}, {158, 273}, {219, 527}, {222, 2219}, {342, 1785}, {387, 3339}, {534, 553}, {658, 897}, {759, 934}, {1020, 1400}, {1086, 1108}, {1122, 1358}, {1365, 2652}, {1367, 3324}, {1434, 2363}, {1445, 1723}, {1461, 1910}, {1486, 1617}, {2003, 2982}, {2217, 2385}
X(3668) = isogonal conjugate of X(2328)
X(3668) = isotomic conjugate of X(1043)
X(3668) = pole wrt polar circle of trilinear polar of X(2322)
X(3668) = X(48)-isoconjugate (polar conjugate) of X(2322)
X(3669) lies on these lines: {1, 3309}, {34, 2424}, {56, 667}, {57, 1022}, {65, 876}, {109, 1308}, {241, 514}, {278, 2401}, {513, 663}, {651, 3257}, {919, 934}, {1015, 1358}, {1019, 1429}, {1407, 2423}, {2526, 2530}, {4394, 4498}
X(3669) = isogonal conjugate of X(644)
X(3669) = isotomic conjugate of X(646)
X(3669) = complement of X(4462)
X(3669) = pole wrt incircle of line X(7)X(8)
X(3669) = trilinear pole of line X(244)X(1357)
X(3670) lies on these lines: {1, 3}, {6, 1759}, {10, 38}, {58, 3218}, {63, 1724}, {72, 3216}, {79, 256}, {81, 849}, {191, 238}, {226, 1393}, {240, 1838}, {244, 1125}, {307, 1210}, {442, 1086}, {518, 3293}, {726, 1089}, {758, 1193}, {975, 3306}, {984, 1698}, {1046, 1203}, {1100, 3285}, {1107, 3125}, {3290, 3294}
Let O* be a circle with center X(2) and variable radius R*. Let LA be the radical axis of O* and the A-excircle, and define LB and LC cyclically. Let A'=LB∩LC, and define B' and C' cyclically. There is a unique value of R* for which triangle A'B'C' is perspective to ABC, and the perspector is X(3671). (Randy Hutson, January 15, 2019)
X(3671) lies on these lines: {1, 7}, {2, 3339}, {10, 12}, {40, 3487}, {56, 551}, {57, 1125}, {142, 960}, {355, 1159}, {388, 519}, {496, 942}, {527, 958}, {938, 1699}, {950, 1836}, {1400, 3294}, {1697, 3475}, {1727, 3338}, {2093, 3085}, {2099, 3244}, {3361, 3616}, {3474, 3601}
X(3671) = complement of X(12526)
X(3671) = anticomplement of X(18249)
X(3672) lies on these lines: {1, 7}, {2, 37}, {6, 144}, {55, 3598}, {63, 2257}, {69, 145}, {81, 2255}, {86, 3445}, {142, 3247}, {190, 3618}, {239, 391}, {307, 938}, {319, 3621}, {320, 3623}, {527, 1449}, {941, 2481}, {999, 1014}, {1266, 3616}, {1423, 2269}, {1429, 2268}
X(3672) = isotomic conjugate of X(1219)
X(3672) = complement of X(4461)
X(3672) = anticomplement of X(2345)
X(3673) lies on these lines: {1, 85}, {2, 277}, {3, 1447}, {4, 7}, {10, 75}, {20, 3598}, {77, 1067}, {150, 1837}, {158, 331}, {169, 673}, {274, 988}, {304, 350}, {312, 1930}, {315, 320}, {318, 1235}, {348, 3086}, {496, 1565}, {517, 3212}, {1434, 3338}
X(3673) = complement of X(25242)
X(3673) = anticomplement of X(25066)
X(3674) lies on these lines: {1, 7}, {10, 3212}, {57, 348}, {65, 760}, {76, 85}, {142, 3061}, {274, 1432}, {527, 2329}, {552, 553}, {942, 1565}, {1125, 1447}, {1358, 2795}, {1441, 1930}, {1446, 2051}, {3598, 3616}
X(3675) lies on these lines: {1, 3}, {11, 1111}, {63, 1083}, {244, 665}, {518, 1026}, {672, 1642}, {764, 1647}, {1027, 2424}, {1357, 2821}, {1463, 1736}, {2310, 3020}, {3218, 3573}
X(3675) = isogonal conjugate of X(5377)Let P1 and P2 be the two points on the Gergonne line whose trilinear polars are parallel to the Gergonne line. P1 and P2 lie on the circumconic centered at X(1086) (hyperbola {{A, B, C, X(2), X(7)}}), and circle {{X(2), X(109), X(675)}}. The midpoint of P1 and P2 is X(1638). X(3676) is the barycentric product P1*P2. (Randy Hutson, January 15, 2019)
X(3676) lies on these lines: {2, 4468}, {7, 6006}, {11, 3323}, {57, 649}, {109, 658}, {241, 514}, {278, 3064}, {513, 676}, {522, 693}, {918, 3239}, {934, 2222}, {1443, 1447}, {3321, 3322}}
X(3676) = isogonal conjugate of X(3939)
X(3676) = isotomic conjugate of X(3699)
X(3676) = complement of X(4468)
X(3676) = anticomplement of X(4521)
X(3676) = anticomplementary conjugate of anticomplement of X(38828)
X(3676) = X(647)-of-intouch-triangle
X(3676) = trilinear pole of line X(1086)X(1358)
X(3677) lies on these lines: {1, 3}, {9, 38}, {42, 3243}, {200, 3242}, {244, 612}, {518, 2999}, {613, 2003}, {1401, 3056}, {1449, 3509}, {2191, 2983}
X(3678) lies on these lines: 1,748 3,2801 8,80 9,943 10,12 35,3219 38,3216 58,1757 78,993 100,191 200,1005 214,2975 386,872 392,3244 502,594 517,546 518,1125 519,960 537,596 551,3555 668,1237 740,3159 762,2295 765,1098 936,1445 942,3635 956,1388 976,1724 997,1420 2292,3293 3057,3625
X(3678) = X(1125)-of-inner-Garcia triangle
X(3678) = excentral-to-ABC barycentric image of X(5)
X(3679) lies on these lines: {1, 2}, {5, 3657}, {9, 80}, {12, 3340}, {30, 40}, {35, 958}, {36, 956}, {46, 529}, {63, 484}, {75, 537}, {100, 993}, {140, 3654}, {148, 1654}, {165, 376}, {210, 381}, {333, 3550}, {341, 1089}, {388, 553}, {516, 3543}, {518, 599}, {524, 3416}, {527, 1478}, {536, 984}, {540, 1046}, {542, 2948}, {544, 1282}, {549, 952}, {730, 3097}, {752, 1757}, {944, 3524}, {946, 3545}, {966, 2321}, {982, 1739}, {996, 1150}, {1213, 3247}, {1377, 3299}, {1378, 3301}, {1479, 2551}, {1573, 2276}, {1574, 2275}, {1697, 1837}, {1743, 2345}, {1768, 3359}, {1788, 3361}, {3208, 3294}, {3434, 3583}, {3436, 3585}, {3534, 3579}
X(3679) = midpoint of X(2) and X(8)
X(3679) = reflection of X(1) in X(2)
X(3679) = isogonal conjugate of X(2163)
X(3679) = isotomic conjugate of complement of X(17488)
X(3679) = isotomic conjugate of anticomplement of X(16590)
X(3679) = complement of X(3241)
X(3679) = anticomplement of X(551)
X(3679) = {X(8),X(10)}-harmonic conjugate of X(1)
X(3679) = harmonic center of incircle and Spieker circle
X(3679) = harmonic center of Conway circle and excircles radical circle
X(3679) = X(165)com[Inverse(n(Hexyl))]
X(3679) = homothetic center of Caelum triangle and cross-triangle of Aquila and anti-Aquila triangles
X(3679) = outer-Garcia-to-ABC similarity image of X(2)
X(3679) = reflection of X(2) in X(10)
X(3679) = excentral-to-ABC barycentric image of X(3576)
X(3679) = homothetic center of Gemini triangle 20 and cross-triangle of Gemini triangles 20 and 28
Let A'B'C' be Triangle T(-1,3) (aka excenters-reflections triangle). Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(3680). (Randy Hutson, March 21, 2019)
Let A' be the orthocenter of BCX(1), and define B' and C' cyclically. A'B'C' is also the anticevian triangle, wrt intouch triangle, of X(1), and also the Garcia reflection triangle (Gemini triangle 8). X(3680) is the orthocenter of A'B'C'. (see also X(1699)) (Randy Hutson, March 21, 2019)
X(3680) lies on the Feuerbach hyperbola and these lines: {1, 474}, {4, 519}, {7, 145}, {8, 3452}, {9, 3057}, {10, 1000}, {21, 1697}, {40, 104}, {57, 1476}, {78, 1320}, {84, 517}, {200, 2098}, {518, 3062}, {728, 2170}, {941, 3169}, {1449, 2298}, {1482, 3577}, {2099, 2900}, {2320, 3601}, {3189, 3244}
X(3680) = isogonal conjugate of X(1420)
X(3680) = isotomic conjugate of X(39126)
X(3680) = anticomplement of X(12640)
X(3680) = X(4)-of-Garcia-reflection-triangle
X(3680) = Garcia-reflection-isogonal conjugate of X(32049)
X(3680) = excentral-to-ABC barycentric image of X(2136)
X(3681) lies on these lines: {1, 748}, {2, 210}, {4, 8}, {9, 1174}, {31, 1757}, {38, 43}, {42, 984}, {55, 1776}, {63, 100}, {65, 3617}, {69, 3263}, {78, 947}, {81, 612}, {144, 3059}, {145, 960}, {150, 2890}, {374, 391}, {390, 1864}, {392, 3241}, {511, 3578}, {561, 668}, {674, 3060}, {846, 2177}, {896, 3550}, {899, 982}, {986, 3214}, {1005, 2900}, {1376, 3218}, {1474, 2287}, {2836, 2895}, {3057, 3621}, {3555, 3616}
X(3681) = isotomic conjugate of isogonal conjugate of X(15624)
X(3681) = isotomic conjugate of polar conjugate of X(17916)
X(3681) = complement of X(4430)
X(3681) = anticomplement of X(354)
X(3681) = centroid of triangle T(a, c - b)
X(3681) = centroid of Gemini triangle 30 (inner-Conway triangle)
X(3681) = excentral-to-ABC barycentric image of X(1699)
X(3682) lies on these lines: {1, 2}, {3, 48}, {9, 581}, {20, 2947}, {35, 1819}, {56, 209}, {58, 2327}, {72, 73}, {97, 2169}, {100, 1816}, {101, 1297}, {210, 2594}, {226, 3191}, {238, 1612}, {255, 394}, {276, 313}, {283, 1794}, {326, 1264}, {329, 1745}, {518, 1066}, {580, 2323}, {603, 3173}, {758, 1042}, {901, 2755}, {908, 1838}, {960, 1064}, {1073, 1260}, {1217, 1826}, {1332, 1792}, {1472, 1918}, {1490, 2324}, {1780, 1801}, {2654, 3419}, {2939, 3101}
X(3682) = isogonal conjugate of X(8747)
X(3682) = isotomic conjugate of polar conjugate of X(71)
X(3682) = X(19)-isoconjugate of X(27)
X(3682) = crossdifference of every pair of points on line X(649)X(7649)
X(3682) = excentral-to-ABC barycentric image of X(1715)
In the plane of a triangle ABC, let
A'B'C' = circumcevian triangle of X(1);
DEF = cevian triangle of X(10);
La = line tfhrough D perpendicular to BC;
A'' = La∩AA', and define B'' and C'' cyclically.
Then X(36;83) is the finite fixed point of the affine transformation that carries A'B'C' onto A''B''C'. (Angel Montesdeoca, November 21, 2022)
X(3683) lies on these lines: {2, 1155}, {3, 1709}, {6, 968}, {9, 55}, {21, 60}, {31, 37}, {38, 1279}, {42, 44}, {45, 612}, {63, 354}, {65, 405}, {144, 3475}, {191, 942}, {212, 1212}, {238, 846}, {381, 1698}, {392, 993}, {430, 1213}, {452, 1837}, {518, 1621}, {553, 1125}, {756, 902}, {940, 1707}, {958, 3057}, {1011, 2182}, {1100, 1962}, {1104, 2292}, {1214, 1456}, {1376, 3305}, {2187, 2267}, {2223, 3294}, {2293, 2318}, {2328, 2361}
X(3683) = complement of X(20292)
X(3683) = excentral-to-ABC barycentric image of X(3219)
X(3684) lies on these lines: {1, 2271}, {2, 2280}, {6, 43}, {8, 41}, {9, 55}, {72, 3496}, {78, 2082}, {100, 672}, {101, 519}, {193, 1958}, {218, 3501}, {219, 3169}, {220, 3208}, {238, 1914}, {239, 385}, {242, 740}, {261, 284}, {294, 2340}, {346, 3217}, {350, 3570}, {391, 2268}, {404, 1475}, {511, 3033}, {518, 910}, {521, 650}, {940, 1449}, {1743, 3550}, {1802, 3189}, {2266, 2550}, {2269, 2287}
X(3684) = excentral-to-ABC barycentric image of X(3509)
X(3685) lies on these lines: {1, 87}, {2, 968}, {8, 9}, {20, 2128}, {21, 261}, {31, 1999}, {55, 312}, {75, 1001}, {100, 2726}, {105, 3263}, {149, 3006}, {190, 518}, {238, 239}, {242, 862}, {321, 1621}, {344, 2550}, {345, 497}, {350, 1281}, {519, 1757}, {522, 663}, {536, 1279}, {595, 2901}, {614, 3210}, {643, 2361}, {664, 1456}, {874, 1921}, {960, 1043}, {1045, 1050}, {1265, 3189}, {1266, 3616}
X(3685) = excentral-to-ABC barycentric image of X(1757)
X(3686) lies on these lines: {1, 966}, {2, 1449}, {6, 10}, {8, 9}, {37, 519}, {44, 594}, {45, 3625}, {69, 142}, {72, 2262}, {75, 527}, {141, 3008}, {145, 3247}, {198, 956}, {239, 1654}, {261, 284}, {306, 2280}, {515, 573}, {553, 3578}, {604, 1150}, {965, 1210}, {997, 3554}, {1100, 1125}, {1107, 2092}, {1743, 2345}, {2238, 2300}, {2287, 2323}, {2324, 2654}
X(3686) = complement of X(3879)
X(3686) = excentral-to-ABC barycentric image of X(37)
X(3687) lies on these lines: {1, 2}, {9, 345}, {57, 69}, {63, 573}, {72, 970}, {75, 226}, {181, 518}, {261, 284}, {312, 2321}, {318, 469}, {319, 2985}, {320, 553}, {321, 908}, {355, 2050}, {440, 2968}, {914, 1150}, {950, 1043}, {960, 1682}, {1211, 2092}, {1376, 1460}, {1738, 2887}, {1812, 2323}, {2895, 3218}
X(3687) = excentral-to-ABC barycentric image of X(171)
X(3688) lies on these lines: {8, 314}, {9, 3056}, {31, 2273}, {37, 674}, {38, 1401}, {39, 1964}, {42, 2300}, {51, 756}, {55, 219}, {71, 2223}, {77, 1362}, {78, 1682}, {181, 612}, {200, 3169}, {511, 984}, {869, 2092}, {1253, 3270}, {1334, 2293}, {1742, 2808}, {2269, 2340}, {2323, 2330}, {2388, 2667}, {3057, 3059}
X(3688) = isogonal conjugate of isotomic conjugate of X(3703)
X(3688) = anticomplement of X(17049)
X(3688) = crosssum of X(7) and X(56)
X(3688) = crosspoint of X(8) and X(55)
X(3688) = excentral-to-ABC barycentric image of X(75)
X(3689) lies on these lines: {8, 2320}, {9, 55}, {37, 2177}, {42, 1100}, {44, 678}, {78, 3057}, {100, 518}, {214, 519}, {354, 1376}, {528, 908}, {612, 3290}, {650, 663}, {899, 1279}, {936, 3303}, {1104, 3214}, {1261, 2194}, {1318, 1320}, {1386, 3240}, {1837, 3189}, {2098, 2136}, {3058, 3452}
X(3689) = isogonal conjugate of isotomic conjugate of X(4723)
X(3689) = crossdifference of every pair of points on line X(57)X(1022)
X(3689) = excentral-to-ABC barycentric image of X(3218)
X(3690) lies on these lines: {3, 1796}, {9, 51}, {10, 3136}, {25, 220}, {31, 2273}, {37, 209}, {42, 213}, {55, 584}, {63, 295}, {71, 228}, {72, 306}, {101, 199}, {181, 756}, {184, 219}, {200, 1018}, {201, 1425}, {210, 430}, {212, 3270}, {373, 3305}, {511, 3219}, {612, 2295}, {1011, 3190}
X(3690) = crosspoint of X(71) and X(72)
X(3690) = excentral-to-ABC barycentric image of X(1746)
X(3691) lies on these lines: {2, 1475}, {8, 9}, {10, 672}, {39, 899}, {41, 958}, {44, 2295}, {210, 1212}, {213, 1573}, {220, 2654}, {239, 1655}, {388, 966}, {405, 2280}, {519, 3294}, {604, 965}, {960, 2170}, {1018, 3626}, {1055, 2975}, {1107, 1193}, {1213, 2260}, {2276, 3214}, {3219, 3496}, {3501, 3617}
X(3691) = excentral-to-ABC barycentric image of X(3294)
X(3692) lies on these lines: {1, 1257}, {6, 3693}, {8, 9}, {19, 5174}, {37, 4513}, {40, 5279}, {48, 4855}, {57, 4869}, {63, 69}, {78, 219}, {100, 610}, {145, 2257}, {190, 322}, {200, 1253}, {220, 3965}, {268, 271}, {281, 6735}, {282, 2057}, {312, 3305}, {326, 1332}, {341, 2322}, {380, 3871}, {519, 1723}, {579, 4684}, {595, 1743}, {644, 2324}, {1018, 1766}, {1034, 7080}, {1073, 3998}, {1229, 4384}, {1441, 3729}, {1445, 3912}, {1792, 1802}, {1809, 2289}, {2264, 3913}, {2285, 3501}, {2897, 3882}, {3436, 8804}, {3713, 4515}, {3731, 9623}, {3926, 7177}, {3949, 3984}, {3951, 4047}, {4073, 4319}, {10325, 10860}
X(3692) = isogonal conjugate of X(1435)X(3693) lies on these lines: {1, 728}, {2, 37}, {8, 1212}, {9, 55}, {44, 765}, {65, 3501}, {78, 220}, {100, 910}, {101, 2751}, {517, 1018}, {518, 672}, {522, 650}, {960, 1334}, {1026, 1642}, {1040, 2324}, {1155, 3509}, {1759, 3579}, {2329, 2646}, {3057, 3061}
X(3693) = isogonal conjugate of X(1462)
X(3693) = isotomic conjugate of X(34018)
X(3693) = excentral-to-ABC barycentric image of X(672)
X(3694) lies on these lines: {1, 2336}, {8, 2335}, {9, 55}, {10, 37}, {71, 72}, {78, 219}, {281, 318}, {306, 307}, {518, 579}, {519, 1108}, {728, 2324}, {997, 2256}, {1761, 3579}, {1792, 2193}, {1793, 2327}, {2260, 3555}, {2345, 3085}, {3061, 3169}
X(3694) = excentral-to-ABC barycentric image of X(579)
X(3695) lies on these lines: {3, 345}, {4, 346}, {5, 312}, {8, 405}, {10, 37}, {12, 1089}, {40, 728}, {72, 306}, {78, 1062}, {100, 2915}, {190, 1330}, {304, 337}, {321, 442}, {519, 1104}, {1215, 3178}, {2049, 2345}, {3159, 3454}
X(3695) = excentral-to-ABC barycentric image of X(580)
X(3696) lies on these lines: {7, 8}, {10, 37}, {40, 1765}, {80, 2805}, {141, 1738}, {192, 3617}, {210, 321}, {239, 1386}, {318, 1882}, {536, 984}, {726, 3626}, {756, 3175}, {872, 3214}, {1150, 1155}, {1214, 2968}, {1376, 1402}, {1861, 1880}
X(3696) = midpoint of X(8) and X(75)
X(3696) = excentral-to-ABC barycentric image of X(991)
X(3697) lies on these lines: {2, 3555}, {8, 392}, {10, 12}, {37, 762}, {200, 405}, {474, 3361}, {517, 3091}, {518, 1698}, {756, 3214}, {936, 956}, {958, 3612}, {997, 1388}, {2551, 3419}, {3057, 3626}, {3219, 3579}, {3295, 3305}
X(3697) = excentral-to-ABC barycentric image of X(631)
X(3698) lies on these lines: {2, 3057}, {8, 354}, {10, 12}, {55, 1706}, {281, 1888}, {374, 1828}, {474, 1319}, {517, 1656}, {518, 3617}, {936, 2099}, {958, 1155}, {1376, 2646}, {1836, 2551}, {1837, 2550}, {3555, 3626}
X(3698) = excentral-to-ABC barycentric image of X(3522)
X(3699) lies on these lines: {1, 1120}, {2, 1280}, {8, 11}, {78, 341}, {100, 190}, {200, 312}, {210, 333}, {537, 1054}, {643, 645}, {644, 1639}, {658, 883}, {664, 668}, {765, 1331}, {1897,4033}, {2899, 3189}
X(3699) = isotomic conjugate of X(3676)X(3700) lies on these lines: {9, 1021}, {37, 647}, {101, 2689}, {321, 850}, {424, 2501}, {522, 650}, {523, 661}, {525, 1577}, {649, 900}, {693, 918}, {824, 3004}, {1635, 2490}, {2533, 3566}, {3910,4391}
X(3700) = anticomplement of X(17069)
X(3700) = isogonal conjugate of X(4565)
X(3700) = isotomic conjugate of X(4573)
X(3700) = perspector of hyperbola {{A,B,C,X(8),X(10)}}
X(3700) = intersection of trilinear polars of X(8) and X(10)
X(3700) = excentral-to-ABC barycentric image of X(661)
X(3701) lies on these lines: {1, 996}, {5, 3006}, {8, 210}, {10, 321}, {12, 313}, {76, 3263}, {281, 318}, {306, 857}, {442, 1230}, {612, 964}, {740, 3214}, {1826, 3610}, {2901, 3293}
X(3701) = isotomic conjugate of X(1014)
X(3701) = excentral-to-ABC barycentric image of X(3216)
X(3702) lies on these lines: {1, 321}, {8, 210}, {21, 261}, {75, 3616}, {78, 1229}, {306, 946}, {430, 1230}, {519, 1089}, {740, 1193}, {1043, 3615}, {1125, 1962}, {1441, 3485}
X(3702) = excentral-to-ABC barycentric image of X(3293)
X(3703) lies on these lines: {2, 1390}, {5, 1089}, {8, 21}, {11, 312}, {38, 141}, {63, 3416}, {200, 1040}, {209, 306}, {321, 2886}, {346, 497}, {594, 2276}, {726, 2887}, {984, 1211}
X(3703) = anticomplement of X(17061)
X(3703) = excentral-to-ABC barycentric image of X(31)
X(3704) lies on these lines: {8, 21}, {10, 37}, {12, 321}, {40, 1503}, {65, 306}, {100, 1791}, {141, 986}, {197, 2915}, {312, 1329}, {346, 2551}, {524, 1046}, {960, 1682}, {1211, 2292}
X(3704) = excentral-to-ABC barycentric image of X(58)
X(3705) lies on these lines: {1, 2}, {11, 312}, {63, 147}, {69, 1447}, {75, 325}, {183, 319}, {262, 321}, {318, 427}, {333, 2194}, {341, 1329}, {345, 497}, {982, 2887}, {1368, 2968}
X(3705) = isotomic conjugate of isogonal conjugate of X(3056)
X(3705) = excentral-to-ABC barycentric image of X(3550)
X(3706) lies on these lines: {1, 2049}, {8, 210}, {38, 536}, {69, 1836}, {75, 354}, {306, 2886}, {321, 518}, {519, 1215}, {984, 3175}, {1043, 2646}, {1386, 3187}, {1456, 1943}, {3416, 3434}
X(3706) = excentral-to-ABC barycentric image of X(42)
X(3707) lies on these lines: {6, 1125}, {8, 9}, {10, 44}, {37, 3244}, {45, 519}, {72, 374}, {142, 320}, {210, 3271}, {226, 1405}, {333, 645}, {966, 1698}, {1449, 3622}, {3247, 3623}
X(3707) = excentral-to-ABC barycentric image of X(45)
X(3708) lies on these lines: {1, 163}, {19, 2159}, {31, 2157}, {115, 1365}, {512, 3022}, {523, 1146}, {525, 1565}, {774, 2179}, {811, 1821}, {1725, 1755}, {2170, 2611}, {2631, 2632}, {2642, 2643}
X(3708) = isogonal conjugate of isotomic conjugate of X(20902)
X(3708) = isogonal conjugate of polar conjugate of X(1109)
X(3708) = polar conjugate of X(23999)
X(3708) = pole wrt polar circle of trilinear polar of X(23999) (line X(662)X(811))
X(3708) = excentral-to-ABC barycentric image of X(150)
X(3708) = X(92)-isoconjugate of X(1101)
X(3709) lies on these lines: {6, 2605}, {9, 3287}, {37, 523}, {44, 2609}, {101, 2701}, {213, 3049}, {512, 798}, {513, 665}, {522, 650}, {647, 661}, {657, 663}, {667, 2484}, {1919, 1960}
X(3709) = isogonal conjugate of X(4573)
X(3709) = complement of X(4374)
X(3709) = anticomplement of X(17066)
X(3709) = excentral-to-ABC barycentric image of X(798)
X(3710) lies on these lines: {8, 9}, {10, 321}, {71, 3610}, {72, 306}, {78, 345}, {145, 1453}, {201, 307}, {318, 341}, {519, 1724}, {946, 3006}, {1792, 1793}, {1834, 3175}
X(3710) = excentral-to-ABC barycentric image of X(1724)
X(3711) lies on these lines: {8, 11}, {9, 55}, {45, 2177}, {214, 956}, {220, 3119}, {518, 3306}, {612, 1100}, {762, 2271}, {899, 3242}, {936, 3304}, {975, 2334}, {1376, 3218}
X(3711) = excentral-to-ABC barycentric image of X(3306)
X(3711) = anticomplement of X(17051)
X(3712) lies on these lines: {8, 21}, {23, 100}, {37, 3291}, {101, 2768}, {109, 2760}, {522, 650}, {524, 896}, {528, 3006}, {536, 3011}, {846, 1211}, {2229, 2276}
X(3712) = isogonal conjugate of X(7316)
X(3712) = anticomplement of X(17070)
X(3712) = excentral-to-ABC barycentric image of X(896)
X(3713) lies on these lines: {6, 8}, {9, 55}, {37, 78}, {72, 1766}, {219, 2321}, {220, 346}, {518, 2285}, {572, 956}, {608, 1861}, {958, 2268}, {1376, 1400}
X(3713) = excentral-to-ABC barycentric image of X(2285)
X(3714) lies on these lines: {4, 3416}, {8, 210}, {10, 37}, {12, 306}, {65, 321}, {72, 1089}, {318, 1859}, {536, 986}, {1220, 1999}, {1840, 1869}, {2292, 3175}
X(3714) = excentral-to-ABC barycentric image of X(386)
X(3715) lies on these lines: {6, 756}, {8, 3058}, {9, 55}, {10, 1836}, {42, 45}, {44, 612}, {518, 3305}, {748, 3242}, {940, 1757}, {960, 2098}, {1376, 3219}
X(3715) = excentral-to-ABC barycentric image of X(3305)
X(3716) lies on these lines: {2, 2254}, {11, 124}, {118, 120}, {522, 650}, {659, 812}, {676, 918}, {900, 3035}, {926, 3041}, {946, 2814}, {1960, 2787}
X(3716) = complement of X(2254)
X(3716) = anticomplement of X(25380)
X(3716) = excentral-to-ABC barycentric image of X(659)
X(3717) lies on these lines: {1, 344}, {8, 9}, {10, 75}, {100, 2751}, {190, 516}, {200, 345}, {238, 519}, {908, 3006}, {1026, 1818}, {1220, 2983}
X(3717) = isogonal conjugate of X(1416)
X(3717) = isotomic conjugate of isogonal conjugate of X(2340)
X(3717) = excentral-to-ABC barycentric image of X(238)
X(3718) lies on these lines: {7, 3263}, {9, 312}, {10, 75}, {69, 72}, {78, 332}, {86, 975}, {190, 1760}, {305, 307}, {322, 668}, {740, 1716}
X(3718) = isogonal conjugate of X(1395)
X(3718) = isotomic conjugate of X(34)
X(3718) = excentral-to-ABC barycentric image of X(1716)
X(3719) lies on these lines: {9, 312}, {63, 69}, {78, 212}, {100, 2365}, {271, 1265}, {326, 394}, {346, 3219}, {740, 1711}, {2185, 3601}, {2269, 2339}
X(3719) = excentral-to-ABC barycentric image of X(1711)
Let A'B'C' be the incentral triangle. Let A" be the intersection, other than the midpoint of BC, of the A-median and the bicevian ellipse of X(1) and X(2). Define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(3720). (Randy Hutson, December 26, 2015)
X(3720) lies on these lines: {1, 2}, {6, 748}, {11, 3136}, {31, 940}, {37, 38}, {55, 750}, {56, 1011}, {57, 968}, {73, 1985}, {77, 2898}, {81, 238}, {86, 310}, {171, 902}, {226, 1458}, {244, 1962}, {291, 1255}, {405, 1468}, {427, 2356}, {497, 2293}, {518, 756}, {649, 2666}, {799, 2107}, {846, 3218}, {851, 2646}, {942, 2292}, {960, 2650}, {991, 1699}, {1042, 3485}, {1100, 2238}, {1197, 3231}, {1279, 2239}, {1376, 2177}, {1818, 2886}, {1836, 3000}
X(3720) = {X(1),X(2)}-harmonic conjugate of X(42)
X(3720) = complement of X(4651)
X(3720) = bicentric sum of PU(84)
X(3720) = PU(84)-harmonic conjugate of X(649)
X(3721) lies on these lines: {1, 32}, {6, 977}, {10, 762}, {37, 65}, {38, 1107}, {42, 2240}, {72, 2238}, {76, 1928}, {213, 758}, {257, 335}, {321, 1237}, {350, 695}, {712, 1930}, {737, 789}, {960, 3290}, {982, 2275}, {986, 2276}, {1574, 1739}, {1964, 2237}
X(3721) = isotomic conjugate of X(38810)
X(3721) = crosspoint of X(1) and X(76)
X(3721) = crosssum of X(1) and X(32)
X(3722) lies on these lines: {1, 88}, {31, 1331}, {37, 2246}, {38, 55}, {42, 1386}, {105, 612}, {200, 748}, {518, 896}, {528, 3120}, {614, 3158}, {756, 1621}, {899, 1279}, {968, 1282}, {976, 1036}, {1283, 2292}, {1496, 1795}, {1647, 3035}, {2346, 2648}
X(3723) lies on these lines: {1, 6}, {86, 536}, {519, 1213}, {551, 2321}, {583, 1334}, {594, 1125}, {902, 1962}, {910, 1953}, {941, 1392}, {966, 3241}, {1055, 2294}, {1319, 2171}, {1388, 2285}, {2178, 3295}, {2280, 3204}, {2345, 3622}, {2667, 3009}
X(3723) = complement of X(5564)
X(3723) = anticomplement of X(28633)
X(3724) lies on these lines: {1, 994}, {3, 2292}, {23, 1283}, {31, 48}, {36, 214}, {42, 181}, {55, 199}, {56, 2650}, {100, 740}, {187, 237}, {215, 2361}, {851, 1284}, {896, 3286}, {2078, 2611}, {2177, 2667}, {2198, 2200}
X(3724) = isogonal conjugate of X(14616)
X(3725) lies on these lines: {1, 333}, {31, 48}, {37, 42}, {43, 312}, {55, 869}, {63, 2274}, {213, 1402}, {228, 1918}, {354, 1201}, {758, 982}, {960, 1193}, {2294, 3290}, {2308, 3248}
X(3726) lies on these lines: {{1, 32}, {37, 38}, {244, 1575}, {335, 350}, {518, 2238}, {519, 3125}, {758, 3230}, {902, 1962}, {940, 3218}, {942, 2295}, {982, 2276}
X(3727) lies on these lines: {1, 32}, {37, 1953}, {55, 3148}, {257, 350}, {517, 2295}, {762, 3626}, {960, 2238}, {986, 2275}, {1100, 2650}, {1107, 2170}, {1125, 3125}, {2276, 3061}
X(3727) = isogonal conjugate of isotomic conjugate of X(21442)
X(3728) lies on these lines: {8, 192}, {9, 2209}, {31, 1778}, {37, 42}, {38, 75}, {518, 2650}, {668, 1221}, {1107, 2309}, {1213, 3122}, {1573, 2388}, {2269, 2310}
X(3728) = anticomplement wrt incentral triangle of X(2667)
X(3729) lies on these lines: {1, 87}, {2, 2415}, {6, 536}, {7, 346}, {8, 144}, {9, 75}, {10, 2996}, {37, 980}, {46, 1089}, {57, 312}, {63, 321}, {69, 527}, {76, 1423}, {78, 990}, {85, 728}, {86, 3247}, {101, 1958}, {141, 545}, {142, 344}, {148, 1654}, {193, 519}, {200, 1721}, {226, 345}, {239, 1278}, {341, 1706}, {385, 3550}, {522, 1027}, {664, 1419}, {940, 3175}, {975, 3159}, {1229, 1445}, {1909, 3208}, {1944, 2324}, {1975, 2329}, {2191, 2414}, {2999, 3210}
X(3729) = crosssum of PU(48)
X(3729) = isogonal conjugate of X(9315)
X(3729) = isotomic conjugate of X(9311)
X(3729) = anticomplement of X(3663)
X(3729) = {X(7),X(346)}-harmonic conjugate of X(3912)
X(3729) = crosspoint of PU'(48), where PU'(48) are the isogonal conjugates of PU(48)
X(3730) lies on these lines: {1, 672}, {2, 2140}, {3, 101}, {4, 9}, {6, 595}, {8, 1018}, {35, 41}, {37, 579}, {39, 995}, {45, 2245}, {55, 218}, {76, 190}, {165, 170}, {213, 386}, {219, 572}, {284, 2911}, {348, 1025}, {514, 3177}, {517, 1212}, {519, 3208}, {644, 2975}, {813, 1083}, {910, 3579}, {993, 2329}, {1011, 3190}, {1030, 3204}, {1400, 3339}, {1726, 3219}, {1743, 2269}, {2203, 2328}, {2260, 3247}, {2267, 2323}, {2275, 3230}
X(3730) = isogonal conjugate of X(14377)
X(3730) = trilinear pole wrt 1st circumperp triangle of antiorthic axis
X(3730) = X(39)-of-excentral-triangle
X(3730) = Cundy-Parry Phi transform of X(103)
X(3730) = Cundy-Parry Psi transform of X(516)
X(3731) lies on these lines: {1, 6}, {2, 2415}, {3, 1696}, {10, 346}, {35, 198}, {71, 2093}, {101, 2268}, {106, 2297}, {165, 846}, {200, 756}, {281, 1785}, {344, 3619}, {391, 519}, {573, 1334}, {610, 3465}, {612, 902}, {966, 2321}, {1400, 3339}, {1698, 1738}, {1707, 1961}, {1742, 3062}, {2182, 3612}, {2285, 3361}, {2999, 3305}
X(3732) lies on these lines: {2, 1565}, {3, 3177}, {85, 169}, {100, 1292}, {101, 514}, {144, 153}, {150, 1146}, {190, 646}, {218, 3212}, {544, 1121}, {651, 653}, {673, 1111}
X(3733) lies on these lines: {6, 798}, {28, 2401}, {36, 238}, {82, 876}, {86, 3253}, {99, 889}, {110, 901}, {512, 1326}, {523, 1325}, {649, 834}, {659, 3004}, {660, 662}, {665, 2483}, {669, 2106}, {688, 875}, {741, 2382}, {759, 2718}, {1474, 2424}
X(3733) = isogonal conjugate of X(3952)
X(3733) = X(92)-isoconjugate of X(4574)
X(3733) = pole wrt circumcircle of line X(1)X(21)
X(3733) = crossdifference of every pair of points on line X(10)X(37)
X(3734) = eigencenter of 1st Brocard triangle (Peter Moses, January 24, 2012); see X(3972)
Let A'B'C' be the 1st Brocard triangle. Let A" be the trilinear pole, with respect to A'B'C', of line BC, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(3734).
X(3734) lies on these lines: {2, 99}, {4, 626}, {6, 538}, {30, 141}, {32, 76}, {39, 1975}, {69, 754}, {83, 194}, {182, 2782}, {183, 187}, {316, 3314}, {350, 2242}, {381, 625}, {1078, 3552}, {1235, 1968}, {1352, 2794}, {1909, 2241}
X(3734) = X(6) of 1st Brocard triangle
X(3734) = 1st-Brocard-isogonal conjugate of X(2)
X(3734) = 1st-Brocard-isotomic conjugate of X(3094)
X(3734) = orthocentroidal-to-1st-Brocard similarity image of X(6)
X(3734) = anti-Artzt isogonal conjugate of X(598)
X(3734) = X(182)-of-anti-Artzt-triangle
X(3735) lies on these lines: {1, 32}, {2, 3125}, {6, 758}, {37, 517}, {38, 2170}, {39, 986}, {75, 712}, {76, 257}, {392, 3290}, {762, 3617}, {766, 3056}, {980, 1959}, {982, 1015}, {984, 1573}, {2549, 2795}, {2809, 3242}
X(3735) = anticomplement of X(24254)
X(3735) = X(75)-of-1st-Brocard-triangle
X(3735) = 1st-Brocard-isogonal conjugate of X(4112)
X(3735) = 1st-Brocard-isotomic conjugate of X(3923)
X(3736) lies on these lines: {1, 75}, {3, 6}, {21, 238}, {35, 1918}, {42, 81}, {43, 333}, {99, 731}, {110, 753}, {869, 984}, {872, 1757}, {995, 1001}, {1014, 1458}, {1408, 2594}, {1412, 1460}, {1707, 2258}, {1818, 2303}
X(3736) = crossdifference of every pair of points on line X(523)X(798)
X(3737) lies on these lines: {1, 523}, {9, 3287}, {36, 238}, {54, 2616}, {59, 110}, {86, 1027}, {112, 2728}, {242, 514}, {284, 1024}, {512, 2959}, {521, 650}, {522, 663}, {741, 2726}, {759, 953}, {832, 1491}, {1443, 1447}
X(3737) = isogonal conjugate of X(4551)
X(3737) = crosssum of Kiepert hyperbola intercepts of antiorthic axis
X(3737) = SS(A → A') of X(2616), where A'B'C' is the excentral triangle
X(3738) lies on these lines: {1, 1769}, {11, 124}, {30, 511}, {100, 109}, {102, 104}, {117, 119}, {151, 153}, {656, 1955}, {885, 3254}, {1027, 1814}, {1317, 1361}, {3035, 3042}, {3036, 3040}
X(3738) = isogonal conjugate of X(2222)
X(3738) = isotomic conjugate of X(35174)
X(3738) = X(2)-Ceva conjugate of X(35128)
X(3738) = X(522)-of-inner-Garcia triangle
X(3738) = X(2804)com(Excentral)
More Combos
Continuing the discussion of points Xcom(T), suppose that T is an arbitrary triangle, and let nT denote the normalization of T. Let NT denote the set of these triangles, as matrices, and let * denote matrix multiplication. Then NT is closed under *. Also, NT is closed under matrix inversion. Consequently, (NT, *) is a group, comparable to the group of stochastic matrices.
Suppose that T1 and T2 are triangles. In many cases, the product T1*T2 is well-defined (e.g., TCCT, page 175). However, n(T1*T2) may not be n(T1)*n(T2) if T1 and T2 are not normalized. Therefore, it is important, when dealing with products, to include the "n" if it is intended.
As a class of examples of triangles defined by matrix products, suppose that T1 is the cevian triangle of a triangle center f : g : h (barycentrics) and that T2 is the cevian triangle of a triangle center u : v : w. The A-vertex of the triangle T3 = (nT1)*(nT2) is given by
u(gu + hu + gv +hw) : hv(u + w) : gw(u + v),
from which it can be shown that T3 is perspective to the triangle ABC, with perspector
P = ghu(v + w) : hfv(w + u) : fgw(u + v).
Also, T3 is perspective to T2, with perspector
Q = u(v + w)(gu + hu + gv + hw) : v(w + u)(hv + fv + hw + fu) : w(u + v)(fw + gw + fu + gv).
Thus, T3 can be constructed directly from the pairs {P, ABC} and {Q, T2}. Regarding the possibility that T3 is also perspective to T1, the concurrence determinant for this condition factors as F1F2F3, where
F1 = u + v + w,
F2 = fu(g + h) + gv(h + f) + hw(f + g),
F3 = (fghuvw)2[(gu)-2 -
(hu)-2 + (hv)-2 - (fv)-2 +
(fw)-2 - (gw)-2].
Consequently, the perspectivity holds if and only if one of the three factors is 0, and from this result, various geometric results can be obtained.
Example: The Incentral and Medial triangles result by taking f : g : h = a : b : c and u : v : w = 1 : 1: 1. The product n(Incentral)*n(Medial) has vertices
A' = b + c : c : b B' = c : c + a : a C' = b : a : a + b
and its inverse has A-vertex A'' = bc + ca + ab : -bc - ca + ab : -bc + ca - ab, from which B'' and C'' are obtained cyclically.
The product n(Medial)*n(Incentral) has A-vertex a(2a + b + c) : b(c + a) : c(a + b), and the inverse of this product has A-vertex (a + b + c)(b + c) : -c(a + c) : -b(a + b).
Having considered the group NT of normalized triangles, we turn next to 3-point combos based on product triangles and inverse triangles. Recall that such a combo, denoted by Xcom(T), is defined, as in the preamble to X(3663), by the matrix product (x y z)*(nT), where nT is the normalization of T.
X(3739) lies on these lines: {2, 37}, {7, 966}, {10, 141}, {11, 126}, {44, 894}, {85, 1418}, {86, 239}, {241, 1441}, {274, 1107}, {320, 1654}, {335, 1268}, {740, 1125}, {742, 3008}, {872, 899}, {984, 1698}, {1010, 1104}, {1086, 1213}, {1368, 2886}, {1958, 2278}, {2234, 2309}
X(3739) = isotomic conjugate of X(32009)
X(3739) = complement of X(37)
X(3739) = anticomplement of X(4698)
X(3739) = polar conjugate of isogonal conjugate of X(22060)
X(3739) = centroid of ABCX(75)
X(3739) = Kosnita(X(75),X(2)) point
X(3739) = {X(2),X(75)}-harmonic conjugate of X(37)
In the plane of a triangle ABC, let
AaAbAc = A-extouch triangle, and define BaBbBc and CaCbCc cyclically
Ab' = AbAc∩BbBc, and define Bc' and Ca' cyclically
Ac' = AbAc∩CbCc, and define Ba' and Cb' cyclically
Then X(3740) = centroid-of-{Ab', Ac', Bc', Ba', Ca', Cb'}. These 6 points lie on the radical circle of the excircles. (Randy Hutson, September 14, 2016)
Let A'B'C' = excentral triangle. Let
Ab" = B'C'∩BbBc, and define Bc'' and Ca'' cyclically
Ac'' = B'C'∩CbCc, and define Ba'' and Cb'' cyclically
Then X(3740) = centroid-of-{Ab",Ac", Bc", Ba", Ca", Cb"}. These 6 points lie on an ellipse. (Randy Hutson, September 14, 2016)
Continuing, let
Ab* = B'C'∩BcBa, and define Bc* and Ca* cyclically
Ac* = B'C'∩CaCb, and define Ba* and Cb* cyclically
Then X(3740) = centroid of {Ab*,Ac*,Bc*,Ba*,Ca*,Cb*}. These 6 points lie on the radical circle of the excircles. (Randy Hutson, September 14, 2016)
X(3740) lies on these lines: {2, 210}, {5, 10}, {9, 165}, {37, 43}, {44, 171}, {55, 3305}, {72, 1698}, {120, 125}, {200, 1001}, {374, 966}, {375, 511}, {405, 2900}, {612, 1386}, {748, 3246}, {756, 899}, {936, 958}, {1051, 1100}, {1107, 2664}, {1155, 3219}, {2801, 3035}, {3057, 3617}, {3555, 3624}
X(3740) = reflection of X(3742) in X(2)
X(3740) = isotomic conjugate of X(32021)
X(3740) = complement of X(354)
X(3740) = anticomplement of X(3848)
X(3741) lies on these lines: {1, 2}, {11, 1211}, {31, 1150}, {38, 321}, {75, 982}, {116, 126}, {141, 674}, {226, 1469}, {238, 333}, {256, 314}, {312, 984}, {516, 1764}, {518, 1215}, {594, 1575}, {964, 1468}, {993, 1011}, {2238, 2300}, {2260, 2345}, {2276, 2321}, {3136, 3454}
X(3741) = complement of X(42)
X(3741) = isotomic conjugate of isogonal conjugate of X(2309)
X(3741) = polar conjugate of isogonal conjugate of X(22065)
X(3741) = {X(2),X(8)}-harmonic conjugate of X(43)
Let A'B'C' be the intouch triangle. Let
AaAbAc = A-extouch triangle, and define BaBbBc and CaCbCc cyclically
Ab' = B'C'∩BbBc, and define Bc' and Ca' cyclically
Ac' = B'C'∩CbCc, and define Ba' and Cb' cyclically
Then X(3742) = centroid of {Ab',Ac',Bc',Ba',Ca',Cb'}. (Randy Hutson, September 14, 2016)
X(3742) lies on these lines: {1, 474}, {2, 210}, {31, 3246}, {37, 982}, {55, 3306}, {57, 1001}, {65, 3616}, {72, 3624}, {86, 1431}, {142, 2886}, {171, 1279}, {244, 1962}, {375, 2810}, {405, 3338}, {517, 549}, {614, 940}, {758, 942}, {958, 3333}, {1155, 1621}, {1698, 3555}, {3057, 3622}
X(3742) = reflection of X(3740) in X(2)
X(3742) = complement of X(210)
X(3742) = X(551)com[Inverse(n(Hexyl))]
X(3743) lies on these lines: {1, 21}, {10, 37}, {40, 2294}, {55, 2915}, {484, 1255}, {756, 3293}, {984, 2667}, {1126, 1757}, {1215, 3159}, {1486, 3295}, {3178, 3454}
X(3743) = complement of X(4647)
X(3743) = complement wrt incentral triangle of X(1)
X(3743) = perspector of Gemini triangle 12 and cross-triangle of Gemini triangles 11 and 12
X(3744) lies on these lines: {1, 3}, {2, 1279}, {8, 1104}, {31, 518}, {37, 82}, {38, 902}, {42, 1386}, {58, 3555}, {63, 3052}, {72, 595}, {78, 1191}, {81, 643}, {145, 345}, {210, 238}, {612, 1001}, {614, 1376}, {748, 3246}, {749, 1100}, {960, 976}, {1455, 3476}, {2298, 2346}, {2886, 3011}, {2999, 3158}
X(3744) = complement of X(5014)
X(3744) = {X(1),X(40)}-harmonic conjugate of X(37549)
X(3744) = {X(55),X(56)}-harmonic conjugate of X(37577)
X(3745) lies on these lines: {1, 3}, {2, 1386}, {6, 210}, {31, 37}, {33, 608}, {42, 1100}, {44, 756}, {81, 518}, {197, 2262}, {200, 1449}, {226, 1456}, {238, 1961}, {902, 1962}, {968, 3052}, {1279, 2239}, {2194, 2303}
X(3745) = complement of X(33075)Let A'B'C' be the incentral triangle. Let La be the reflection of line BC in line B'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines A'A", B'B", C'C" concur in X(3746). (Randy Hutson, September 14, 2016)
X(3746) lies on these lines: {1, 3}, {5, 3058}, {10, 1621}, {11, 3628}, {12, 546}, {21, 519}, {42, 595}, {58, 902}, {62, 1250}, {79, 516}, {80, 943}, {100, 1125}, {140, 3582}, {145, 993}, {191, 518}, {238, 3293}, {386, 2177}, {388, 3529}, {390, 1479}, {404, 551}, {442, 528}, {495, 3585}, {496, 632}, {497, 498}, {499, 1058}, {575, 2330}, {576, 3056}, {612, 1995}, {1001, 1698}, {1124, 3594}, {1126, 2308}, {1253, 1497}, {1283, 2292}, {1335, 3592}, {1376, 3624}, {1478, 3146}, {1500, 1914}, {2066, 3301}, {2241, 2276}, {2975, 3244}
X(3746) = incentral-isogonal conjugate of X(1)
X(3746) = trilinear pole, wrt incentral triangle, of line that is incentral isogonal conjugate of incentral inellipse
X(3746) = {X(1),X(35)}-harmonic conjugate of X(36)
X(3747) lies on these lines: {1, 21}, {6, 2667}, {9, 2209}, {37, 1918}, {42, 213}, {55, 869}, {100, 2664}, {187, 237}, {190, 714}, {238, 239}, {756, 3294}, {922, 3285}, {1914, 2210}, {2195, 3512}, {2245, 3122}, {2300, 2309}
X(3747) = isogonal conjugate of X(18827)X(3748) lies on these lines: {1, 3}, {37, 2280}, {42, 1279}, {105, 1255}, {210, 1001}, {226, 3058}, {390, 1836}, {518, 1621}, {672, 1100}, {910, 1953}, {954, 1864}, {968, 3242}, {1108, 2266}, {1362, 2772}, {1365, 3021}, {1453, 2334}, {1962, 2611}
X(3749) lies on these lines: {1, 3}, {9, 983}, {31, 1331}, {43, 3158}, {63, 902}, {100, 614}, {200, 238}, {214, 1811}, {345, 519}, {518, 1707}, {612, 1621}, {943, 989}, {1279, 1376}, {1449, 2276}, {3011, 3434}
X(3750) lies on these lines: {1, 3}, {2, 2177}, {42, 238}, {43, 1001}, {81, 902}, {99, 2668}, {256, 2346}, {333, 519}, {518, 846}, {748, 3240}, {941, 983}, {968, 984}
Let A'B'C' denote the inverse of the normalized hexyl triangle. Then A', expressed in homogeneous barycentrics (i.e., not normalized), is given by
2a(b + c) : a2 + b2 - c2 : a2 - b2 + c2
In general, if a triangle DEF is perspective to a given triangle GHI from a point P and also perspector to another triangle JKL from a point Q, then, clearly, lines joining P to G, H, I, and line joining Q to J,K,L concur in pairs to form DEF. Thus, if constructions are known for G,H,I,P,J,K,I,Q, then a construction for DEF easily follows. In particular, the triangle A'B'C' can be constructed in several ways, as it is perspective to the following triangles with perspectors:
ABC from X(4), orthic triangle from X(4), medial triangle from X(2), intouch triangle from X(65), extouch triangle from X(10), Euler triangle X(4)
X(3751) lies on these lines: {1, 6}, {7, 1738}, {8, 193}, {10, 69}, {31, 1331}, {33, 1957}, {40, 511}, {42, 63}, {43, 57}, {46, 3293}, {55, 1707}, {58, 1792}, {78, 1468}, {81, 612}, {141, 1698}, {165, 1350}, {171, 200}, {182, 3576}, {210, 940}, {240, 2331}, {314, 989}, {355, 3564}, {386, 988}, {517, 1351}, {519, 1992}, {524, 3416}, {614, 3315}, {651, 2263}, {872, 2274}, {896, 2177}, {899, 3306}, {942, 1722}, {952, 1353}, {968, 3219}, {971, 1721}, {978, 3333}, {982, 2999}, {990, 1814}, {1125, 3618}, {1420, 1428}, {1445, 1458}, {1697, 3056}, {2330, 3601}, {2647, 3340}, {2648, 3577}, {2854, 2948}, {3094, 3097}, {3158, 3550}, {3216, 3338}, {3218, 3240}, {3589, 3624}
X(3751) = reflection of X(1) in X(6)
X(3751) = {X(1),X(6)}-harmonic conjugate of X(16475)
X(3752) lies on these lines: {1, 474}, {2, 37}, {3, 1104}, {6, 57}, {31, 1155}, {34, 1466}, {38, 210}, {39, 1212}, {40, 1191}, {42, 244}, {43, 518}, {44, 63}, {55, 614}, {56, 197}, {65, 1193}, {72, 3216}, {81, 88}, {142, 2092}, {165, 3052}, {171, 1054}, {200, 3242}, {216, 1108}, {226, 1086}, {241, 2275}, {386, 942}, {517, 995}, {595, 3579}, {673, 893}, {940, 1100}, {958, 988}, {960, 978}, {980, 1107}, {1122, 2347}, {1201, 3057}, {1203, 3336}, {1210, 1834}, {1333, 1817}, {1403, 3185}, {1421, 3256}, {1435, 2331}, {1455, 1470}, {1616, 1697}, {1638, 3310}, {1738, 2886}, {1764, 2300}, {1876, 3192}, {3293, 3555}
X(3752) = isotomic conjugate of X(32017)
X(3752) = complementary conjugate of X(21244)
X(3752) = polar conjugate of isogonal conjugate of X(22344)
X(3752) = complement of X(312)
X(3752) = crosssum of X(6) and X(9)
X(3752) = crosspoint of X(2) and X(57)
X(3753) lies on these lines: {1, 474}, {2, 392}, {5, 1519}, {7, 3421}, {8, 443}, {9, 2093}, {10, 12}, {21, 3579}, {37, 1018}, {40, 405}, {46, 958}, {57, 956}, {80, 3255}, {142, 1145}, {318, 1148}, {354, 519}, {355, 377}, {374, 2835}, {375, 2390}, {404, 1385}, {406, 1902}, {475, 1829}, {518, 599}, {551, 2802}, {856, 1214}, {860, 1824}, {936, 3340}, {960, 1698}, {993, 1155}, {997, 2099}, {999, 3306}, {1012, 3359}, {1125, 3057}, {1378, 2362}, {1439, 1441}, {1737, 2886}, {1845, 3040}, {1861, 1905}, {2550, 3419}, {2650, 3214}
X(3753) = X(2)-of-inverse(n(hexyl triangle))
Let OA be the circle centered at the A-vertex of the 2nd Conway triangle and passing through A; define OB and OC cyclically. X(3754) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(3754) lies on these lines: {1, 88}, {5, 2800}, {8, 2891}, {10, 12}, {21, 484}, {40, 1006}, {46, 993}, {80, 2475}, {140, 517}, {354, 3244}, {355, 2801}, {474, 2099}, {518, 3626}, {519, 942}, {551, 3057}, {860, 1825}, {997, 3340}, {1193, 1739}, {2295, 3125}, {2650, 3293}, {2975, 3336}, {3555, 3625}
X(3754) = X(389) of Fuhrmann triangle
X(3755) lies on these lines: {1, 142}, {4, 2331}, {6, 516}, {10, 37}, {40, 387}, {42, 226}, {43, 3452}, {65, 1439}, {386, 946}, {497, 2999}, {515, 990}, {519, 599}, {528, 1386}, {908, 3240}, {1001, 3008}, {1886, 2266}, {2177, 3011}
Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Nagel line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, B' = Mc∩Ma, C' = Ma∩Mb. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Nagel line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(3756); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)
X(3756) lies on these lines: {1, 1145}, {2, 1280}, {8, 1120}, {11, 244}, {88, 149}, {106, 952}, {513, 1357}, {528, 1054}, {1015, 1146}, {1616, 1788}
X(3756) = centroid of (degenerate) pedal triangle of X(106)
X(3756) = crossdifference of every pair of points on line X(101)X(1293)
X(3757) lies on these lines: {1, 2}, {25, 92}, {31, 894}, {55, 75}, {69, 3475}, {183, 322}, {192, 968}, {238, 1215}, {274, 2223}, {312, 1001}, {314, 2346}, {321, 1621}, {333, 518}, {675, 831}, {726, 846}, {1104, 1220}, {1281, 1283}, {1402, 1441}, {1909, 1965}
Central Triangles and More Combos
Many triangle centers can be defined (as just above) by the form Xcom(nT), where T denotes a central triangle. In order to present such centers, it is helpful to introduce the notation T(f(a,b,c), g(b,c,a)) for central triangles. Following TCCT, pages 53-54, suppose that each of f(a,b,c) and g(a,b,c) is a center-function or the zero function, and that one of these three conditions holds:
the degree of homogeneity of g equals that of f;
f is the zero function and g is not the zero function;
g is the zero function and f is not the zero function.
There are two cases to be considered: If g(a,b,c)=g(a,c,b), then the central triangle T(f(a,b,c), g(b,c,a)) is defined by the following 3x3 matrix (whose rows give homogeneous coordinates for the A-, B-, C- vertices, respectively):
f(a,b,c) g(b,c,a)
g(c,a,b)
g(a,b,c) f(b,c,a)
g(c,a,b)
g(a,b,c) g(b,c,a)
f(c,a,b)
If g(a,b,c) is not equal to g(a,c,b), then the central triangle T(f(a,b,c),g(b,c,a)) is defined by the following 3x3 matrix:
f(a,b,c) g(b,c,a)
g(c,b,a)
g(a,c,b) f(b,c,a)
g(c,a,b)
g(a,b,c) g(b,a,c)
f(c,a,b)
If the homogenous coordinates are chosen to be barycentric, as they are for present purposes, then the new notation for central triangles is typified by these examples:
Medial triangle = T(1, 0)
Anticomplementary triangle = T(-1, 1)
Incentral triangle = T(a, 0)
Excentral triangle = T(-a, b)
Euler triangle = T(2(b4 + c4 -
b2c2 - c2a2 -
a2b2), (b2 + c2 -
a2)(a2 + b2 - c2))
2nd Euler triangle = T(2a2(b4 +
c4 - 2b2c2 - c2a2
- a2b2), (a2 - b2 +
c2)(a4 + b4 + c4 -
2a2c2 - 2b2c2))
3rd Euler triangle = T((b - c)2, c2 -
a2 - bc)
4th Euler triangle = T((b + c)2, c2 -
a2 + bc)
5th Euler triangle = T(2(b2 + c2),
b2 + c2 - a2 )
The vertices of the five Euler triangles lie on the nine-point circle. (See C. Kimberling, "Twenty-one points on the nine-point circle," Mathematical Gazette 92 (2008) 29-38.) Randy Hutson observes (January 8, 2015) that the 2nd Euler triangle is the complement of the circumorthic triangle, the 2nd Euler triangle is the reflection of the orthic triangle in X(5); the 3rd Euler triangle is the complement of the 1st circumperp triangle, the 4th Euler triangle is the complement of the 2nd circumperp triangle; the 4th Euler triangle the triangle whose sidelines are the radical axes of each of the Odehnal tritangent circles (defined at X(6176)) and the corresponding excircle; the 5th Euler triangle is the complement of the circummedial triangle, and the vertices of the 5th Euler triangle are the intersections, other than midpoints of the sides of ABC, of the nine-point circle and the medians.
The notation T(f(a,b,c),g(b,c,a)) can be used to define several more triangles; in each case, two of the perspectivities can be used to construct the triangle.
T(bc, b2) is perspective to ABC with perspector X(6); other such pairs: incentral, X(192), excentral, X(1045); tangential, X(6); anticomplementary, X(192)
T(-bc, b2) is perspective to ABC with perspector X(6); incentral, X(2); tangential, X(6); T(bc,b2), X(6)
T(bc, c2) is perspective to ABC with perspector X(76); anticomplementary, X(8); 4th Brocard, X(76), Fuhrmann, X(8); cevian(X(350)),X(1)
T(-bc, c2) is perspective to ABC with perspector X(76); cevian(X(321)),X(75); anticevian(X(8)), X(2)
T(a2, a2 - b2) is perspective to the medial triangle with perspector X(3); tangential , X(3); symmedial, X(2); cevian(X(287)), X(98)
T(a2, a2 - c2) is perspective to the tangential triangle with perspector X(25); orthic , X(25); medial, X(6); 1st Lemoine, X(1383); circummedial, X(251)
T(a2, a2 + b2) is perspective to ABC with perspector X(83); medial, X(6); 1st Neuberg, X(182)
T(a2, a2 + c2) is perspective to ABC with perspector X(141); medial, X(3); 1st Neuberg, X(182); T(a2,a2 - c2), X(2); cevian(X(69)), X(6); cevian(X(76)), X(2); cevian(X(3313)),X(39)
The triangles IT1 and IT2 are triply perspective to ABC (as are T1 and T3). Order-label IT1 as A'B'C'. The three perspectivities are then given by
AA'∩BB'∩CC' = X(3862)
AB'∩BC'∩CA' = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =
ab(a2 - bc)(c2 - ab)(b2 + bc +
c2)
AC'∩BA'∩CB' = f(a,c,b) : f(b,a,c) : f(c,b,a)
Peter Moses noted (12/23/2011) that T(bc, b2) is triply perspective to ABC. With T(bc, b2) order-labeled as A'B'C', the three perspectivities with perspectors are as follows:
AA'∩BB'∩CC' = X(6); AB'∩BC'∩CA' = P(6);
AC'∩BA'∩CB' = U(6). (The notation P(k) and U(k) refers to a
bicentric pair of points; see Tables at the top of ETC.)
Likewise, T(bc, c2) is triply perspective to ABC, with
perspectors: X(76), P(10), and U(10).
(This section was added to ETC on 12/23/2011.)
X(3758) lies on these lines: {1, 190}, {2, 44}, {6, 75}, {7, 3618}, {8, 1992}, {9, 86}, {41, 662}, {43, 2234}, {81, 312}, {83, 3673}, {85, 651}, {87, 1964}, {192, 1100}, {193, 319}, {256, 749}, {291, 751}, {318, 648}, {524, 3661}, {594, 3629}, {597, 1086}, {645, 1509}, {750, 3570}, {872, 1740}, {1449, 3644}, {1760, 2285}, {3589, 3662}
X(3758) = complement of X(4741)
X(3758) = anticomplement of X(17237)
X(3758) = {X(6),X(75)}-harmonic conjugate of X(3759)
X(3759) lies on these lines: 1, 872}, {2, 319}, {6, 75}, {7, 1992}, {8, 1386}, {43, 1964}, {44, 192}, {86, 1449}, {87, 2234}, {145, 344}, {190, 1743}, {193, 320}, {256, 751}, {273, 648}, {291, 749}, {312, 3187}, {524, 3662}, {594, 597}, {604, 662}, {664, 1445}, {1043, 1453}, {1086, 3629}, {1740, 3248}, {1760, 2082}, {3589, 3661}
X(3759) = complement of X(17373)
X(3759) = anticomplement of X(17231)
X(3759) = {X(6),X(75)}-harmonic conjugate of X(3758)
X(3760) lies on these lines: {1, 76}, {33, 1235}, {35, 183}, {36, 1975}, {69, 1479}, {75, 1089}, {172, 3734}, {274, 3624}, {304, 1111}, {312, 1930}, {313, 3293}, {315, 3583}, {339, 1062}, {384, 609}, {538, 2275}, {668, 3632}
X(3760) = isotomic conjugate of X(749)
X(3760) = {X(1),X(76)}-harmonic conjugate of X(3761)
X(3761) lies on these lines: {1, 76}, {34, 1235}, {35, 1975}, {36, 183}, {43, 310}, {69, 1478}, {75, 537}, {85, 1930}, {274, 1698}, {304, 1089}, {315, 3585}, {339, 1060}, {385, 609}, {538, 2276}, {693, 1022}, {975, 1228}, {1018, 3729}, {1914, 3734}
X(3761) = isotomic conjugate of X(751)
X(3761) = {X(1),X(76)}-harmonic conjugate of X(3760)
The trilinear polar of X(3762) passes through X(1647). (Randy Hutson, June 7, 2019)
X(3762) lies on these lines: {1, 3716}, {4, 2814}, {10, 2254}, {80, 3738}, {153, 2827}, {190, 646}, {514, 661}, {659, 2787}, {900, 1145}, {918, 1086}, {1121, 2481}
X(3762) = isogonal conjugate of X(32665)
X(3762) = isotomic conjugate of X(3257)
X(3762) = complement of X(21222)
X(3762) = anticomplement of X(3960)
X(3762) = X(663)-of-inner-Garcia triangle
X(3762) = crossdifference of every pair of points on line X(31)X(692)
X(3763) lies on these lines: {2, 6}, {3, 2916}, {5, 1350}, {10, 3242}, {49, 182}, {66, 154}, {125, 2930}, {140, 1352}, {159, 1853}, {316, 2076}, {381, 3098}, {511, 1656}, {518, 1698}, {631, 1503}, {632, 3564}, {1086, 2345}, {1125, 3416}, {1386, 3624}, {2097, 3452}
X(3764) lies on these lines: {1, 3122}, {2, 2228}, {6, 560}, {31, 2245}, {39, 3271}, {42, 2183}, {75, 256}, {291, 751}, {320, 982}, {511, 2274}, {573, 1918}, {674, 869}, {1837, 2310}, {1964, 2277}, {2275, 3248}
X(3764) = isogonal conjugate of isotomic conjugate of X(25760)
X(3765) lies on these lines: {2, 330}, {4, 8}, {6, 313}, {75, 1654}, {76, 239}, {668, 3661}, {730, 869}, {894, 3596}, {1230, 3187}
X(3765) = anticomplement of X(37596)
X(3766) lies on these lines: {2, 665}, {75, 900}, {150, 928}, {244, 1111}, {316, 512}, {334, 876}, {335, 918}, {513, 3261}, {514, 661}, {668, 891}, {885, 2481}
X(3766) = isogonal conjugate of X(34067)
X(3766) = isotomic conjugate of X(660)
X(3766) = complement of polar conjugate of isogonal conjugate of X(23188)
X(3766) = anticomplementary conjugate of X(39353)
X(3766) = anticomplement of X(665)
X(3766) = crossdifference of every pair of points on line X(31)X(1911)
X(3767) is the center of the inellipse that is the barycentric square of the orthic axis. The Brianchon point (perspector) of the inellipse is X(393). (Randy Hutson, October 15, 2018)
X(3767) lies on these lines: {2, 39}, {3, 230}, {4, 32}, {5, 6}, {20, 187}, {25, 2353}, {30, 3053}, {53, 1598}, {69, 626}, {148, 3552}, {172, 1478}, {193, 625}, {216, 3547}, {232, 3542}, {235, 2138}, {315, 385}, {382, 1384}, {388, 2242}, {393, 800}, {427, 1184}, {497, 2241}, {498, 2276}, {499, 2275}, {574, 631}, {609, 3585}, {637, 1504}, {638, 1505}, {946, 1572}, {1015, 3086}, {1204, 1562}, {1368, 1611}, {1479, 1914}, {1500, 3085}, {1506, 3090}, {1714, 2238}, {1834, 2271}, {1899, 2450}, {1976, 2909}, {2023, 3095}, {3054, 3526}
X(3767) = complement of X(3926)
X(3767) = anticomplement of X(3788)
X(3767) = polar conjugate of X(34405)
X(3767) = {X(5),X(6)}-harmonic conjugate of X(2548)
X(3768) lies on these lines: {44, 513}, {190, 646}, {292, 3572}, {573, 2827}, {813, 3257}, {888, 2667}, {1015, 1960}
X(3768) = isogonal conjugate of X(4607)
X(3769) lies on these lines: {2, 1386}, {31, 312}, {43, 1964}, {55, 1999}, {75, 171}, {100, 2352}, {145, 2646}, {190, 1707}, {239, 1376}, {333, 612}, {740, 3550}, {940, 3757}, {1155, 3210}, {1748, 1897}, {2162, 2235}, {3052, 3685}
X(3770) lies on these lines: {6, 76}, {37, 1655}, {75, 1654}, {81, 1230}, {99, 1030}, {239, 1269}, {256, 714}, {274, 1213}, {310, 2238}, {313, 894}, {314, 524}, {319, 321}, {350, 1100}, {384, 2220}, {385, 1333}, {538, 2092}, {594, 668}, {1228, 2303}, {1330, 3416}
X(3771) lies on these lines: {1, 2}, {55, 2887}, {345, 726}, {752, 3052}, {1575, 3694}
X(3772) is the center of the inconic that is the polar conjugate of the isotomic conjugate of the incircle. (Randy Hutson, October 15, 2018)
X(3772) lies on these lines: {1, 442}, {2, 37}, {4, 1104}, {6, 226}, {11, 33}, {27, 1333}, {31, 1836}, {44, 329}, {55, 3011}, {56, 225}, {57, 1020}, {72, 1714}, {172, 379}, {220, 3008}, {223, 3554}, {278, 393}, {387, 3487}, {475, 3086}, {497, 1279}, {516, 3052}, {528, 3749}, {851, 2352}, {899, 2318}, {908, 2911}, {946, 1191}, {1072, 3428}, {1125, 2049}, {1201, 2654}, {1284, 3185}, {1329, 1722}, {1376, 1738}, {1612, 3651}, {2218, 3145}, {2887, 3416}, {2999, 3553}, {3191, 3216}, {3434, 3744}
X(3772) = complement of X(345)
X(3772) = barycentric product X(1)*X(17861)
X(3772) = polar conjugate of X(34406)
X(3772) = trilinear pole of the polar, wrt the Fuhrmann circle, of the perspector of the Fuhrmann circle
X(3773) lies on these lines: {8, 238}, {10, 37}, {141, 726}, {306, 1215}, {313, 1089}, {319, 1757}, {321, 2887}, {519, 597}, {752, 3416}, {984, 3661}, {3703, 3741}
X(3774) lies on these lines: {10, 37}, {39, 518}, {42, 3121}, {100, 1908}, {192, 1921}, {213, 872}, {984, 2276}, {1045, 3508}, {1574, 3739}
X(3775) lies on these lines: {1, 319}, {10, 141}, {11, 1211}, {238, 1654}, {594, 726}, {984, 3661}, {1100, 1125}, {2308, 3578}
X(3775) = reflection of X(1100) in X(1125)
X(3775) = complement of X(4649)
X(3775) = QA-P21 (Reflection of QA-P16 in QA-P1) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/49-qa-p21.html)
X(3776) lies on these lines: {241, 514}, {321, 693}
X(3776) = isotomic conjugate of X(4621)
X(3777) lies on these lines: {10, 514}, {513, 663}, {659, 905}, {891, 1734}
X(3778) lies on these lines: {6, 560}, {10, 3728}, {31, 579}, {37, 3122}, {39, 2309}, {42, 181}, {63, 1716}, {71, 3747}, {75, 700}, {141, 2228}, {142, 244}, {209, 3725}, {256, 291}, {313, 714}, {573, 2209}, {674, 1964}, {723, 789}, {749, 751}, {869, 2277}, {982, 2887}, {1445, 1758}, {1918, 2245}, {2085, 3670}, {2275, 3056}, {3009, 3688}, {3123, 3663}
X(3778) = isogonal conjugate of isotomic conjugate of X(2887)Let A'B'C' be the extangents triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B" and C" cyclically. Let A* be the trilinear pole, wrt A'B'C', of line B"C", and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(3779). (Randy Hutson, December 2, 2017)
X(3779) lies on these lines: {1, 3688}, {6, 31}, {7, 8}, {19, 1843}, {40, 511}, {56, 1818}, {181, 200}, {210, 966}, {269, 1362}, {291, 1740}, {579, 2223}, {869, 2277}, {968, 3690}, {1400, 2340}, {1402, 3190}, {1486, 2911}, {1631, 2174}, {1654, 3681}, {1721, 2808}, {1743, 3271}, {1964, 2275}, {2093, 2810}, {2294, 3728}, {3169, 3174}
X(3779) = homothetic center of extangents triangle and reflection of intangents triangle in X(6)
X(3779) = perspector of extangents triangle and Mandart-excircles triangle
X(3779) = X(7)-of-extangents-triangle if ABC is acute
X(3780) lies on these lines: {1, 1573}, {6, 8}, {37, 3691}, {39, 3293}, {42, 1107}, {43, 2275}, {44, 1334}, {72, 3727}, {145, 2176}, {172, 3684}, {213, 519}, {239, 1909}, {518, 3721}, {956, 2271}, {992, 1100}, {1015, 3216}, {1230, 3187}, {1475, 1575}, {1724, 2241}, {1743, 3208}, {3230, 3244}, {3555, 3726}
X(3781) lies on these lines: {1, 3688}, {3, 48}, {9, 511}, {35, 2175}, {51, 3305}, {58, 2273}, {63, 295}, {69, 72}, {182, 2323}, {200, 3033}, {209, 940}, {220, 1350}, {238, 3056}, {386, 2300}, {517, 2550}, {674, 1001}, {936, 970}, {984, 1469}, {991, 3730}, {2276, 3736}, {2979, 3219}, {3098, 3220}
X(3782) lies on these lines: {1, 30}, {2, 45}, {5, 3670}, {7, 940}, {11, 982}, {12, 986}, {38, 2886}, {75, 1211}, {141, 321}, {210, 1738}, {226, 1465}, {306, 536}, {312, 3662}, {320, 1999}, {516, 3744}, {524, 3187}, {726, 2887}, {908, 3752}, {1266, 3687}, {1699, 3677}, {3242, 3434}
X(3782) = complement of X(32933)
X(3782) = polar conjugate of isogonal conjugate of X(23154)
X(3782) = pole of de Longchamps line wrt incircle
X(3783) lies on these lines: {1, 2}, {69, 1740}, {72, 695}, {87, 193}, {100, 2239}, {238, 1914}, {291, 518}, {319, 1964}, {320, 2234}, {350, 740}, {668, 730}, {672, 1282}, {752, 2230}, {788, 1491}, {984, 2276}, {1654, 2309}
X(3784) lies on these lines: {1, 1401}, {3, 73}, {36, 1397}, {51, 3306}, {57, 511}, {63, 295}, {165, 1362}, {171, 1469}, {182, 2003}, {200, 2810}, {394, 1473}, {517, 3474}, {982, 3056}, {1040, 1364}, {1350, 1407}, {2979, 3218}
X(3785) lies on these lines: {2, 32}, {3, 69}, {4, 183}, {20, 76}, {39, 193}, {99, 3522}, {140, 1007}, {141, 3053}, {187, 3620}, {217, 394}, {316, 3091}, {317, 3088}, {325, 631}, {376, 1975}, {1384, 3619}
X(3785) = isotomic conjugate of X(8801)
X(3785) = anticomplement of X(2548)
X(3786) lies on these lines: {1, 3728}, {8, 314}, {9, 21}, {29, 1827}, {58, 1757}, {72, 894}, {81, 612}, {86, 518}, {210, 333}, {228, 3219}, {511, 1654}, {869, 984}, {960, 1043}, {1045, 2292}
X(3787) lies on these lines: {32, 394}, {39, 3051}, {51, 3231}, {184, 187}, {323, 1627}, {511, 1196}, {800, 3289}, {1194, 2979}, {1351, 1611}, {1501, 3292}, {1692, 1993}, {2056, 2076}, {3053, 3167}, {3060, 3291}}
X(3788) lies on these lines: {2, 39}, {3, 114}, {4, 625}, {5, 3734}, {32, 325}, {69, 1692}, {115, 1975}, {140, 141}, {187, 315}, {316, 3552}, {754, 3053}, {1007, 2548}, {1078, 3314}
X(3788) = complement of X(3767)
X(3788) = anticomplement of X(7886)
X(3789) lies on these lines: {1, 1573}, {2, 210}, {8, 350}, {9, 1282}, {10, 2140}, {120, 141}, {984, 2276}, {1001, 2280}, {1211, 2886}, {1654, 3056}, {3452, 3741}
X(3790) lies on these lines: {8, 9}, {10, 192}, {11, 312}, {190, 3416}, {341, 3704}, {495, 3695}, {726, 3662}, {984, 3661}, {1278, 1738}, {3596, 3701}, {3681, 3690}
X(3791) lies on these lines: {1, 333}, {6, 1215}, {10, 3745}, {31, 740}, {43, 1964}, {171, 239}, {238, 1999}, {321, 2308}, {519, 3703}, {1386, 3741}, {3244, 3748}
X(3792) lies on these lines: {1, 674}, {31, 2979}, {36, 2245}, {238, 511}, {320, 758}, {748, 3060}, {788, 1491}, {984, 1469}, {1216, 3072}, {1944, 2607}, {2175, 3098}
X(3793) lies on these lines: {3, 193}, {30, 148}, {32, 141}, {69, 1384}, {187, 524}, {194, 548}, {230, 625}, {441, 3580}, {574, 3629}, {2080, 3564}
X(3794) lies on these lines: {1, 21}, {2, 51}, {29, 1828}, {86, 1431}, {210, 333}, {404, 1730}, {1010, 3753}, {1330, 2478}, {1403, 3286}, {3056, 3705}
X(3795) lies on these lines: {1, 1575}, {2, 740}, {6, 2108}, {43, 55}, {100, 985}, {291, 1002}, {518, 3097}, {678, 3240}, {984, 2276}
X(3796) lies on these lines: {2, 154}, {3, 49}, {6, 22}, {25, 182}, {206, 1619}, {458, 1629}, {1184, 1691}, {1350, 1993}, {3051, 3053}
X(3796) = isogonal conjugate of X(8801)
X(3796) = isotomic conjugate of complement of X(39025)
X(3796) = crossdifference of every pair of points on line X(826)X(2501)
X(3797) lies on these lines: {2, 37}, {190, 742}, {194, 304}, {238, 239}, {335, 726}, {518, 2113}, {824, 3250}, {984, 3661}
X(3798) lies on these lines: {239, 514}, {812, 3676}, {900, 2487}, {2254, 3667}, {2488, 3309}, {2786, 3239}
X(3799) lies on these lines: {100, 101}, {190, 513}, {668, 891}, {2802, 3679}, {2809, 3681}, {3507, 3747}
X(3800) lies on these lines: {30, 511}, {419, 2501}, {879, 3527}, {1734, 3004}
X(3800) = isogonal conjugate of X(907)
X(3801) lies on these lines: {514, 659}, {523, 656}, {826, 1089}
X(3802) lies on these lines: {1, 39}, {238, 239}, {1682, 3022}
X(3803) lies on these lines: {36, 238}, {649, 3309}, {650, 830}
X(3804) lies on these lines: {187, 237}, {1499, 3265}, {2525, 3566}
X(3805) is the infinite point of perspectrices of every pair of {ABC, Gemini triangle 32, Gemini triangle 34}. (Randy Hutson, January 15, 2019)
X(3805) lies on these lines: {30, 511}, {38, 661}
X(3805) = isogonal conjugate of X(30670)
X(3805) = crossdifference of every pair of points on line X(6)X(256)
X(3805) = ideal point of PU(35)
X(3806) lies on these lines: {230, 231}, {826, 2474}
X(3807) lies on these lines: {100, 190}, {514, 668}
X(3808) lies on this line: {30, 511}
X(3808) = isogonal conjugate of X(8684)
X(3808) = crossdifference of every pair of points on line X(6)X(983)
X(3809) lies on this line: {1, 2}
X(3810) lies on this line: {30, 511}
X(3810) = isogonal conjugate of X(8685)
X(3811) lies on these lines: {1, 2}, {3, 518}, {4, 2900}, {6, 3694}, {9, 943}, {12, 3419}, {21, 3681}, {33, 2901}, {35, 63}, {37, 2271}, {40, 758}, {46, 100}, {47, 1331}, {55, 72}, {56, 3555}, {58, 1792}, {65, 3689}, {84, 2801}, {101, 1973}, {158, 1897}, {169, 3684}, {210, 405}, {214, 1420}, {218, 3693}, {354, 474}, {392, 3303}, {404, 3338}, {443, 3475}, {500, 524}, {516, 1490}, {595, 3749}, {726, 990}, {908, 1479}, {912, 1158}, {942, 1376}, {954, 3059}, {956, 2646}, {960, 3295}, {993, 3601}, {1046, 3550}, {1066, 1818}, {1389, 3680}, {1612, 1723}, {1706, 3754}, {2136, 2802}, {2177, 2292}, {2321, 3553}, {2550, 3487}, {2551, 3488}, {2975, 3612}, {3243, 3333}, {3421, 3486}, {3697, 3711}
X(3811) = intouch-to-ABC barycentric image of X(3)
X(3812) lies on these lines: {1, 474}, {2, 65}, {6, 1722}, {7, 2551}, {8, 354}, {9, 3339}, {10, 141}, {21, 1155}, {29, 1888}, {37, 986}, {40, 1001}, {44, 1046}, {46, 405}, {56, 3306}, {57, 958}, {72, 1698}, {75, 3714}, {140, 517}, {171, 1104}, {226, 1329}, {241, 1254}, {377, 1837}, {392, 3624}, {404, 2646}, {442, 1737}, {452, 3474}, {475, 1905}, {595, 3246}, {758, 3634}, {899, 2650}, {908, 3649}, {938, 2550}, {956, 3338}, {1210, 2886}, {1738, 1834}, {1836, 2478}, {1858, 2476}, {1875, 1940}, {2295, 3290}, {2496, 3309}, {2802, 3636}, {3057, 3616}, {3452, 3671}, {3555, 3679}
X(3812) = centroid of ABCX(65)
X(3812) = Kosnita(X(65),X(2)) point
X(3812) = complement of X(960)
X(3812) = perspector of Gemini triangle 21 and cross-triangle of Gemini triangles 21 and 23
X(3813) lies on these lines: {1, 442}, {2, 3303}, {3, 528}, {4, 529}, {5, 519}, {8, 11}, {10, 496}, {12, 145}, {21, 3058}, {56, 3434}, {149, 2975}, {377, 3304}, {452, 497}, {495, 3244}, {499, 3035}, {518, 946}, {522, 596}, {535, 3627}, {943, 1001}, {956, 1479}, {1213, 3169}, {1376, 3086}, {1484, 2802}, {1698, 2136}, {2476, 3241}, {3158, 3624}, {3189, 3616}, {3679, 3680}, {3702, 3703}, {3704, 3705}
X(3814) lies on these lines: {2, 36}, {4, 2077}, {5, 10}, {11, 519}, {12, 1125}, {30, 3035}, {100, 3583}, {115, 1575}, {119, 214}, {121, 3259}, {381, 1376}, {403, 1861}, {404, 3585}, {427, 1878}, {442, 1155}, {484, 1698}, {495, 551}, {496, 3244}, {498, 2478}, {499, 3436}, {516, 1532}, {758, 908}, {958, 1656}, {1107, 1506}, {1621, 3584}, {1739, 3120}, {2392, 3142}, {2550, 3545}, {2551, 3090}, {3836, 4129}
X(3814) = isotomic conjugate of isogonal conjugate of X(20962)
X(3814) = polar conjugate of isogonal conjugate of X(22059)
X(3814) = {X(5),X(10)}-harmonic conjugate of X(25639)
X(3814) = nine-point-circle-inverse of X(10)
X(3814) = complement of X(36)
X(3814) = complementary conjugate of X(214)
X(3814) = inverse-in-excircles-radical-circle of X(970)
X(3815) lies on these lines: {2, 6}, {3, 2548}, {5, 39}, {11, 2276}, {12, 2275}, {25, 160}, {30, 574}, {32, 140}, {53, 232}, {187, 549}, {216, 1368}, {233, 1196}, {251, 2965}, {262, 1513}, {381, 2549}, {495, 1015}, {496, 1500}, {543, 3363}, {566, 858}, {594, 3705}, {631, 1285}, {1107, 1329}, {1575, 2886}, {1595, 3199}, {2207, 3541}, {2963, 3108}
X(3815) = crosspoint of X(2) and X(262)
X(3815) = crosssum of X(6) and X(182)
X(3815) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,230), (5,39,5254), (395,396,597)
X(3815) = X(6)-of-5th-Euler triangle
X(3815) = inverse-in-{circumcircle, nine-point circle}-inverter of X(352)
X(3815) = insimilicenter of nine-point and (1/2)-Moses circles; the exsimilicenter is X(5254)
X(3816) lies on these lines: {1, 1329}, {2, 11}, {5, 515}, {8, 1997}, {10, 496}, {12, 1388}, {38,1647}, {56, 2478}, {142, 1538}, {226, 3660}, {354, 908}, {392, 1737}, {405, 499}, {442, 3586}, {474, 1479}, {495, 551}, {518, 3452}, {529, 999}, {958, 3086}, {960, 1210}, {978, 1834}, {982, 3756}, {1532, 3576}, {1836, 3306}, {3304, 3436}
X(3816) = complement of X(1376)
In the plane of a triangle ABC, let I = X(1) and
L = line through midpoint of CA perpendicular to BI
L' = line through midpoint of AB perpendicular to CI
L'' = line through midpoint of AI perpendicular to BC
The lines L, L', L'' concur in a point, A'; define B' and C' cyclically. Ten X(3817) = X(2)-of-A'B'C'.
Also, A'B'C' = complement of the excentral triangle, and A'B'C' = extraversion triangle of X(10). (Randy Hutson, September 14, 2016)
Let A'B'C' be the intouch triangle, and AaAbAc, BaBbBc, CaCbCc the A-, B-, and C-extouch triangles. Let Ab' = B'C'∩BcBa and Ac' = B'C'∩CaCb. Define Bc' and Ba', Ca', Cb' cyclically. Then X(3817) is the centroid of {Ab',Ac',Bc',Ba',Ca',Cb'} is X(3817). The six points lie on a common ellipse. (Randy Hutson, September 14, 2016)
Let Wa be the inverter of the B- and C-excircles, and define Wb and Wc cyclically; see X(5577). Let Ia be Wa-inverse of the incircle, and define Ib and Ic cyclically. The radical center of circles Ia, Ib, Ic is X(3817). (Randy Hutson, September 14, 2016)
Let A'B'C' be the circumcevian triangle of X(1). Let L1 be the Simson line of A', and define L2 and L3 cyclically. Let A'' = L2∩L3, and define B'' and C'' cyclically. Then X(3817) = X(2)-of-A''B''C''. (Angel Montesdeoca, June 28, 1017)
X(3817) is the centroid of AbAcBcBaCaCb in the construction of the perspeconic of the Ursa-minor and Ursa-major triangles. (Randy Hutson, June 27, 2018)
X(3817) lies on these lines: {1, 3091}, {2, 165}, {4, 1125}, {5, 10}, {11, 118}, {20, 3624}, {40, 3090}, {57, 1776}, {142, 1538}, {262, 726}, {355, 3244}, {381, 515}, {519, 3545}, {546, 1385}, {748, 1754}, {908, 3681}, {944, 3636}, {962, 1698}, {971, 3742}, {1210, 3671}, {1323, 2898}, {1482, 3625}, {1709, 3306}, {3057, 3614}, {3579, 3628}
X(3817) = complement of X(165)
X(3817) = centroid of 3rd Euler triangle
X(3817) = homothetic center of 3rd Euler triangle and extraversion triangle of X(10)
X(3818) lies on these lines: {2, 1495}, {3, 2916}, {4, 69}, {5, 182}, {6, 13}, {30, 141}, {98, 3407}, {125, 1995}, {147, 262}, {302, 383}, {303, 1080}, {343, 428}, {376, 3619}, {382, 1350}, {403, 1974}, {546, 576}, {575, 3091}, {1469, 3585}, {1539, 2781}, {1619, 1853}, {3056, 3583}, {3060, 3410}, {3543, 3620}, {3545, 3618}
X(3818) = intersection of Fermat axes of ABC and 1st Brocard triangle
X(3818) = inverse-in-Kiepert-hyperbola of X(5309)
X(3818) = X(7)-of-Ehrmann-vertex-triangle if ABC is acute
X(3818) = X(6)-of-Ehrmann-mid-triangle
X(3818) = {X(13),X(14)}-harmonic conjugate of X(5309)
X(3818) = perspector of Ehrmann vertex-triangle and Ehrmann mid-triangle
X(3818) = intersection, other than X(4), of the Brocard circles of the 1st and 2nd Ehrmann circumscribing triangles
Let A'B'C' be the medial triangle. Let Ab and Ac be the circumcircle intercepts of line B'C'. Define Bc and Ca cyclically, and define Ba and Cb cyclically. X(3819) is the centroid of AbAcBcBaCaCb. (Randy Hutson, July 31 2018)
X(3819) lies on these lines: {2, 51}, {3, 64}, {25, 3098}, {39, 1613}, {52, 3526}, {122, 129}, {140, 389}, {141, 1368}, {182, 394}, {185, 3523}, {187, 1915}, {210, 2810}, {474, 970}, {575, 1993}, {674, 3742}, {960, 2390}, {982, 3688}, {984, 1401}, {1194, 3231}, {1196, 3094}, {1843, 3619}, {3218, 3690}, {3533, 3567}
X(3819) = complement of X(51)
X(3819) = X(2) of polar triangle of complement of polar circle
X(3820) lies on these lines: {2, 495}, {3, 1603}, {5, 10}, {8, 496}, {9, 119}, {11, 3679}, {12, 57}, {30, 1376}, {116, 121}, {140, 958}, {210, 1737}, {329, 442}, {355, 936}, {381, 2550}, {474, 3436}, {549, 993}, {908, 3753}, {952, 997}, {1484, 3036}, {1596, 1861}, {2885, 3454}
X(3821) lies on these lines: {1, 2896}, {2, 846}, {3, 142}, {10, 75}, {45, 3634}, {86, 1326}, {141, 740}, {182, 2792}, {226, 1403}, {519, 599}, {579, 1761}, {752, 1386}, {1352, 2784}, {1698, 3729}, {2085, 3670}, {2092, 3454}, {2887, 3666}, {3094, 3735}
X(3821) = complement of X(3923)
X(3821) = anticomplement of X(24295)
X(3821) = X(10) of 1st Brocard triangle
X(3821) = 1st-Brocard-isogonal conjugate of X(24267)
X(3821) = 1st-Brocard-isotomic conjugate of X(24275)
Suppose that ABC is an acute triangle. Let LA be the circle that is externally tangent to the nine-point circle and to the sidelines AB and AC. Define LB and LC cyclically. The three circles are here named the Odehnal tritangent circles. Their radical center is X(3822). The center of the Apollonian circle of LA, LB, LC is X(6167). (Boris Odehnal, "A Triad of Tritangent Circles, Journal for Geometry and Graphics 18 (2014) no. 1, 61-71)
X(3822) lies on these lines: {1, 2476}, {2, 36}, {5, 515}, {10, 12}, {11, 551}, {21, 3585}, {35, 2475}, {37, 115}, {63, 1698}, {100, 3584}, {116, 119}, {377, 498}, {381, 1001}, {495, 519}, {496, 3636}, {1329, 3634}, {1621, 3583}, {2901, 3178}
X(3823) lies on these lines: {2, 1279}, {10, 141}, {44, 966}, {238, 1698}, {536, 1738}, {1086, 3717}, {2887, 3740}, {3246, 3634}
X(3823) = complement of X(1279)
X(3823) = anticomplement of X(24715)
X(3824) lies on these lines: {5, 142}, {30, 1125}, {79, 3683}, {442, 942}, {527, 3634}, {1698, 3715}, {3454, 3739}
X(3825) lies on these lines: {2, 35}, {5, 515}, {10, 11}, {12, 551}, {37, 1506}, {56, 535}, {404, 3583}, {495, 3636}, {496, 519}, {499, 993}, {758, 1210}, {946, 3754}, {1001, 1656}, {2476, 3624}, {2886, 3634}, {2975, 3582}, {3452, 3678}
X(3826) lies on these lines: {2, 11}, {5, 516}, {7, 12}, {9, 46}, {10, 141}, {37, 1738}, {226, 3740}, {427, 1890}, {443, 958}, {498, 954}, {984, 1086}, {1269, 3701}, {1386, 3008}, {1827, 1861}, {1836, 3305}, {3243, 3679}
X(3826) = complement of X(1001)
X(3826) = X(6)-of-4th-Euler-triangle
X(3826) = inverse-in-Feuerbach-hyperbola of X(3058)
X(3826) = centroid of trapezoid X(1)X(7)X(8)X(9)
X(3827) lies on these lines: {1, 159}, {6, 19}, {30, 511}, {66, 72}, {141, 960}, {154, 354}, {197, 1763}, {206, 942}, {210, 1853}, {1214, 3185}, {1279, 2195}, {1633, 3100}, {1824, 1836}, {1828, 1837}, {1843, 1858}, {1854, 3057}
X(3827) = crossdifference of every pair of points on line X(6)X(521)
X(3828) lies on these lines: {1, 2}, {12, 553}, {40, 3545}, {115, 121}, {165, 3543}, {375, 2392}, {381, 516}, {515, 549}, {517, 547}, {537, 3739}, {756, 1739}, {758, 3740}, {903, 1268}, {946, 3654}, {1656, 3656}, {3526, 3653}
X(3828) = midpoint of X(2) and X(10)
X(3828) = complement of X(551)
X(3828) = {X(1),X(2)}-harmonic conjugate of X(19883)
X(3828) = centroid of mid-triangle of Gemini triangles 19 and 20
X(3829) lies on these lines: {2, 11}, {5, 519}, {12, 3241}, {145, 3614}, {381, 529}, {496, 551}, {1329, 3679}
As a point on the Euler line, X(3830) has Shinagawa coefficients (1, -9).
Let T=A'B'C' be a triangle directly similar to ABC and inscribed in the circumcircle. As T varies, the isogonal conjugate of the trilinear product A'*B'*C' traces the Yff hyperbola, centered at X(381) with vertices at X(2) and X(4), and foci at X(3) and X(3830). (Randy Hutson, June 27, 2018)
X(3830) lies on these lines: {2, 3}, {115, 1384}, {265, 541}, {394, 1531}, {399, 1539}, {515, 3656}, {516, 3654}, {542, 1351}, {946, 3655}, {999, 3583}, {1159, 1836}, {1478, 3058}, {1853, 2777}, {3295, 3585}, {3521, 3527}
X(3830) = anticomplement of X(8703)
X(3830) = QA-P24 (AntiComplement of QA-P1 wrt the Morley Triangle) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/52-qa-p24.html)
X(3831) lies on these lines: {1, 2}, {5, 2887}, {38, 3701}, {141, 1329}, {312, 986}, {726, 1089}, {750, 964}, {942, 1215}, {992, 3686}, {1861, 3144}, {2277, 2321}, {2392, 3142}, {3663, 3760}, {3714, 3752}
As a point on the Euler line, X(3832) has Shinagawa coefficients (1, 4).
X(3832) lies on these lines: {2, 3}, {6, 1131}, {8, 1699}, {11, 3600}, {12, 390}, {64, 3066}, {145, 946}, {355, 3621}, {489, 3595}, {490, 3593}, {515, 3622}, {962, 3617}, {3085, 3583}, {3086, 3585}
X(3832) = {X(1131),X(1132)}-harmonic conjugate of X(6)
X(3833) lies on these lines: {2, 758}, {10, 354}, {116, 119}, {140, 517}, {519, 3742}, {547, 2771}, {551, 2802}, {942, 3634}, {993, 3306}, {1698, 3681}, {1739, 3720}, {2836, 3589}, {3244, 3698}, {3336, 3647}
X(3833) = complement of X(10176)
X(3834) lies on these lines: {2, 44}, {10, 141}, {37, 3662}, {238, 3624}, {244, 2228}, {513, 3716}, {524, 3008}, {536, 1086}, {545, 2325}, {752, 1125}, {1279, 3616}, {2887, 3742}, {3589, 3664}, {3631, 3686}
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X(3834) = isotomic conjugate of X(32012)
X(3834) = complement of X(44)
X(3834) = polar conjugate of isogonal conjugate of X(22067)
X(3834) = complementary conjugate of X(16594)
Let A'B'C' be the Aquila triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. A"B"C" is homothetic to ABC at X(3835) and to A'B'C' at X(10). (Randy Hutson, June 27, 2018)
X(3835) lies on these lines: {2, 649}, {116, 3259}, {226, 3676}, {512, 625}, {513, 3716}, {514, 661}, {522, 1491}, {650, 812}, {788, 3741}, {802, 3709}, {824, 3004}, {1538, 3309}, {1848, 3064}, {2254, 3667}
X(3835) = isotomic conjugate of X(4598)
X(3835) = complement of X(649)
X(3835) = anticomplement of X(31286)
X(3835) = trilinear product of vertices of the Aquila triangle
X(3835) = polar conjugate of isogonal conjugate of X(22090)
X(3835) = X(1)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles
X(3835) = center of the perspeconic of the Ursa-minor and Ursa-major triangles
X(3836) lies on these lines: {2,31}, {5,15310}, {8,17232}, {9,4655}, {10,141}, {11,4871}, {12,1463}, {37,3821}, {42,18139}, {43,18134}, {44,1213}, {57,4438}, {75,3773}, {120,17793}, {190,17767}, {239,17772}, {244,3006}, {305,18067}, {312,17889}, {320,1698}, {334,1921}, {344,24248}, {346,7613}, {442,3831}, {513,3814}, {516,4432}, {519,4864}, {536,6541}, {537,3717}, {614,4865}, {726,1086}, {740,1738}, {742,25357}, {756,17184}, {899,3936}, {902,24542}, {908,4892}, {940,25453}, {946,21626}, {960,25108}, {984,3662}, {1001,4660}, {1089,1269}, {1125,1279}, {1193,17674}, {1215,5249}, {1326,25536}, {1329,20258}, {1376,3771}, {1386,17356}, {1836,4011}, {1961,19786}, {2140,12263}, {2886,3840}, {3008,4974}, {3011,4434}, {3120,4358}, {3123,22220}, {3246,19862}, {3290,4071}, {3305,4703}, {3416,16825}, {3452,4138}, {3663,4078}, {3666,24169}, {3670,21035}, {3685,17266}, {3696,17231}, {3703,24165}, {3705,17063}, {3720,4972}, {3726,4119}, {3741,3925}, {3751,17298}, {3782,3971}, {3814,4129},{3842,4357}, {3909,20962}, {3923,5880}, {3944,18743}, {3993,17243}, {4009,21093}, {4136,20271}, {4167,21951}, {4359,15523}, {4362,24789}, {4407,17227}, {4417,16569}, {4422,17768}, {4431,4535}, {4446,24046}, {4527,17233}, {4649,17300}, {4663,17376}, {4672,17353}, {4676,17341}, {4697,5294}, {4698,25354}, {4710,18040}, {4716,6542}, {4743,17241}, {4980,6535}, {5087,11814}, {5205,17719}, {5220,7232}, {5248,16299}, {5263,17283}, {5268,25527}, {5695,17267}, {5852,7238}, {5853,17059}, {5988,9478}, {6685,17056}, {7295,16048}, {7800,25500}, {8287,20546}, {9470,20341}, {9780,17238}, {12047,25079}, {12782,24190}, {15481,17345}, {16415,25440}, {16468,17352}, {16830,17291}, {17047,24206}, {17237,25352}, {17260,24697}, {17263,24723}, {17284,24693}, {17289,24342}, {17307,19856}, {17357,24295}, {17748,20227}, {19557,19559}, {19877,20072}, {20337,20339}, {20340,20343}, {20432,20703}, {20491,20532}, {20530,20531}, {20947,24731}, {24176,24180}, {24358,24699}
X(3836) = complement of X(238)
X(3836) = anticomplement of X(31289)
X(3837) lies on these lines: {2, 659}, {5, 2826}, {10, 891}, {11, 244}, {120, 2977}, {325, 523}, {334, 876}, {513, 3716}, {661, 1639}, {814, 905}, {946, 2821}, {1125, 1960}, {1577, 2530}
X(3837) = isotomic conjugate of X(8709)
X(3837) = polar conjugate of isogonal conjugate of X(22092)
X(3837) = complementary conjugate of X(38989)
X(3838) lies on these lines: {2, 1155}, {11, 3742}, {30, 1125}, {65, 2476}, {142, 1538}, {149, 3748}, {226, 518}, {442, 960}, {908, 3740}, {1001, 1699}, {1376, 3256}, {2475, 2646}, {3120, 3666}
X(3838) = complement of X(4640)
As a point on the Euler line, X(3839) has Shinagawa coefficients (1, 6).
X(3839) lies on these lines: {2, 3}, {145, 3656}, {147, 671}, {390, 3583}, {519, 1699}, {598, 3424}, {946, 3241}, {962, 3679}, {1131, 1328}, {1132, 1327}, {3585, 3600}, {3622, 3655}
X(3839) = circumcircle-inverse of X(37957)
X(3839) = {X(2),X(3)}-harmonic conjugate of X(15721)
X(3839) = trisector nearest X(4) of segment X(2)X(4)
X(3839) = endo homothetic center of Ehrmann side-triangle and submedial triangle; the homothetic center is X(5055)
X(3840) lies on these lines: {1, 2}, {11, 2887}, {226, 1401}, {244, 321}, {312, 726}, {350, 3663}, {354, 1215}, {740, 3752}, {748, 1150}, {1575, 2321}, {2810, 3038}, {3551, 3662}
X(3840) = isotomic conjugate of X(32011)X(3841) lies on these lines: {2, 35}, {5, 516}, {10, 12}, {79, 3219}, {377, 993}, {484, 1698}, {495, 3626}, {496, 1125}, {535, 958}, {626, 3739}, {759, 833}, {1770, 3647}
X(3842) lies on these lines: {1, 872}, {2, 38}, {10, 37}, {75, 1089}, {86, 1757}, {333, 1961}, {518, 1125}, {726, 3634}, {1237, 1921}, {2664, 3736}, {2667, 3293}
X(3842) = complement of X(24325)As a point on the Euler line, X(3843) has Shinagawa coefficients (1, 7).
Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). Let A'B'C' be the Euler-of-Euler triangle. X(3843) is the radical center of the circumcircles of triangles A'AeAi, B'BeBi, C'CeCi. (Randy Hutson, June 7, 2019)
X(3843) lies on these lines: {2, 3}, {68, 3531}, {265, 3527}, {355, 3625}, {946, 3635}, {999, 3585}, {1159, 1837}, {1352, 3630}, {1482, 1699}, {3295, 3583}, {3426, 3521}
X(3843) = complement of X(17538)
X(3843) = anticomplement of X(15712)
X(3843) = homothetic center of Ehrmann vertex-triangle and anti-incircle-circles triangle
X(3843) = homothetic center of Ehrmann mid-triangle and X3-ABC reflections triangle
X(3843) = endo-homothetic center of Ehrmann mid-triangle and anti-Euler triangle; the homothetic center is X(3091)
X(3843) = {X(2043),X(2044)}-harmonic conjugate of X(547)
X(3844) lies on these lines: {2, 1386}, {6, 1698}, {8, 3619}, {10, 141}, {120, 125}, {594, 1738}, {599, 3751}, {3242, 3679}, {3589, 3634}, {3661, 3696}
As a point on the Euler line, X(3845) has Shinagawa coefficients (1, 9).
X(3845) is the point P which divides segment X(4)X(5) in the ratio PX(4)/PX(5) = -1/2. (Randy Hutson, June 27, 2018)
X(3845) lies on these lines: {2, 3}, {6, 1327}, {343, 1531}, {399, 1994}, {495, 3058}, {496, 3585}, {541, 1539}, {542, 1353}, {946, 1483}, {952, 1699}
X(3845) = circumcircle-inverse of X(37958)
X(3845) = orthocentroidal-circle-inverse of X(3830)
X(3845) = Ehrmann-side-to-orthic similarity image of X(2)
X(3845) = Johnson-to-Ehrmann-mid similarity image of X(381)
X(3845) = trisector nearest X(4) of segment X(4)X(5)
X(3845) = X(2)-of-Ehrmann-mid-triangle
X(3845) = X(10246)-of-orthic-triangle if ABC is acute
X(3845) = {X(2),X(4)}-harmonic conjugate of X(3830)
X(3845) = {X(13595),X(13596)}-harmonic conjugate of X(3)
X(3846) lies on these lines: {2,31}, {5,10}, {9,4438}, {11,1211}, {42,5741}, {43,4085}, {57,4655}, {63,4703}, {75,3944}, {114,124}, {141,3816}, {142,4138}, {226,24325}, {244,17184}, {312,3773}, {325,4357}, {612,4865}, {726,4415}, {740,3687}, {756,3006}, {899,4972}, {908,1215}, {978,16062}, {984,3705}, {1001,3771}, {1100,10026}, {1104,1125}, {1193,5051}, {1203,25441}, {1376,4660}, {1836,3980}, {1921,7018}, {1999,17772}, {3120,4359}, {3178,6051}, {3218,4683}, {3266,21415}, {3662,17063}, {3666,4425}, {3670,14815}, {3702,20653}, {3703,3971}, {3706,21085}, {3717,4096}, {3720,3936}, {3727,4167}, {3739,3838}, {3752,3821}, {3757,17719}, {3772,16825}, {3782,24165}, {3831,4187}, {3834,3848}, {3844,11814}, {3923,24703}, {3925,5241}, {3931,17748}, {3961,4514}, {3966,4362}, {4011,4679}, {4023,4685}, {4026,6685}, {4038,17778}, {4104,4847}, {4199,21321}, {4358,15523}, {4383,25453}, {4384,17064}, {4389,17591}, {4418,5057}, {4429,16569}, {4647,6042}, {4854,4970}, {4892,5249}, {5015,5293}, {5211,17598}, {5259,25645}, {5272,25527}, {5278,24892}, {5739,11269}, {5814,17733}, {5955,12699}, {7741,10479}, {7988,18229}, {8167,16846}, {9284,16584}, {10453,24217}, {11680,21242}, {15507,18235}, {15825,19863}, {16610,24169}, {16739,17203}, {16817,24161}, {17202,23659}, {17234,25502}, {17274,18193}, {17355,17747}, {17357,21249}, {17596,24723}, {17605,25385}, {17889,19804}, {19543,25440}, {19563,20333}, {21856,25066}
X(3846) = complement of X(171)
X(3846) = isotomic conjugate of isogonal conjugate of X(23659)
X(3846) = polar conjugate of isogonal conjugate of X(22447)
X(3847) lies on these lines: {2,5217}, {3,6667}, {4,6691}, {5,515}, {8,11}, {10,3829}, {12,3622}, {55,6931}, {56,5187}, {496,3244}, {528,9669}, {529,3086}, {550,6681}, {958,6919}, {1001,3090}, {1210,5087}, {1479,3035}, {1656,6690}, {1698,1706}, {2478,4999}, {3614,3616}, {3628,5248}, {3812,3817}, {3820,4691}, {3838,9843}, {3878,6797}, {3913,5274}, {4189,7294}, {4423,6933}, {5046,5303}, {5057,7098}, {5690,6702}, {5842,6959}, {5880,6831}, {6253,6979}, {6856,8167}, {6882,7681}, {6971,7680}
X(3848) lies on these lines: {2, 210}, {140, 517}, {142, 1538}, {165, 1001}, {171, 3246}, {960, 3624}, {3149, 3576}, {3622, 3698}
X(3848) = complement of X(3740)
X(3849) lies on these lines: {2, 187}, {30, 511}, {325, 2482}, {381, 2080}, {385, 671}, {597, 2030}, {599, 3734}, {1992, 2549}
X(3849) = isogonal conjugate of X(6323)
X(3849) = infinite point of Brocard axis of the 3rd pedal triangle of X(6)
As a point on the Euler line, X(3850) has Shinagawa coefficients (3, 7).
X(3850) lies on these lines: {2, 3}, {355, 3633}, {399, 1199}, {946, 3625}, {952, 3635}, {1327, 3594}, {1328, 3592}, {3583, 3614}
X(3850) = midpoint of X(4) and X(140)
X(3850) = {X(4),X(5)}-harmonic conjugate of X(140)
X(3850) = X(13624)-of-orthic-triangle if ABC is acute
As a point on the Euler line, X(3851) has Shinagawa coefficients (3, 5)
X(3851) lies on these lines: {2, 3}, {355, 3244}, {946, 3626}, {1352, 3629}, {1479, 3614}, {1482, 3632}, {3519, 3527}
X(3851) = reflection of X(15700) in X(2)
X(3851) = complement of X(3528)
X(3851) = anticomplement of X(14869)
X(3851) = inverse-in-orthocentroidal-circle of X(550)
X(3851) = homothetic center of Euler triangle and mid-triangle of medial and anticomplementary triangles
X(3851) = homothetic center of 2nd isogonal triangle of X(4) and mid-triangle of orthic and circumorthic triangles
X(3851) = homothetic center of X(140)-altimedial and X(4)-anti-altimedial triangles
X(3851) = {X(2),X(4)}-harmonic conjugate of X(550)
X(3851) = {X(2),X(5)}-harmonic conjugate of X(5079)
X(3852) lies on these lines: {6, 695}, {30, 511}, {32, 206}, {66, 315}, {141, 3491}, {159, 3499}
As a point on the Euler line, X(3853) has Shinagawa coefficients (1, -9).
X(3853) lies on these lines: {2, 3}, {1327, 3592}, {1328, 3594}
X(3853) = anticomplement of X(33923)
X(3853) = {X(4),X(5)}-harmonic conjugate of X(3861)
As a point on the Euler line, X(3854) has Shinagawa coefficients (3, 8).
X(3854) lies on these lines: {2, 3}, {946, 3621}, {1699, 3617}
As a point on the Euler line, X(3855) has Shinagawa coefficients (2, 5).
X(3855) lies on these lines: {2, 3}, {944, 3636}, {946, 3632}
X(3855) = anticomplement of X(15720)
X(3855) = {X(2),X(4)}-harmonic conjugate of X(3529)
X(3855) = {X(2),X(20)}-harmonic conjugate of X(3530)
X(3855) = orthocentroidal circle inverse of X(3529)
As a point on the Euler line, X(3856) has Shinagawa coefficients (5, 17).
X(3856) lies on this line: {2, 3}
As a point on the Euler line, X(3857) has Shinagawa coefficients (5, 13)
X(3857) lies on this line: {2, 3}
As a point on the Euler line, X(3858) has Shinagawa coefficients (3, 11)
X(3858) lies on this line: {2, 3}
As a point on the Euler line, X(3859) has Shinagawa coefficients (7, 19)
X(3859) lies on this line: {2, 3}
As a point on the Euler line, X(3860) has Shinagawa coefficients (7, 27)
X(3860) lies on this line: {2, 3}
As a point on the Euler line, X(3861) has Shinagawa coefficients (1, 13).
X(3861) lies on this line: {2, 3}
X(3861) = {X(4),X(5)}-harmonic conjugate of X(3853)
Inverse Triangles and More Combos
The Euler triangle and the 2nd, 3rd, 4th, and 5th Euler triangles are discussed in the preamble to X(3758), along with eight other central triangles using the notation T(f(a,b,c), g(b,c,a)). Recall that the inverse of a normalized central triangle T, denoted by Inverse(nT) or Inverse(n(T)), is also a normalized central triangle. Barycentrics for the inverse of each of the 13 triangles and properties of these triangles are given (Peter Moses, December 2011) as follows:
Inverse(n(Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = (b2 - c2)2 +
a2(3a2 - 4b2 - 4c2)
g(b,c,a) = b4 - (c2 -
a2)2
Inverse(n(2nd Euler triangle)) = T(f(a,b,c), g(b,c,a)),
where
f(a,b,c) = a2(a2 - b2 -
c2)(a6 + b6 + c6 -
a4b2 - a4c2 -
a2b4 - a2c4 -
2a2b2c2 - b4c2 -
b2c4)
g(b,c,a) = b2(b2 - c2 -
a2(a6 + b6 - c6 -
a4b2 - 3a4c2 -
a2b4 + 3a2c4 +
2a2b2c2 - 3b4c2
+ 3b2c4)
Inverse(n(3rd Euler triangle)) = T(f(a,b,c), g(b,c,a)),
where
f(a,b,c) = a(b2 + c2 - ab - ac + bc)
g(b,c,a) = b(a2 - ab - ac + bc - c2)
Inverse(n(4th Euler triangle)) = T(f(a,b,c), g(b,c,a)),
where
f(a,b,c) = a(a - b - c)(b2 + c2 + ab + ac)
g(b,c,a) = b(-a + b + c)(a2 + ab - ac - bc -
c2)
Inverse(n(5th Euler triangle)) = T(f(a,b,c), g(b,c,a)),
where
f(a,b,c) = (a2 - 2b2 -
2c2)(3a2 + b2 + c2)
g(b,c,a) = -(a2 - b2 -
c2)(2a2 - b2 + 2c2)
Inverse(n(T(bc,b2)) = T(f(a,b,c), g(b,c,a)),
where
f(a,b,c) = -bc(a2 - bc)(b2 + c2 +
bc)
g(b,c,a) = b2(ab - c2)(a2 +
c2 + ac)
Inverse(n(T(-bc,b2)) = T(f(a,b,c), g(b,c,a)),
where
f(a,b,c) = bc(a2 - bc)(b2 + c2 -
bc)
g(b,c,a) = b2(ab + c2)(a2 +
c2 - ac)
Inverse(n(T(bc,c2)) = T(f(a,b,c), g(b,c,a)),
where
f(a,b,c) = -a(a2 - bc)(b2 + c2 +
bc)
g(b,c,a) = (ab - c2)(a2 + c2 + ac)
Inverse(n(T(-bc,c2)) = T(f(a,b,c), g(b,c,a)),
where
f(a,b,c) = -a(a2 - bc)(b2 + c2 -
bc)
g(b,c,a) = (ab + c2)(a2 + c2 - ac)
Inverse(n(T(a2,a2 - b2)) =
T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = (3a2 - b2 -
c2)(b4 + c4 -
b2c2)
g(b,c,a) = (a2 - 3b2 +
c2)(a2b2 - 2b2c2
+ c4)
Inverse(n(T(a2,a2 - c2)) =
T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = a2(b2 + c2 -
3a2)
g(b,c,a) =(a2 - c2)(a2 -
3b2 + c2)
Inverse(n(T(a2,a2 + b2)) =
T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = -(3a2 + b2 +
c2)(b4 + c4 +
b2c2)
g(b,c,a) = (a2 + 3b2 +
c2)(a2b2 + c4)
Inverse(n(T(a2,a2 + c2)) =
T(f(a,b,c), g(b,c,a)), where
f(a,b,c) = -a2(a2 + b2 +
c2)(3a2 + b2 + c2)
g(b,c,a) = (a2 + c2)(a2 +
b2 - c2)(a2 + 3b2 +
c2)
For present purposes, label these last triangles as IT1, IT2, IT3, IT4, IT5, IT6, IT7, IT8; then all except IT5 and IT6 are perspective to the reference triangle ABC; perspectors are described at X(3862)-X(3866). Label the corresponding original triangles T1, T2, T3, T4, T5, T6, T7, T8; then the following pairs are perspective: (T1, IT1), (T2, IT2), (T3,IT3), (T4, IT4), (T7, IT7), (T8,IT8); coordinates for the perspectors are long and omitted.
The triangles IT1 and IT2 are triply perspective to ABC (as are T1 and T3). Order-label IT1 as A'B'C'. The three perspectivities are then given by
AA'∩BB'∩CC' = X(3862)
AB'∩BC'∩CA' = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =
ab(a2 - bc)(c2 - ab)(b2 + bc +
c2)
AC'∩BA'∩CB' = f(a,c,b) : f(b,a,c) : f(c,b,a)
Note that the second two perspectors are a bicentric pair (not triangle centers; see TCCT, page 47); likewise for the other triple perspectivity, with IT3 order-labeled as A'B'C':
AA'∩BB'∩CC' = X(3864)
AB'∩BC'∩CA' = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) =
a(a2 - bc)(c2 - ab)(b2 + bc +
c2)
AC'∩BA'∩CB' = g(a,c,b) : g(b,a,c) : g(c,b,a)
If you have The Geometer's Sketchpad, you can view X(3862).
X(3862) lies on these lines: {6,292}, {37,256}, {75,141}, {291,518}, {660,2235}, {753,813}, {984,3094}, {1333,2311}, {1755,2076}, {2276,3116}
{X(694),X(1581)}-harmonic conjugate of X(3863)X(3863) lies on these lines: {6,893}, {37,256}, {75,257}, {904,1964}, {1178,1333}, {3056,3116}, {3061,3094}
X(3863) = {X(694),X(1581)}-harmonic conjugate of X(3862)
X(3864) lies on these lines: {1,39}, {10,257}, {76,334}, {295,3497}, {335,726}, {511,1757}, {761,813}, {984,3094}
X(3864) = {X(1581),X(1916)}-harmonic conjugate of X(3865)
X(3865) lies on these lines: {1,256}, {10,257}, {39,893}, {904,995}, {986,3095}, {3061,3094}
X(3865) = {X(1581),X(1916)}-harmonic conjugate of X(3864)
X(3866) lies on these lines: {69,194}, {83,419}
X(3867) lies on these lines: {4,6}, {25,3589}, {141,427}, {428,597}, {511,1595}, {1350,3088}
X(3868) lies on these lines: {1, 21}, {2, 72}, {3, 3218}, {4, 912}, {6, 977}, {7, 8}, {10, 3681}, {20, 145}, {27, 3187}, {34, 651}, {41, 3509}, {42, 986}, {46, 100}, {56, 1259}, {57, 78}, {84, 1320}, {101, 1729}, {144, 452}, {146, 149}, {171, 976}, {193, 1829}, {200, 3339}, {226, 2476}, {239, 379}, {244, 978}, {286, 2997}, {326, 1014}, {329, 938}, {354, 960}, {355, 2888}, {386, 3670}, {392, 3622}, {405, 3219}, {412, 1897}, {497, 1858}, {527, 950}, {579, 2198}, {580, 1331}, {596, 994}, {664, 3188}, {894, 964}, {908, 1210}, {936, 3306}, {971, 3146}, {982, 1193}, {997, 3338}, {1012, 1482}, {1445, 1467}, {1475, 3061}, {1490, 1998}, {1697, 3243}, {1698, 3678}, {1870, 1993}, {2095, 3149}, {2176, 3726}, {2280, 3496}, {2475, 2894}, {2802, 3633}, {2886, 3649}, {3057, 3241}, {3189, 3474}, {3617, 3753}, {3679, 3754}
X(3868) = isogonal conjugate of X(2218)
X(3868) = isotomic conjugate of X(2997)
X(3868) = anticomplement of X(72)
X(3868) = perspector of Conway triangle and Gemini triangle 29
X(3868) = {X(1),X(63)}-harmonic conjugate of X(21)
X(3868) = {X(7),X(8)}-harmonic conjugate of X(377)
X(3868) = X(4)com[Inverse(n(4th Euler triangle))]
X(3869) lies on these lines: {1, 21}, {2, 65}, {3, 3417}, {4, 8}, {6, 3727}, {9, 1405}, {10, 908}, {19, 2287}, {22, 3556}, {28, 1748}, {40, 78}, {41, 3496}, {46, 404}, {48, 1761}, {56, 3218}, {90, 1320}, {101, 1759}, {144, 145}, {146, 2778}, {210, 3617}, {213, 3735}, {219, 608}, {221, 394}, {314, 2995}, {354, 3622}, {391, 2262}, {392, 942}, {672, 3061}, {693, 1938}, {912, 944}, {934, 2365}, {936, 2093}, {956, 1482}, {958, 2099}, {982, 1201}, {986, 1193}, {995, 3670}, {1005, 1697}, {1156, 3680}, {1259, 3428}, {1610, 1812}, {1630, 2327}, {1698, 3754}, {1776, 2098}, {1836, 2475}, {1854, 3100}, {1898, 3621}, {2176, 3721}, {2390, 2979}, {2771, 3648}, {2802, 3632}, {3241, 3555}, {3306, 3339}, {3678, 3679}, {3698, 3740}
X(3869) = isogonal conjugate of X(2217)
X(3869) = isotomic conjugate of X(2995)
X(3869) = anticomplement of X(65)
X(3869) = antigonal conjugate of X(34242)
X(3869) = X(80)-of-inner-Garcia triangle
X(3869) = X(4)-of-inner-Conway-triangle
X(3869) = radical center of circles centered at excenters and passing through corresponding vertex of anticomplementary triangle
X(3870) lies on these lines: {1, 2}, {3, 3555}, {6, 3692}, {7, 3174}, {9, 1174}, {31, 1331}, {33, 92}, {38, 2177}, {55, 63}, {57, 100}, {65, 224}, {72, 3295}, {149, 1699}, {165, 3218}, {193, 2293}, {210, 1001}, {226, 2900}, {329, 390}, {354, 1376}, {388, 3189}, {404, 3333}, {495, 3419}, {497, 908}, {528, 1836}, {664, 1088}, {902, 1707}, {950, 3436}, {960, 3303}, {962, 1490}, {968, 984}, {1005, 1697}, {1320, 3577}, {1445, 1617}, {1708, 2078}, {1709, 2801}, {2136, 3340}, {2550, 3475}, {2975, 3601}, {3242, 3666}, {3421, 3488}, {3711, 3740}
X(3870) = isogonal conjugate of X(2191)
X(3870) = anticomplement of X(4847)
X(3870) = X(11550)-of-excentral-triangle
X(3870) = perspector of Gemini triangle 29 and tangential triangle, wrt ABC, of {ABC, Gemini 29}-circumconic
X(3871) lies on these lines: {1, 88}, {2, 496}, {3, 145}, {5, 149}, {8, 21}, {10, 1621}, {12, 528}, {35, 519}, {36, 3244}, {41, 644}, {56, 3241}, {60, 643}, {72, 1005}, {78, 1697}, {238, 3214}, {380, 3692}, {390, 2478}, {405, 3617}, {411, 517}, {452, 1260}, {474, 3622}, {495, 2475}, {595, 3293}, {943, 3419}, {956, 3621}, {960, 3689}, {993, 3632}, {999, 3623}, {1036, 1261}, {1329, 3058}, {1334, 3684}, {1376, 3303}, {1468, 3550}, {1470, 1476}, {2136, 3601}, {2268, 3169}, {2280, 3501}, {2346, 2550}, {2476, 3085}, {3218, 3555}, {3685, 3701}
X(3871) = intouch-to-ABC barycentric image of X(12)
X(3872) lies on these lines: {1, 2}, {9, 644}, {21, 1697}, {34, 318}, {40, 2975}, {63, 517}, {72, 1482}, {75, 77}, {100, 3576}, {104, 3359}, {149, 3586}, {280, 1219}, {355, 1532}, {391, 2324}, {392, 3305}, {404, 1420}, {515, 3434}, {518, 2099}, {529, 1836}, {908, 3421}, {946, 3436}, {952, 3419}, {958, 3057}, {960, 2098}, {993, 2802}, {996, 998}, {999, 3306}, {1060, 2968}, {1100, 3713}, {1120, 2191}, {1259, 3295}, {1319, 1376}, {2082, 2329}, {2093, 3218}, {2136, 3601}, {2345, 3554}, {2550, 3476}, {3338, 3754}
X(3872) = {X(1),X(8)}-harmonic conjugate of X(78)X(3873) lies on these lines: {2, 210}, {6, 3726}, {7, 3434}, {8, 443}, {42, 982}, {43, 244}, {51, 2810}, {55, 3218}, {57, 100}, {65, 145}, {72, 3616}, {78, 3333}, {149, 1836}, {200, 3306}, {376, 517}, {404, 3338}, {614, 3315}, {674, 2979}, {748, 1757}, {938, 3436}, {940, 3242}, {960, 3622}, {962, 1071}, {984, 3720}, {999, 1260}, {1001, 3219}, {1150, 3757}, {1699, 2801}, {2280, 3509}, {2771, 3656}, {3057, 3623}, {3083, 3640}, {3084, 3641}, {3240, 3752}, {3550, 3722}, {3624, 3678}, {3632, 3754}
X(3873) = complement of X(4661)
X(3873) = anticomplement of X(210)
X(3873) = centroid of Gemini triangle 29
X(3873) = centroid of cross-triangle of Gemini triangles 1 and 13
X(3874) lies on these lines: {1, 21}, {2, 3678}, {4, 2801}, {7, 2894}, {8, 2891}, {10, 141}, {35, 3218}, {40, 3243}, {42, 3670}, {56, 214}, {65, 519}, {72, 354}, {78, 3338}, {92, 1844}, {100, 3336}, {145, 2802}, {210, 3634}, {213, 3726}, {244, 3216}, {295, 2809}, {386, 982}, {392, 3636}, {404, 3337}, {516, 1071}, {517, 550}, {551, 960}, {726, 2901}, {912, 946}, {997, 3333}, {1482, 2800}, {1698, 3681}, {1739, 3214}, {1759, 2280}, {3057, 3635}, {3626, 3753}
X(3874) = complement of X(5904)
X(3874) = X(13419)-of-excentral-triangle
X(3875) lies on these lines: {1, 75}, {2, 2321}, {6, 536}, {7, 145}, {8, 3672}, {9, 192}, {43, 350}, {57, 1999}, {63, 3187}, {69, 519}, {190, 1743}, {193, 527}, {269, 664}, {273, 1897}, {312, 2999}, {313, 3293}, {319, 3632}, {320, 3633}, {322, 3673}, {344, 3008}, {545, 3629}, {596, 969}, {614, 3263}, {712, 1572}, {726, 3751}, {870, 2663}, {894, 1278}, {1269, 3761}, {1423, 3169}, {1447, 3158}, {2136, 3212}, {2258, 3112}, {3216, 3264}, {3244, 3664}
X(3875) = isotomic conjugate of X(34860)
X(3875) = anticomplement of X(2321)
X(3876) lies on these lines: {1, 748}, {2, 72}, {3, 3219}, {8, 210}, {9, 21}, {10, 908}, {38, 978}, {43, 2292}, {63, 404}, {65, 3740}, {81, 975}, {145, 392}, {238, 976}, {329, 377}, {474, 3218}, {517, 3091}, {518, 3616}, {631, 912}, {651, 1038}, {750, 1046}, {758, 1698}, {857, 3661}, {899, 986}, {958, 3715}, {970, 3690}, {971, 3522}, {984, 1193}, {997, 2975}, {1071, 3523}, {1468, 1757}, {3061, 3691}, {3555, 3622}, {3687, 3710}
X(3877) lies on these lines: {1, 21}, {2, 392}, {8, 210}, {9, 644}, {40, 404}, {65, 3616}, {72, 145}, {78, 1697}, {100, 997}, {149, 3419}, {377, 962}, {405, 1482}, {518, 1992}, {519, 3681}, {942, 3622}, {946, 2476}, {956, 3219}, {958, 2098}, {982, 1149}, {986, 1201}, {999, 3218}, {1000, 3421}, {1001, 2099}, {1334, 3061}, {2093, 3306}, {2176, 3727}, {2771, 3655}, {2800, 3576}, {2802, 3679}, {3230, 3735}, {3555, 3623}, {3624, 3754}, {3632, 3678}
X(3878) lies on these lines: {1, 21}, {2, 3754}, {3, 214}, {5, 10}, {8, 80}, {9, 1389}, {40, 997}, {65, 392}, {72, 519}, {101, 3496}, {210, 3626}, {213, 3727}, {354, 3636}, {404, 484}, {405, 2099}, {518, 3244}, {551, 942}, {759, 1098}, {944, 2801}, {956, 2098}, {958, 1482}, {986, 995}, {1201, 3670}, {1320, 3467}, {1829, 1904}, {2176, 3735}, {2922, 3556}, {3061, 3730}, {3230, 3721}, {3555, 3635}, {3632, 3681}, {3634, 3753}
X(3878) = complement of X(5903)
X(3878) = X(80)-of-X(1)-Brocard triangle
X(3878) = X(10)-of-inner-Garcia triangle
X(3878) = X(13403)-of-excentral-triangle
X(3878) = outer-Garcia-to-inner-Garcia similarity image of X(10)
X(3879) lies on these lines: {1, 69}, {2, 1449}, {6,3912} {7, 145}, {9, 193}, {10, 86}, {35, 1444}, {37, 524}, {44, 3629}, {57, 3169}, {75, 519}, {81, 306}, {100, 1014}, {141, 1100}, {142, 239}, {171, 332}, {192, 527}, {226, 1943}, {307, 1442}, {314, 1909}, {317, 1785}, {320, 3244}, {344, 1743}, {518, 3688}, {553, 3210}, {664, 3668}, {894, 2321}, {940, 3687}, {1264, 3717}, {3008, 3759}, {3241, 3672}, {3630, 3723}
X(3879) = anticomplement of X(3686)
X(3880) lies on these lines: {1, 474}, {8, 210}, {10, 496}, {30, 511}, {36, 2932}, {37, 3169}, {65, 145}, {72, 3586}, {78, 2098}, {100, 1319}, {354, 3241}, {392, 3679}, {644, 2348}, {942, 3244}, {958, 1697}, {1145, 1737}, {1212, 3208}, {1318, 1320}, {1616, 1722}, {1875, 1897}, {1898, 3621}, {2170, 3693}, {3555, 3633}, {3616, 3698}, {3635, 3754}
X(3880) = isogonal conjugate of X(8686)
X(3880) = crossdifference of every pair of points on line X(6)X(4394)
X(3881) lies on these lines: {1, 21}, {10, 354}, {65, 1317}, {72, 551}, {79, 149}, {100, 3337}, {244, 3293}, {517, 548}, {518, 1125}, {519, 942}, {535, 950}, {537, 3159}, {596, 740}, {946, 2801}, {960, 3636}, {2550, 3296}, {3218, 3746}, {3243, 3333}, {3624, 3681}, {3625, 3753}, {3634, 3742}
X(3882) lies on these lines: {1, 3122}, {9, 1654}, {40, 1330}, {63, 2895}, {69, 573}, {100, 109}, {101, 1310}, {163, 662}, {190, 646}, {193, 579}, {524, 2245}, {583, 3629}, {645, 3570}, {664, 1020}, {740, 1756}, {1423, 3169}, {1655, 3208}, {1763, 3719}, {1792, 3430}, {2897, 3692}
X(3882) = anticomplement of X(17197)
X(3883) lies on these lines: {1, 69}, {8, 9}, {10, 82}, {55, 3687}, {75, 516}, {141, 1279}, {226, 3757}, {239, 3755}, {306, 1621}, {333, 643}, {519, 751}, {528, 3696}, {960, 3688}, {1001, 3416}, {1211, 3744}, {1891, 2354}, {2975, 3220}, {3058, 3706}, {3683, 3703}
X(3884) lies on these lines: {1, 21}, {10, 11}, {35, 214}, {65, 551}, {72, 3244}, {140, 517}, {210, 3625}, {405, 2098}, {518, 3635}, {519, 960}, {942, 3636}, {997, 1697}, {1000, 2551}, {1001, 1482}, {1149, 3670}, {1385, 2800}, {2170, 3294}, {3230, 3727}, {3633, 3681}
X(3884) = X(10)-of-X(1)-Brocard triangle
X(3885) lies on these lines: {1, 88}, {8, 210}, {20, 145}, {21, 1697}, {65, 3241}, {72, 3621}, {78, 2136}, {149, 355}, {392, 3617}, {496, 1145}, {518, 3644}, {644, 2082}, {758, 3633}, {942, 3623}, {1482, 3149}, {2170, 3208}, {3622, 3753}, {3632, 3681}
X(3885) lies on these lines: {1, 75}, {2, 3755}, {8, 9}, {10, 344}, {55, 3706}, {69, 516}, {78, 1229}, {145, 894}, {200, 312}, {306, 3434}, {497, 3687}, {518, 3729}, {519, 1992}, {528, 3416}, {536, 3242}, {1001, 3696}, {1757, 3632}, {3210, 3677}
X(3887) lies on these lines: {1, 2254}, {10, 3716}, {11, 116}, {30, 511}, {80, 885}, {100, 101}, {103, 104}, {118, 119}, {149, 150}, {152, 153}, {214, 3126}, {663, 1734}, {1022, 1280}, {1317, 1362}, {3032, 3033}, {3036, 3041}, {3045, 3046}
X(3887) = isogonal conjugate of X(1308)
X(3887) = isotomic conjugate of X(35171)
X(3887) = X(2)-Ceva conjugate of X(35125)
X(3887) = crossdifference of every pair of points on line X(6)X(244)
X(3887) = X(514)-of-inner-Garcia triangle
X(3888) lies on these lines: {1, 3123}, {2, 3271}, {8, 2810}, {69, 2876}, {99, 815}, {100, 109}, {110, 833}, {190, 513}, {295, 1281}, {320, 674}, {662, 1492}, {789, 805}, {883, 926}, {1227, 2877}, {1332, 1633}, {2227, 3510}, {3056, 3662}
X(3889) lies on these lines: {1, 21}, {2, 3555}, {8, 354}, {65, 3241}, {72, 3622}, {78, 3243}, {100, 3338}, {145, 942}, {404, 3333}, {517, 3522}, {518, 3616}, {1125, 3681}, {1320, 3340}, {3218, 3295}, {3621, 3753}, {3633, 3754}
X(3890) lies on these lines: {1, 21}, {2, 3057}, {8, 392}, {65, 3622}, {72, 3241}, {100, 1697}, {145, 960}, {210, 3621}, {517, 631}, {518, 3623}, {986, 1149}, {1001, 2098}, {1698, 2802}, {3218, 3304}, {3633, 3678}
X(3891) lies on these lines: {1, 321}, {2, 1390}, {31, 726}, {38, 1150}, {76, 3112}, {100, 1403}, {145, 388}, {192, 1621}, {239, 3681}, {278, 1280}, {518, 3187}, {536, 3744}, {1255, 3616}, {1279, 3175}, {1824, 3555}
X(3892) lies on these lines: {1, 21}, {10, 3742}, {65, 3635}, {72, 3636}, {145, 3754}, {210, 1125}, {214, 999}, {354, 519}, {518, 551}, {872, 995}, {942, 3244}, {997, 3243}, {2802, 3241}, {3158, 3333}, {3616, 3678}
X(3893) lies on these lines: {1, 3689}, {8, 210}, {55, 2136}, {65, 519}, {72, 2802}, {145, 354}, {200, 2098}, {382, 517}, {392, 3626}, {518, 1278}, {942, 3633}, {1697, 3683}, {1706, 3304}, {3244, 3753}
X(3894) lies on these lines: {1, 21}, {46, 3158}, {65, 3632}, {72, 3624}, {78, 3337}, {210, 942}, {517, 3534}, {518, 599}, {912, 1699}, {2093, 3174}, {3555, 3633}
X(3895) lies on these lines: {1, 88}, {8, 9}, {40, 145}, {46, 3244}, {57, 3241}, {63, 519}, {78, 3057}, {960, 3711}, {1706, 3616}, {1837, 3036}, {2320, 3601}, {3305, 3679}, {3333, 3623}, {3338, 3635}
X(3896) lies on these lines: {2, 3696}, {10, 1962}, {38, 519}, {42, 321}, {55, 3187}, {65, 145}, {100, 1402}, {192, 3681}, {239, 1621}, {306, 3755}, {312, 3240}, {386, 3702}, {2901, 3293}
X(3897) lies on these lines: {1, 21}, {2, 355}, {8, 2320}, {100, 3612}, {214, 1698}, {388, 1319}, {404, 3576}, {452, 3487}, {515, 2476}, {958, 3715}, {1001, 1388}, {1320, 1697}, {2136, 3601}
X(3898) lies on these lines: {1, 21}, {2, 2802}, {10, 496}, {55, 214}, {65, 3636}, {72, 3635}, {145, 3678}, {210, 392}, {517, 549}, {960, 3244}, {997, 3158}, {1125, 3057}, {3616, 3754}
X(3899) lies on these lines: {1, 21}, {8, 3583}, {65, 3624}, {72, 3586}, {165, 2800}, {210, 381}, {392, 3742}, {484, 997}, {960, 1698}, {1376, 3245}, {2802, 3681}, {3057, 3633}
X(3900) lies on these lines: {1, 905}, {8, 885}, {30, 511}, {55, 1946}, {650, 663}, {764, 3680}, {1146, 3022}, {1960, 2516}, {2192, 2431}, {2254, 3669}, {3064, 3700}, {3158, 3251}
X(3900) = isogonal conjugate of X(934)
X(3900) = isotomic conjugate of X(4569)
X(3900) = crossdifference of every pair of points on line X(6)X(57)
X(3900) = X(2)-Ceva conjugate of X(35508)
X(3900) = medial-isotomic conjugate of X(35508)
X(3901) lies on these lines: {1, 21}, {65, 3679}, {72, 1698}, {78, 3336}, {79, 3419}, {517, 1657}, {518, 3632}, {519, 1770}, {942, 3624}, {997, 3337}, {3681, 3754}
X(3902) lies on these lines: {8, 210}, {75, 3241}, {314, 1320}, {321, 519}, {528, 1227}, {1089, 3625}, {1441, 2099}, {2170, 2321}
X(3903) lies on these lines: {1, 1581}, {99, 512}, {256, 1320}, {257, 3057}, {643, 3573}, {893, 3744}, {1120, 1431}, {1280, 1432}
X(3903) = isogonal conjugate of X(4367)
X(3903) = isotomic conjugate of X(4374)
X(3903) = trilinear pole of line X(9)X(43)
X(3904) lies on these lines: {190, 644}, {323, 401}, {514, 661}, {676, 3616}, {1320, 2804}, {2254, 2785}, {2401, 2990}
X(3904) = isogonal conjugate of X(32675)
X(3904) = isotomic conjugate of X(655)
X(3905) lies on these lines: {1, 75}, {32, 712}, {192, 2329}, {239, 3061}, {726, 1975}, {1429, 3210}, {1959, 3187}
X(3906) lies on these lines: {30, 511}, {39, 647}, {76, 850}, {262, 2394}, {879, 3431}
X(3906) = isogonal conjugate of X(11636)
X(3906) = X(2)-Ceva conjugate of X(17416)
X(3906) = infinite point of radical axis of Brocard circle and orthocentroidal circle
X(3906) = crossdifference of every pair of points on line X(6)X(23)
X(3907) lies on these lines: {1, 810}, {30, 511}, {663, 3716}, {1027, 1222}, {1459, 2517}
X(3907) = crossdifference of every pair of points on line X(6)X(893)
X(3908) lies on these lines: {100, 110}, {513, 644}, {668, 892}
X(3909) lies on these lines: {100, 109}, {2836, 2895}
X(3909) = anticomplement of X(18191)
X(3910) lies on these lines: {30, 511}, {3700,4391}
X(3910) = isogonal conjugate of X(8687)
X(3910) = isotomic conjugate of X(6648)
X(3910) = crossdifference of every pair of points on line X(2)X(12)
X(3910) = X(830)com[Inverse(n(4th Euler triangle))]
X(3911) lies on these lines: {1, 631}, {2, 7}, {3, 950}, {8, 1420}, {10, 56}, {11, 516}, {12, 3634}, {36, 80}, {40, 3086}, {46, 499}, {65, 392}, {73, 3216}, {88, 655}, {100, 2078}, {108, 1861}, {109, 238}, {140, 942}, {165, 497}, {201, 3670}, {208, 475}, {214, 519}, {216, 1108}, {241, 514}, {244, 3011}, {319, 2985}, {333, 1412}, {376, 3586}, {388, 1698}, {405, 1466}, {468, 1876}, {484, 3582}, {496, 3579}, {498, 3338}, {517, 1387}, {518, 3035}, {551, 2099}, {580, 3075}, {602, 1771}, {603, 1724}, {604, 1150}, {673, 927}, {750, 1471}, {899, 1458}, {902, 1647}, {914, 1813}, {936, 1467}, {938, 3523}, {993, 1470}, {1016, 1429}, {1054, 1738}, {1279, 3756}, {1357, 1463}, {1371, 1659}, {1376, 1617}, {1388, 3244}, {1402, 3741}, {1436, 1751}, {1519, 2950}, {1621, 3256}, {1699, 3474}, {1768, 1776}, {1838, 1940}, {3085, 3333}, {3321, 3323}, {3339, 3485}, {3340, 3616}, {3476, 3679}, {3487, 3525}, {3488, 3524}
X(3911) = isogonal conjugate of X(2316)
X(3911) = isotomic conjugate of X(4997)
X(3911) = complement of X(908)
X(3911) = {X(2),X(7)}-harmonic conjugate of X(5219)
X(3911) = {X(2),X(57)}-harmonic conjugate of X(226)
X(3911) = inverse-in-circumconic-centered-at-X(9) of X(57)
X(3911) = perspector of Gemini triangle 10 and cross-triangle of ABC and Gemini triangle 10
X(3911) = trilinear pole of line X(900)X(1317) (the perspectrix of ABC and Gemini triangle 9)
X(3912) is the perspector of the circumconic passing through the isogonal conjugates of PU(48). (Randy Hutson, August 29, 2018)
X(3912) lies on the cubics K766 and K868 and these lines: {1, 2}, {5, 29331}, {6, 3879}, {7, 346}, {9, 69}, {11, 20486}, {21, 24632}, {35, 21511}, {36, 21495}, {37, 141}, {39, 37596}, {40, 36698}, {44, 524}, {45, 599}, {55, 11343}, {56, 21477}, {57, 345}, {58, 16050}, {63, 3730}, {71, 16574}, {72, 30810}, {73, 28777}, {75, 142}, {76, 85}, {77, 28739}, {81, 5280}, {86, 4758}, {88, 34892}, {99, 35163}, {100, 2725}, {101, 2862}, {115, 20337}, {144, 3161}, {171, 33158}, {172, 5337}, {190, 320}, {192, 3662}, {193, 1743}, {194, 24215}, {213, 37676}, {220, 23151}, {238, 5847}, {241, 3693}, {244, 32848}, {281, 27384}, {297, 1785}, {302, 5242}, {303, 5243}, {307, 27396}, {313, 18040}, {318, 37448}, {319, 3686}, {321, 1930}, {322, 20946}, {325, 26012}, {326, 28753}, {329, 15490}, {333, 1174}, {334, 350}, {335, 726}, {354, 3703}, {391, 18230}, {440, 10381}, {469, 5342}, {480, 30620}, {487, 32556}, {488, 32555}, {514, 661}, {515, 6996}, {516, 3685}, {518, 3717}, {522, 30188}, {528, 4702}, {536, 1086}, {537, 4439}, {545, 4908}, {553, 32007}, {573, 21371}, {594, 3739}, {597, 16666}, {637, 31562}, {638, 31561}, {664, 35158}, {666, 2338}, {668, 3975}, {672, 18206}, {673, 5853}, {674, 4553}, {740, 1738}, {748, 32852}, {750, 33156}, {752, 4432}, {756, 33081}, {760, 20715}, {846, 33085}, {894, 3664}, {903, 28301}, {940, 17750}, {942, 3695}, {944, 7397}, {946, 7377}, {948, 9312}, {950, 7270}, {952, 19512}, {960, 30847}, {966, 17270}, {968, 26034}, {980, 2276}, {982, 12782}, {984, 4078}, {988, 16043}, {991, 12618}, {993, 16367}, {999, 21526}, {1001, 3416}, {1009, 37609}, {1015, 35126}, {1016, 1429}, {1018, 20367}, {1043, 16054}, {1089, 13407}, {1100, 3589}, {1121, 4555}, {1146, 34852}, {1150, 16788}, {1155, 3712}, {1211, 6537}, {1213, 4698}, {1214, 18639}, {1229, 1441}, {1265, 11523}, {1269, 4043}, {1270, 30412}, {1271, 30413}, {1278, 4072}, {1279, 5846}, {1329, 30812}, {1330, 12572}, {1332, 2323}, {1334, 17137}, {1375, 5440}, {1376, 16412}, {1388, 31230}, {1432, 32017}, {1438, 3684}, {1442, 28780}, {1445, 3692}, {1449, 3618}, {1453, 13742}, {1468, 33819}, {1475, 27109}, {1500, 3666}, {1574, 31198}, {1621, 33078}, {1654, 17260}, {1708, 3719}, {1724, 11342}, {1742, 21629}, {1757, 34379}, {1763, 8897}, {1781, 27059}, {1818, 1861}, {1826, 18065}, {1836, 4387}, {1848, 17442}, {1921, 20496}, {1931, 6629}, {1948, 6335}, {1962, 32781}, {1978, 18891}, {1992, 16670}, {1997, 30827}, {2183, 3882}, {2223, 4447}, {2227, 23462}, {2228, 3122}, {2238, 16782}, {2239, 3747}, {2243, 4760}, {2294, 18697}, {2324, 27509}, {2329, 5745}, {2345, 4470}, {2550, 3886}, {2796, 24692}, {2887, 24210}, {2895, 27065}, {2901, 19791}, {2951, 9801}, {3061, 3452}, {3120, 31041}, {3175, 3782}, {3208, 21281}, {3210, 24177}, {3218, 3977}, {3219, 4001}, {3243, 4901}, {3246, 28538}, {3247, 3619}, {3262, 4858}, {3263, 3930}, {3264, 4033}, {3271, 9025}, {3295, 21514}, {3303, 21496}, {3304, 21519}, {3305, 5739}, {3306, 17740}, {3314, 24241}, {3454, 26601}, {3475, 3974}, {3596, 17786}, {3601, 24609}, {3620, 3731}, {3629, 16669}, {3630, 15492}, {3631, 16814}, {3644, 4398}, {3668, 25252}, {3672, 17304}, {3673, 20173}, {3675, 22116}, {3689, 26007}, {3694, 16608}, {3696, 3826}, {3701, 33839}, {3704, 3812}, {3706, 3925}, {3707, 17335}, {3710, 3868}, {3713, 25878}, {3714, 25466}, {3723, 17045}, {3746, 21516}, {3748, 4030}, {3750, 33079}, {3752, 20255}, {3755, 4429}, {3758, 4667}, {3759, 17341}, {3760, 21073}, {3763, 4657}, {3772, 3905}, {3773, 24325}, {3775, 3842}, {3778, 21330}, {3779, 35892}, {3781, 10477}, {3790, 5542}, {3814, 26019}, {3821, 3993}, {3823, 28581}, {3844, 4026}, {3873, 32862}, {3875, 4000}, {3876, 4101}, {3888, 29353}, {3914, 25957}, {3933, 25066}, {3944, 4138}, {3945, 5749}, {3946, 4360}, {3963, 20891}, {3965, 25067}, {3966, 4423}, {3969, 4359}, {3971, 33064}, {3973, 20080}, {3986, 17238}, {3991, 37597}, {3994, 32856}, {3995, 17184}, {3998, 18636}, {4007, 20195}, {4011, 32946}, {4019, 25065}, {4021, 17291}, {4022, 21035}, {4029, 4389}, {4037, 24081}, {4038, 32780}, {4053, 22047}, {4054, 4671}, {4058, 4699}, {4060, 5564}, {4070, 24685}, {4080, 6549}, {4082, 32937}, {4085, 31342}, {4095, 17062}, {4098, 4704}, {4102, 32015}, {4104, 33084}, {4109, 18140}, {4110, 30090}, {4144, 4465}, {4150, 18044}, {4273, 30906}, {4292, 7283}, {4297, 37416}, {4312, 24280}, {4314, 17691}, {4339, 33198}, {4361, 17265}, {4363, 4675}, {4366, 17766}, {4370, 4715}, {4383, 16502}, {4385, 21620}, {4395, 4727}, {4409, 28322}, {4415, 35652}, {4419, 17274}, {4434, 26629}, {4440, 4887}, {4441, 30949}, {4445, 17259}, {4461, 31995}, {4464, 4852}, {4473, 20072}, {4475, 20590}, {4488, 20059}, {4513, 5228}, {4515, 21258}, {4552, 22464}, {4642, 26562}, {4649, 33159}, {4656, 27184}, {4659, 4873}, {4665, 4688}, {4670, 17359}, {4681, 17235}, {4686, 7263}, {4687, 5224}, {4690, 17330}, {4693, 24715}, {4700, 16786}, {4708, 4755}, {4716, 31252}, {4719, 25914}, {4725, 4969}, {4741, 17333}, {4767, 36240}, {4779, 30332}, {4780, 27480}, {4855, 24580}, {4856, 17121}, {4859, 17151}, {4864, 9053}, {4884, 21342}, {4890, 25144}, {4936, 20111}, {4968, 17672}, {4970, 24169}, {4974, 17772}, {4975, 30384}, {5010, 21508}, {5015, 17681}, {5081, 26003}, {5174, 37389}, {5204, 16431}, {5217, 16436}, {5219, 28808}, {5223, 27549}, {5226, 31994}, {5227, 26130}, {5232, 5296}, {5233, 5316}, {5247, 33821}, {5266, 7819}, {5273, 37655}, {5278, 16783}, {5279, 16566}, {5284, 33075}, {5295, 8728}, {5299, 32911}, {5435, 32003}, {5461, 30823}, {5563, 21540}, {5587, 36662}, {5590, 6351}, {5591, 6352}, {5603, 7402}, {5687, 37272}, {5691, 7406}, {5695, 5880}, {5717, 13740}, {5718, 30818}, {5814, 11108}, {5839, 37650}, {5977, 16609}, {6174, 32043}, {6185, 14942}, {6329, 16668}, {6510, 36949}, {6547, 16610}, {6550, 24133}, {6557, 10405}, {6631, 37758}, {6646, 17261}, {6651, 17770}, {6656, 13161}, {6767, 21529}, {7155, 27429}, {7232, 17262}, {7237, 17470}, {7280, 21537}, {7291, 16550}, {7308, 14555}, {7373, 21542}, {7384, 19925}, {7490, 7718}, {8055, 30695}, {8069, 37344}, {8162, 21528}, {8193, 11350}, {8287, 21237}, {8359, 37599}, {8362, 37592}, {8369, 37589}, {9028, 16560}, {9294, 21191}, {9310, 25940}, {9441, 28849}, {10026, 20529}, {10164, 28909}, {10445, 10446}, {10452, 21061}, {10481, 25242}, {11329, 25440}, {11340, 37557}, {11353, 26687}, {11712, 26231}, {11997, 18252}, {12263, 30982}, {12410, 37269}, {12699, 36731}, {13006, 36212}, {13466, 16594}, {13476, 21865}, {14001, 37552}, {14839, 20358}, {14943, 33675}, {15488, 19542}, {15507, 33845}, {15668, 17293}, {16060, 37573}, {16061, 37607}, {16484, 33076}, {16524, 37673}, {16578, 37796}, {16602, 21868}, {16603, 17046}, {16671, 32455}, {16672, 17325}, {16675, 17253}, {16676, 21356}, {16704, 36954}, {16738, 27032}, {16779, 17338}, {16781, 37679}, {16784, 37680}, {16785, 32779}, {16887, 25092}, {17050, 17143}, {17053, 27633}, {17056, 21024}, {17063, 32855}, {17066, 21958}, {17067, 17133}, {17077, 27514}, {17116, 26806}, {17120, 20090}, {17122, 33160}, {17123, 32861}, {17126, 35263}, {17144, 20257}, {17178, 27073}, {17197, 30939}, {17256, 17271}, {17258, 17273}, {17290, 17301}, {17305, 17320}, {17307, 17322}, {17331, 17343}, {17334, 17345}, {17336, 17347}, {17337, 17348}, {17339, 17350}, {17340, 17351}, {17358, 17368}, {17370, 17380}, {17371, 17381}, {17382, 17395}, {17383, 17396}, {17385, 17398}, {17463, 35552}, {17484, 32106}, {17490, 24175}, {17495, 24191}, {17499, 17778}, {17592, 33174}, {17647, 37233}, {17670, 24178}, {17671, 21075}, {17682, 24588}, {17698, 37594}, {17717, 30869}, {17719, 30837}, {17720, 24281}, {17754, 30962}, {17781, 32859}, {17787, 30097}, {17790, 21091}, {17791, 18151}, {17792, 21746}, {17798, 21989}, {17861, 20171}, {17862, 20237}, {17977, 26884}, {18073, 21094}, {18135, 30961}, {18149, 19567}, {18150, 22031}, {18480, 36728}, {18692, 26163}, {18714, 20932}, {18904, 20363}, {19582, 27129}, {19623, 25536}, {19786, 34064}, {20078, 25734}, {20132, 24295}, {20172, 32941}, {20262, 28778}, {20333, 20530}, {20336, 22021}, {20343, 20688}, {20541, 25383}, {20549, 21830}, {20610, 20785}, {20881, 30379}, {20895, 20905}, {20911, 21808}, {20912, 21417}, {20927, 20930}, {20942, 20943}, {20956, 21591}, {21010, 21985}, {21025, 25102}, {21026, 33136}, {21060, 27538}, {21069, 24220}, {21072, 30007}, {21074, 29965}, {21076, 25660}, {21101, 33931}, {21257, 25140}, {21331, 35101}, {21454, 32098}, {21810, 24090}, {22016, 22019}, {22020, 22028}, {22267, 37574}, {23058, 28827}, {23536, 33840}, {23681, 30699}, {23682, 30969}, {23829, 24290}, {24050, 27569}, {24214, 25264}, {24479, 36800}, {24629, 33089}, {24635, 25082}, {24803, 35160}, {25019, 26540}, {25099, 26543}, {25245, 26573}, {25378, 25760}, {25681, 30826}, {25772, 25856}, {25959, 33134}, {25961, 32860}, {26580, 31017}, {26738, 30629}, {26764, 27017}, {27003, 33168}, {27186, 28605}, {27254, 27267}, {27269, 27288}, {27431, 27497}, {27495, 31323}, {27523, 36854}, {27751, 35121}, {28526, 32857}, {28603, 33920}, {28606, 33172}, {30808, 36205}, {30824, 34362}, {30856, 35141}, {30857, 36236}, {31053, 31060}, {31225, 33298}, {32024, 33066}, {32913, 33164}, {32919, 33115}, {32925, 33069}, {32926, 33124}, {32928, 33123}, {32930, 32949}, {32936, 33067}, {32942, 33073}, {32943, 33072}, {33942, 34847}, {34234, 34932}, {35075, 35122}, {35085, 35134}, {35153, 35177}, {35168, 36915}, {35595, 37656}, {36605, 38255}, {37176, 37554}
X(3912) = midpoint of X(i) and X(j) for these {i,j}: {2, 17310}, {44, 17374}, {190, 320}, {238, 32846}, {239, 6542}, {335, 3797}, {350, 4562}, {1086, 3943}, {3685, 4645}, {3717, 4684}, {3932, 4966}, {4693, 24715}, {4908, 31138}, {10026, 35080}, {17264, 17297}
X(3912) = reflection of X(i) in X(j) for these {i,j}: {44, 4422}, {190, 2325}, {239, 3008}, {1086, 3834}, {1266, 1086}, {1738, 3836}, {3717, 3932}, {4440, 4887}, {4480, 190}, {4684, 4966}, {4899, 3717}, {4974, 31289}, {9436, 35094}, {35119, 20530}
X(3912) = isogonal conjugate of X(1438)
X(3912) = isotomic conjugate of X(673)
X(3912) = complement of X(239)
X(3912) = anticomplement of X(3008)
X(3912) = complementary conjugate of X(20333)
X(3912) = antitomic conjugate of X(9436)
X(3912) = inverse-in-Steiner-circumellipse of X(8)
X(3912) = inverse-in-Steiner-inellipse of X(10)
X(3912) = medial isogonal conjugate of X(20333)
X(3912) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1280, 69}, {1477, 7}, {1810, 4329}, {6078, 513}, {35160, 21285}, {35355, 150}, {36807, 6327}
X(3912) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 20333}, {2, 20542}, {6, 17793}, {31, 17755}, {100, 27854}, {171, 39080}, {172, 19563}, {213, 35068}, {291, 141}, {292, 10}, {295, 18589}, {334, 626}, {335, 2887}, {649, 38989}, {660, 3835}, {667, 35119}, {692, 27929}, {694, 3846}, {741, 3739}, {813, 513}, {875, 1086}, {876, 116}, {904, 39044}, {1911, 2}, {1922, 37}, {1967, 4357}, {2196, 3}, {2311, 960}, {3252, 120}, {3572, 11}, {4444, 21252}, {4518, 21244}, {4562, 21260}, {4583, 21262}, {4584, 512}, {4639, 23301}, {4876, 1329}, {5378, 27076}, {7077, 3452}, {7122, 5976}, {7233, 17046}, {9468, 1107}, {14598, 39}, {18265, 1212}, {18268, 1125}, {18827, 21240}, {18893, 8265}, {18895, 21235}, {18897, 16584}, {22116, 20540}, {30663, 20541}, {34067, 514}, {35352, 21253}, {37128, 3741}, {37207, 788}
X(3912) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 17755}, {75, 4712}, {92, 20431}, {334, 10}, {350, 726}, {3263, 3717}, {3570, 2786}, {4562, 514}, {4876, 17760}, {5378, 4568}, {9436, 4899}, {14942, 3729}, {18025, 8}, {18031, 75}, {18157, 3263}, {30941, 518}, {36807, 2}
X(3912) = X(i)-cross conjugate of X(j) for these (i,j): {518, 9436}, {672, 1861}, {3675, 23829}, {3693, 3717}, {3930, 518}, {3932, 3263}, {4712, 75}, {4966, 30941}, {16593, 2}, {17464, 1}, {20860, 6}
X(3912) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1438}, {3, 8751}, {4, 32658}, {6, 105}, {9, 1416}, {19, 36057}, {25, 1814}, {31, 673}, {32, 2481}, {48, 36124}, {55, 1462}, {56, 294}, {57, 2195}, {58, 18785}, {101, 1027}, {109, 1024}, {112, 10099}, {513, 919}, {514, 32666}, {560, 18031}, {604, 14942}, {649, 36086}, {650, 32735}, {651, 884}, {663, 36146}, {666, 667}, {885, 1415}, {927, 3063}, {1015, 5377}, {1106, 6559}, {1292, 2440}, {1333, 13576}, {1397, 36796}, {1407, 28071}, {1492, 29956}, {1911, 6654}, {1973, 31637}, {1980, 36803}, {2175, 34018}, {2223, 6185}, {3290, 15382}, {3309, 32644}, {4724, 36138}, {4762, 32724}, {6169, 9316}, {23696, 32674}
X(3912) = cevapoint of X(i) and X(j) for these (i,j): {1, 16550}, {2, 20533}, {6, 20871}, {350, 33677}, {518, 3693}, {672, 1818}, {3675, 24290}, {3930, 3932}, {6084, 27918}
X(3912) = crosspoint of X(i) and X(j) for these (i,j): {2, 335}, {75, 18031}, {85, 35160}, {4600, 4639}, {9436, 10029}, {18157, 30941}
X(3912) = crosssum of X(i) and X(j) for these (i,j): {6, 1914}, {31, 9454}, {41, 8647}, {1015, 8659}, {1458, 9316}
X(3912) = trilinear pole of line X(918)X(2254)
X(3912) = crossdifference of PU(48)
X(3912) = barycentric square root of X(4437)
X(3912) = polar conjugate of X(36124)
X(3912) = pole wrt polar circle of trilinear polar of X(36124) (line X(19)X(1024))
barycentric product X(i)*X(j) for these {i,j}: {1, 3263}, {7, 3717}, {8, 9436}, {10, 30941}, {37, 18157}, {69, 1861}, {75, 518}, {76, 672}, {85, 3693}, {86, 3932}, {92, 25083}, {99, 4088}, {190, 918}, {241, 312}, {264, 1818}, {274, 3930}, {304, 5089}, {305, 2356}, {306, 15149}, {310, 20683}, {313, 3286}, {321, 18206}, {334, 8299}, {335, 17755}, {341, 34855}, {345, 5236}, {350, 22116}, {522, 883}, {561, 2223}, {665, 1978}, {668, 2254}, {673, 4437}, {693, 1026}, {799, 24290}, {926, 4572}, {1025, 4391}, {1268, 4966}, {1458, 3596}, {1502, 9454}, {1876, 3718}, {1921, 3252}, {1928, 9455}, {1969, 20752}, {2283, 35519}, {2284, 3261}, {2340, 6063}, {2414, 4468}, {2481, 4712}, {2991, 20431}, {3161, 10029}, {3262, 36819}, {3264, 34230}, {3675, 7035}, {3952, 23829}, {4238, 14208}, {4373, 4899}, {4447, 7018}, {4684, 5936}, {5383, 23773}, {6184, 18031}, {6385, 39258}, {14439, 20568}, {16593, 36807}, {20504, 35574}, {31637, 34337}
X(3912) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 105}, {2, 673}, {3, 36057}, {4, 36124}, {6, 1438}, {8, 14942}, {9, 294}, {10, 13576}, {19, 8751}, {37, 18785}, {48, 32658}, {55, 2195}, {56, 1416}, {57, 1462}, {63, 1814}, {69, 31637}, {75, 2481}, {76, 18031}, {85, 34018}, {100, 36086}, {101, 919}, {109, 32735}, {120, 1738}, {190, 666}, {200, 28071}, {239, 6654}, {241, 57}, {312, 36796}, {344, 31638}, {346, 6559}, {513, 1027}, {518, 1}, {521, 23696}, {522, 885}, {536, 36816}, {650, 1024}, {651, 36146}, {656, 10099}, {663, 884}, {664, 927}, {665, 649}, {672, 6}, {673, 6185}, {692, 32666}, {765, 5377}, {883, 664}, {918, 514}, {926, 663}, {1025, 651}, {1026, 100}, {1292, 36041}, {1362, 1458}, {1458, 56}, {1642, 2246}, {1738, 14267}, {1818, 3}, {1861, 4}, {1876, 34}, {1978, 36803}, {2223, 31}, {2254, 513}, {2283, 109}, {2284, 101}, {2340, 55}, {2356, 25}, {2414, 37206}, {2808, 23694}, {3126, 2254}, {3239, 28132}, {3250, 29956}, {3252, 292}, {3263, 75}, {3286, 58}, {3675, 244}, {3693, 9}, {3699, 36802}, {3717, 8}, {3930, 37}, {3932, 10}, {4088, 523}, {4238, 162}, {4447, 171}, {4468, 2402}, {4518, 33676}, {4553, 35333}, {4554, 34085}, {4684, 3616}, {4712, 518}, {4899, 145}, {4925, 3667}, {4966, 1125}, {5089, 19}, {5236, 278}, {5257, 14625}, {6168, 6180}, {6184, 672}, {8299, 238}, {8693, 36138}, {9436, 7}, {9451, 9453}, {9454, 32}, {9455, 560}, {9502, 910}, {10029, 27818}, {14439, 44}, {14626, 2334}, {15149, 27}, {16593, 3008}, {16728, 18206}, {17435, 2170}, {17464, 3290}, {17755, 239}, {17794, 33674}, {18157, 274}, {18206, 81}, {20482, 21956}, {20504, 23770}, {20683, 42}, {20749, 20780}, {20752, 48}, {20778, 7193}, {21959, 21051}, {22116, 291}, {23102, 4712}, {23773, 21138}, {23829, 7192}, {24290, 661}, {25083, 63}, {25302, 34063}, {26706, 36111}, {27919, 4366}, {28850, 14197}, {30941, 86}, {34230, 106}, {34253, 1429}, {34337, 1861}, {34855, 269}, {35293, 1155}, {36101, 9503}, {36819, 104}, {36905, 14189}, {37908, 2299}, {38989, 27846}, {39046, 20470}, {39258, 213}
X(3912) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2, 17023}, {1, 3679, 36479}, {1, 17284, 2}, {1, 17316, 29574}, {1, 29573, 17316}, {1, 29579, 29596}, {1, 29583, 29601}, {1, 29598, 26626}, {1, 29602, 29585}, {1, 29637, 1125}, {1, 29660, 551}, {1, 29674, 10}, {1, 29960, 30038}, {2, 8, 4384}, {2, 10, 24603}, {2, 145, 5222}, {2, 239, 3008}, {2, 306, 3687}, {2, 3187, 26723}, {2, 3616, 29603}, {2, 3621, 24599}, {2, 3661, 10}, {2, 4393, 17367}, {2, 5308, 16831}, {2, 6542, 239}, {2, 10453, 17026}, {2, 16815, 31211}, {2, 16816, 29628}, {2, 16826, 1125}, {2, 17135, 24592}, {2, 17230, 3661}, {2, 17244, 29571}, {2, 17284, 29596}, {2, 17292, 29604}, {2, 17316, 1}, {2, 17367, 31191}, {2, 20016, 29590}, {2, 20055, 16816}, {2, 26113, 25510}, {2, 26247, 3011}, {2, 26575, 24982}, {2, 26581, 24987}, {2, 26593, 25006}, {2, 26594, 25007}, {2, 26595, 24996}, {2, 26597, 24997}, {2, 26599, 25011}, {2, 26610, 25005}, {2, 26626, 29598}, {2, 26774, 27044}, {2, 26821, 26982}, {2, 29569, 16826}, {2, 29570, 17397}, {2, 29572, 17244}, {2, 29573, 29574}, {2, 29577, 29594}, {2, 29579, 17284}, {2, 29582, 29600}, {2, 29583, 17316}, {2, 29585, 26626}, {2, 29586, 29614}, {2, 29587, 17292}, {2, 29589, 29569}, {2, 29590, 29607}, {2, 29591, 29610}, {2, 29592, 29609}, {2, 29593, 29576}, {2, 29595, 29612}, {2, 29599, 29581}, {2, 29610, 3634}, {2, 29611, 17308}, {2, 29612, 19862}, {2, 29616, 8}, {2, 29618, 3244}, {2, 29621, 5308}, {2, 29624, 3616}, {2, 29625, 3636}, {2, 30967, 4871}, {2, 31027, 3741}, {2, 31028, 3840}, {2, 31038, 11019}, {2, 31093, 3006}, {2, 32858, 306}, {2, 34255, 11679}, {2, 36845, 24600}, {6, 4851, 3879}, {6, 17267, 17279}, {6, 17279, 17353}, {6, 17311, 4851}, {7, 346, 3729}, {7, 4869, 17298}, {8, 29616, 17294}, {8, 29627, 2}, {8, 29966, 30030}, {9, 69, 4416}, {9, 344, 25101}, {9, 17296, 69}, {10, 29571, 2}, {10, 29594, 3661}, {10, 29600, 29571}, {10, 29968, 30063}, {11, 20486, 20544}, {37, 141, 4357}, {37, 17231, 141}, {37, 17237, 4364}, {42, 30821, 2}, {43, 30822, 2}, {45, 599, 4643}, {69, 344, 9}, {75, 142, 24199}, {75, 2321, 4431}, {75, 17233, 2321}, {75, 17234, 142}, {75, 17240, 17233}, {75, 17241, 17234}, {76, 312, 4044}, {81, 33157, 5294}, {86, 17285, 17289}, {86, 17289, 5750}, {141, 4364, 17237}, {141, 17243, 37}, {142, 2321, 75}, {142, 17233, 4431}, {144, 3161, 25728}, {145, 5222, 16834}, {145, 29986, 30036}, {145, 30833, 2}, {190, 17264, 2325}, {190, 17297, 320}, {192, 3662, 3663}, {192, 17232, 3662}, {192, 17242, 3950}, {193, 26685, 1743}, {200, 30813, 2}, {226, 7146, 3674}, {226, 21071, 4044}, {239, 17266, 2}, {239, 17310, 6542}, {239, 29607, 29590}, {241, 3693, 25083}, {306, 25935, 8}, {312, 7146, 17760}, {312, 18134, 226}, {312, 20917, 76}, {319, 17263, 17277}, {319, 17277, 3686}, {320, 2325, 4480}, {320, 17264, 190}, {321, 3970, 22048}, {321, 18139, 5249}, {321, 20913, 20888}, {344, 17296, 4416}, {345, 18141, 57}, {346, 4869, 7}, {594, 3739, 4967}, {594, 17245, 3739}, {894, 17268, 17280}, {894, 17280, 17355}, {894, 17300, 3664}, {894, 17312, 17300}, {1001, 3416, 3883}, {1100, 17357, 3589}, {1125, 29604, 2}, {1229, 1441, 20236}, {1999, 16826, 30168}, {2321, 17234, 24199}, {2345, 4648, 10436}, {3006, 29824, 26015}, {3161, 21296, 144}, {3178, 3831, 5530}, {3218, 32849, 3977}, {3219, 32863, 4001}, {3247, 17306, 17321}, {3262, 37788, 4858}, {3452, 4035, 4417}, {3589, 17390, 1100}, {3616, 29624, 29597}, {3619, 17321, 17306}, {3620, 17257, 17272}, {3631, 17332, 17344}, {3632, 31183, 16833}, {3661, 17230, 29594}, {3661, 17244, 2}, {3661, 29571, 24603}, {3661, 29572, 29571}, {3661, 29576, 29593}, {3661, 29577, 17230}, {3661, 29581, 29576}, {3661, 29582, 17244}, {3662, 17232, 21255}, {3662, 17242, 192}, {3663, 3950, 192}, {3663, 21255, 3662}, {3664, 17355, 894}, {3686, 6666, 17277}, {3687, 29968, 24603}, {3723, 17384, 17045}, {3729, 17298, 7}, {3731, 17272, 17257}, {3739, 17229, 594}, {3758, 17342, 17354}, {3758, 17378, 4667}, {3758, 17387, 17378}, {3759, 17341, 17352}, {3759, 17386, 17377}, {3763, 16777, 4657}, {3834, 3943, 1266}, {3840, 29671, 24239}, {3844, 15569, 4026}, {3875, 17282, 4000}, {3879, 17353, 6}, {3932, 4684, 4899}, {3936, 4358, 908}, {3950, 21255, 3663}, {4000, 17314, 3875}, {4043, 18143, 1269}, {4360, 16706, 3946}, {4360, 17283, 16706}, {4361, 17265, 17278}, {4361, 17309, 17299}, {4363, 17269, 17281}, {4363, 17313, 4675}, {4364, 17237, 4357}, {4384, 17294, 8}, {4393, 17389, 3244}, {4393, 29629, 31191}, {4416, 25101, 9}, {4417, 18743, 3452}, {4431, 24199, 75}, {4437, 16593, 17755}, {4445, 17259, 17275}, {4447, 8299, 2223}, {4475, 20703, 20590}, {4511, 28757, 26006}, {4664, 17227, 4389}, {4665, 34824, 4688}, {4670, 17359, 17369}, {4671, 31019, 4054}, {4675, 17281, 4363}, {4681, 17235, 17246}, {4687, 5224, 5257}, {4687, 17228, 5224}, {4693, 31151, 24715}, {4698, 17239, 1213}, {4704, 17236, 17247}, {4727, 31243, 4395}, {4851, 17267, 17353}, {4851, 17279, 6}, {4852, 17356, 17366}, {4852, 17388, 4464}, {4873, 6173, 4659}, {5233, 30829, 5316}, {5308, 29611, 2}, {6376, 18743, 30830}, {6542, 17266, 3008}, {6542, 29590, 20016}, {7081, 29839, 13405}, {7232, 17262, 17276}, {7270, 19815, 37086}, {10324, 11019, 8}, {10436, 17286, 2345}, {10453, 29641, 4847}, {14829, 33116, 5745}, {15668, 17293, 17303}, {16284, 18743, 30854}, {16672, 21358, 17325}, {16706, 17315, 4360}, {16814, 17344, 17332}, {16815, 29626, 2}, {16816, 20055, 29617}, {16826, 17292, 2}, {16826, 29575, 29569}, {16826, 29587, 29604}, {16826, 29589, 29606}, {16826, 29609, 29592}, {16831, 17308, 2}, {16834, 29605, 145}, {17023, 29574, 1}, {17023, 29596, 2}, {17023, 29601, 29574}, {17034, 30173, 10}, {17045, 34573, 17384}, {17160, 27191, 37756}, {17229, 17245, 4967}, {17230, 17244, 10}, {17230, 29572, 2}, {17230, 29579, 29674}, {17230, 29582, 29571}, {17230, 29599, 29593}, {17230, 29600, 24603}, {17231, 17243, 4357}, {17232, 17242, 3663}, {17233, 17234, 75}, {17233, 17241, 142}, {17234, 17240, 2321}, {17238, 27268, 17248}, {17240, 17241, 75}, {17244, 29572, 29600}, {17244, 29576, 29581}, {17244, 29577, 3661}, {17244, 29581, 29599}, {17244, 29582, 29572}, {17244, 29594, 24603}, {17244, 29674, 29596}, {17248, 27268, 3986}, {17260, 17287, 1654}, {17261, 17288, 6646}, {17263, 17277, 6666}, {17263, 17295, 3686}, {17265, 17309, 4361}, {17266, 17310, 239}, {17267, 17311, 6}, {17268, 17300, 17355}, {17268, 17312, 894}, {17269, 17313, 4363}, {17277, 17295, 319}, {17278, 17299, 4361}, {17279, 17311, 3879}, {17280, 17300, 894}, {17280, 17312, 3664}, {17283, 17315, 3946}, {17284, 17316, 17023}, {17284, 29573, 1}, {17284, 29583, 29574}, {17284, 29602, 29598}, {17285, 17317, 5750}, {17289, 17317, 86}, {17290, 17318, 17301}, {17291, 17319, 17302}, {17292, 29569, 1125}, {17292, 29575, 16826}, {17292, 29637, 29596}, {17302, 17319, 4021}, {17310, 27757, 6633}, {17316, 26626, 29585}, {17316, 29573, 29601}, {17316, 29579, 2}, {17316, 29583, 29573}, {17316, 29585, 29602}, {17335, 17346, 3707}, {17335, 17360, 17346}, {17336, 17361, 17347}, {17337, 17362, 17348}, {17338, 17363, 17349}, {17339, 17364, 17350}, {17340, 17365, 17351}, {17341, 17386, 3759}, {17342, 17387, 3758}, {17348, 17372, 17362}, {17349, 17373, 17363}, {17350, 17375, 17364}, {17351, 17376, 17365}, {17352, 17377, 3759}, {17354, 17378, 3758}, {17354, 17387, 4667}, {17358, 17379, 17368}, {17366, 17388, 4852}, {17367, 17389, 4393}, {17367, 29618, 17389}, {17367, 29629, 2}, {17368, 17391, 17379}, {17369, 17392, 4670}, {17370, 17393, 17380}, {17371, 17394, 17381}, {17385, 28639, 17398}, {17389, 29629, 17367}, {17397, 29570, 551}, {17397, 29613, 2}, {17758, 21070, 20888}, {17763, 29632, 3011}, {17786, 20923, 3596}, {18040, 18137, 313}, {18139, 20913, 17758}, {18230, 32099, 391}, {19791, 37096, 23537}, {20016, 29590, 239}, {20055, 29617, 3625}, {20337, 23947, 115}, {21371, 22370, 573}, {24899, 24947, 2}, {25957, 32915, 3914}, {26540, 26669, 25019}, {26626, 29585, 1}, {26626, 29593, 29659}, {26626, 29598, 17023}, {27113, 27166, 2}, {27191, 37756, 17067}, {27399, 27526, 6745}, {27474, 27475, 27478}, {28422, 28713, 26006}, {28734, 28789, 2}, {28740, 28795, 2}, {28742, 28797, 2}, {28751, 28805, 2}, {28756, 28812, 2}, {28757, 28813, 2}, {28761, 28816, 2}, {28808, 30828, 5219}, {28813, 28982, 26006}, {29569, 29575, 29606}, {29569, 29579, 29637}, {29569, 29587, 2}, {29569, 29589, 29575}, {29571, 29594, 10}, {29571, 29600, 17244}, {29572, 29577, 10}, {29572, 29593, 29599}, {29573, 29579, 17023}, {29573, 29598, 29602}, {29574, 29596, 17023}, {29574, 29601, 17316}, {29575, 29587, 1125}, {29576, 29581, 2}, {29576, 29593, 10}, {29577, 29582, 2}, {29578, 29591, 3634}, {29578, 29610, 2}, {29579, 29583, 1}, {29580, 29586, 3636}, {29580, 29614, 29586}, {29581, 29599, 29571}, {29584, 29588, 3635}, {29584, 29619, 29588}, {29585, 29602, 29574}, {29587, 29589, 16826}, {29590, 29607, 3008}, {29592, 29609, 1125}, {29593, 29599, 2}, {29594, 29600, 2}, {29595, 29608, 19862}, {29596, 29601, 1}, {29597, 29603, 3616}, {29598, 29602, 1}, {29604, 29606, 16826}, {29608, 29612, 2}, {29608, 29622, 29612}, {29610, 29620, 29578}, {29611, 29621, 16831}, {29612, 29622, 29595}, {29613, 29623, 29570}, {29614, 29625, 29580}, {29615, 29626, 16815}, {29616, 29627, 4384}, {29617, 29628, 16816}, {29618, 29629, 4393}, {29619, 29630, 29584}, {29643, 30942, 29639}, {29988, 30059, 30109}, {31017, 31035, 26580}
Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the same side of BC as A. Define OB and OC cyclically. X(3913) is the radical center of the circles OA, OB, OC. See the reference at X(1001).
X(3913) lies on these lines: {1, 474}, {2, 3303}, {3, 519}, {4, 528}, {6, 979}, {8, 21}, {10, 1001}, {12, 3434}, {20, 529}, {35, 956}, {36, 3633}, {40, 518}, {43, 1191}, {46, 3555}, {56, 100}, {65, 224}, {72, 2900}, {78, 3057}, {81, 2334}, {200, 960}, {218, 1018}, {220, 3208}, {341, 3685}, {390, 480}, {404, 3241}, {405, 3679}, {497, 1329}, {521, 3157}, {535, 1657}, {950, 1260}, {978, 1616}, {986, 3242}, {993, 3625}, {999, 3244}, {1104, 3749}, {1279, 1722}, {1320, 1392}, {1466, 3476}, {1482, 2802}, {1621, 3617}, {2082, 3693}, {2264, 3692}, {2269, 3713}, {2478, 3058}, {2550, 2894}, {2886, 3085}, {2975, 3621}, {3035, 3086}, {3243, 3339}, {3698, 3748}
X(3913) = X(6247)-of-excentral-triangle
X(3913) = intouch-to-ABC barycentric image of X(10)
X(3914) lies on these lines: {1, 224}, {2, 968}, {4, 1039}, {6, 1836}, {10, 321}, {11, 3752}, {31, 516}, {38, 3663}, {42, 226}, {43, 908}, {46, 1076}, {55, 3011}, {58, 1770}, {63, 1711}, {65, 225}, {75, 305}, {141, 3706}, {142, 3720}, {228, 1284}, {278, 2263}, {306, 740}, {307, 3778}, {354, 1086}, {387, 1838}, {497, 614}, {517, 1072}, {518, 3782}, {528, 3744}, {536, 3703}, {612, 2550}, {748, 3008}, {774, 1210}, {851, 1402}, {899, 3452}, {942, 1070}, {946, 1193}, {1211, 3696}, {1279, 3058}, {1699, 2999}, {1722, 2478}, {1837, 1853}, {1842, 3556}, {1851, 2082}, {2006, 3256}, {2092, 3136}, {2650, 3671}, {2886, 3666}, {3210, 3705}
X(3914) = complement of X(32929)
X(3915) lies on these lines: {1, 21}, {3, 902}, {6, 1334}, {8, 238}, {10, 748}, {32, 3230}, {35, 995}, {40, 614}, {41, 1914}, {42, 3295}, {46, 244}, {48, 3285}, {55, 1191}, {56, 1149}, {65, 1279}, {78, 3749}, {100, 978}, {109, 1106}, {171, 3616}, {212, 1104}, {213, 2241}, {221, 1458}, {386, 2177}, {392, 1472}, {404, 3550}, {517, 582}, {519, 1724}, {601, 1385}, {603, 1319}, {750, 1125}, {944, 3073}, {946, 3011}, {960, 976}, {1001, 1918}, {1042, 1617}, {1055, 3053}, {1253, 1697}, {1388, 1399}, {1451, 2099}, {1829, 2212}, {1834, 3058}, {1935, 3476}, {2098, 2361}, {2268, 2300}
X(3915) = isogonal conjugate of X(34860)
X(3915) = crosspoint of X(109) and X(765)
X(3915) = crosssum of X(i) and X(j) for these {i,j}: {244, 522}, {512, 16614}
X(3915) = crossdifference of every pair of points on line X(661)X(4521)
X(3916) lies on these lines: {1, 3052}, {3, 63}, {8, 376}, {9, 474}, {10, 535}, {20, 3419}, {21, 942}, {35, 518}, {36, 191}, {40, 956}, {44, 3216}, {46, 958}, {55, 3555}, {56, 392}, {57, 405}, {58, 3666}, {65, 993}, {140, 908}, {144, 3523}, {283, 1789}, {329, 631}, {404, 3219}, {411, 971}, {517, 2975}, {553, 1125}, {603, 1214}, {758, 2646}, {896, 1193}, {988, 1707}, {1001, 3338}, {1104, 3670}, {1158, 3428}, {1212, 1759}, {1376, 3697}, {1437, 1444}, {1465, 1935}, {1466, 1708}, {1724, 3752}, {1748, 1871}, {1770, 2886}, {2094, 3616}, {2915, 3220}, {3337, 3742}
X(3916) = isogonal conjugate of polar conjugate of X(4359)
X(3916) = isotomic conjugate of isogonal conjugate of X(23201)
X(3916) = isotomic conjugate of polar conjugate of X(1100)
X(3916) = X(19)-isoconjugate of X(1255)
X(3917) lies on these lines: {2, 51}, {3, 49}, {22, 1495}, {25, 1350}, {38, 1401}, {39, 3051}, {52, 140}, {63, 295}, {69, 305}, {125, 343}, {141, 427}, {143, 632}, {165, 2807}, {181, 750}, {182, 1993}, {187, 1501}, {212, 1364}, {216, 3289}, {219, 1473}, {228, 1818}, {354, 674}, {389, 631}, {404, 970}, {418, 2972}, {426, 577}, {549, 1154}, {575, 1994}, {599, 1853}, {612, 1469}, {614, 3056}, {626, 2450}, {748, 3271}, {851, 1764}, {991, 1011}, {1038, 1425}, {1040, 3270}, {1194, 1613}, {1196, 3231}, {1352, 1370}, {1915, 2076}, {2810, 3681}, {2896, 3491}, {3525, 3567}
X(3917) = isogonal conjugate of X(32085)X(3918) lies on these lines: {10, 12}, {354, 3625}, {474, 1388}, {484, 3647}, {517, 3628}, {942, 3626}, {1125, 1387}, {3635, 3742}
X(3919) lies on these lines: {1, 1392}, {10, 12}, {354, 2802}, {517, 549}, {942, 3244}, {1159, 1376}, {1621, 3245}
X(3920) lies on these lines: {1, 2}, {6, 3681}, {22, 55}, {23, 3743}, {25, 3295}, {31, 984}, {33, 390}, {37, 82}, {38, 171}, {81, 518}, {86, 3263}, {100, 3666}, {105, 1255}, {172, 1627}, {210, 1386}, {238, 756}, {388, 1370}, {427, 495}, {611, 1993}, {750, 982}, {846, 902}, {940, 3242}, {968, 3749}, {1038, 3600}, {1056, 1060}, {1150, 3769}, {1172, 2346}, {1180, 2276}, {1194, 1500}, {1203, 3678}, {1469, 2979}, {1757, 2308}, {1921, 3112}, {1962, 3722}, {1995, 3303}, {3056, 3060}, {3290, 3723}, {3306, 3677}, {3315, 3742}
X(3920) = complement of X(33090)
X(3921) lies on these lines: {10, 12}, {392, 3679}, {405, 3158}, {517, 3545}, {1698, 3555}, {1962, 3214}
X(3922) lies on these lines: {10, 12}, {145, 354}, {517, 3526}, {942, 3632}, {1706, 3689}, {3057, 3616}
X(3923) lies on these lines: {1, 87}, {2, 846}, {4, 9}, {6, 740}, {8, 1757}, {20, 2944}, {31, 321}, {44, 3696}, {55, 1215}, {58, 314}, {63, 3741}, {75, 238}, {76, 1966}, {171, 312}, {182, 2783}, {190, 984}, {226, 3771}, {256, 1008}, {386, 1045}, {519, 1992}, {536, 1386}, {537, 3242}, {752, 3416}, {896, 1150}, {936, 1721}, {964, 2292}, {988, 1125}, {990, 997}, {996, 2802}, {1009, 1284}, {1352, 2792}, {1468, 3702}, {1836, 2887}, {2308, 3187}, {2795, 3734}, {3175, 3745}
X(3923) = complement of X(24248)
X(3923) = anticomplement of X(3821)
X(3923) = 1st-Brocard-isotomic conjugate of X(3735)
X(3923) = X(1)-of-1st-Brocard-triangle
X(3924) lies on these lines: {1, 2}, {4, 3120}, {6, 2294}, {7, 2647}, {21, 986}, {28, 2206}, {31, 65}, {32, 3125}, {34, 207}, {38, 958}, {40, 902}, {56, 244}, {81, 409}, {227, 1319}, {405, 2292}, {517, 582}, {604, 1880}, {607, 2170}, {748, 960}, {758, 1724}, {942, 1468}, {977, 1220}, {982, 2975}, {993, 3670}, {1106, 1455}, {1148, 1870}, {1191, 2099}, {1253, 1279}, {1453, 2308}, {1616, 2098}, {1837, 3772}, {1854, 2310}, {2331, 3554}, {2646, 3752}
Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the medial triangle at X(3925). (Randy Hutson, December 2, 2017)
X(3925) lies on these lines: {2, 11}, {5, 40}, {8, 3475}, {9, 1836}, {10, 12}, {19, 427}, {38, 1086}, {56, 443}, {71, 1213}, {75, 3703}, {125, 3611}, {141, 3779}, {142, 354}, {306, 3696}, {329, 3715}, {377, 958}, {429, 1869}, {495, 3679}, {496, 3624}, {516, 3683}, {594, 2294}, {612, 3772}, {756, 3120}, {858, 3101}, {908, 3740}, {984, 3782}, {1212, 1855}, {1329, 2476}, {1738, 3666}, {1842, 1883}, {1853, 3197}, {1859, 1861}, {3189, 3616}
X(3926) lies on these lines: {2, 39}, {3, 69}, {4, 325}, {5, 1007}, {20, 99}, {32, 193}, {110, 2366}, {115, 2996}, {183, 631}, {187, 439}, {264, 1217}, {276, 1502}, {304, 345}, {316, 3146}, {326, 1264}, {339, 3548}, {350, 3086}, {441, 1073}, {491, 1587}, {492, 1588}, {498, 3761}, {499, 3760}, {524, 3053}, {574, 3620}, {626, 2549}, {1078, 3523}, {1102, 3719}, {1235, 3541}, {1310, 3556}, {1909, 3085}, {2548, 3734}
X(3926) = isogonal conjugate of X(2207)
X(3926) = isotomic conjugate of X(393)
X(3926) = complement of X(6392)
X(3926) = anticomplement of X(3767)
X(3926) = X(92)-isoconjugate of X(1974)
X(3926) = trilinear pole of line X(520)X(3265)
X(3926) = trilinear product of vertices of pedal triangle of X(2) reflected in X(2)
X(3926) = pole wrt polar circle of trilinear polar of X(6524) (line X(2489)X(2508))
X(3926) = polar conjugate of X(6524)
X(3926) = barycentric square of X(69)
X(3926) = trilinear product of vertices of X(2)-anti altimedial triangle
X(3927) lies on these lines: {1, 3683}, {3, 63}, {4, 144}, {5, 329}, {8, 30}, {9, 942}, {10, 527}, {40, 971}, {46, 210}, {55, 191}, {69, 3695}, {200, 3579}, {201, 222}, {219, 3157}, {382, 3419}, {405, 3219}, {474, 3218}, {500, 3190}, {518, 3295}, {595, 3242}, {758, 958}, {894, 2049}, {896, 976}, {908, 1656}, {956, 1482}, {960, 999}, {984, 1046}, {986, 1757}, {1376, 3678}, {1698, 3715}, {1714, 3782}
X(3927) = Conway-triangle-to-inner-Conway-triangle similarity image of X(3)
X(3928) lies on these lines: {1, 3052}, {2, 7}, {30, 84}, {31, 3677}, {40, 376}, {46, 529}, {55, 3243}, {165, 518}, {191, 3338}, {200, 1155}, {219, 1407}, {222, 2323}, {528, 1768}, {536, 1764}, {551, 3333}, {614, 896}, {758, 3576}, {940, 3247}, {956, 2093}, {958, 3339}, {960, 3361}, {982, 1707}, {988, 1046}, {1449, 3666}, {1453, 3670}, {1697, 3241}, {1743, 3752}, {1761, 2257}, {2975, 3340}, {3359, 3654}
X(3928) = isogonal conjugate of isotomic conjugate of X(21605)
X(3928) = centroid of Gemini triangle 24
X(3928) = intouch-to-excentral similarity image of X(2)
X(3929) lies on these lines: {1, 3683}, {2, 7}, {10, 3474}, {30, 40}, {44, 2999}, {72, 3601}, {81, 3247}, {84, 376}, {165, 210}, {201, 1394}, {219, 2003}, {220, 222}, {238, 3677}, {333, 3729}, {519, 1697}, {612, 896}, {728, 3719}, {846, 3751}, {912, 3576}, {940, 3731}, {958, 3340}, {960, 1420}, {984, 1707}, {991, 2318}, {1155, 3715}, {1214, 1419}, {1621, 3243}, {1743, 3666}, {3158, 3681}, {3534, 3587}
X(3929) = 2nd-extouch-to-excentral similarity image of X(2)
X(3930) lies on these lines: {1, 1390}, {9, 1174}, {37, 42}, {38, 2276}, {72, 1334}, {100, 3509}, {101, 2752}, {145, 3061}, {226, 306}, {244, 1575}, {518, 672}, {519, 2170}, {523, 661}, {594, 2294}, {678, 2243}, {758, 1018}, {899, 3290}, {910, 3689}, {1400, 3694}, {1475, 3555}, {1500, 2292}, {1914, 3722}, {1926, 1978}, {2246, 3684}, {2295, 2650}, {2340, 2356}, {2345, 3475}, {3294, 3678}
X(3931) lies on these lines: {1, 3}, {2, 3702}, {4, 941}, {10, 37}, {38, 3555}, {42, 72}, {210, 3293}, {226, 227}, {386, 960}, {392, 1193}, {405, 968}, {595, 1386}, {756, 3214}, {975, 1376}, {976, 2177}, {984, 1716}, {1089, 3175}, {1100, 2241}, {1125, 3752}, {1181, 1409}, {1427, 3671}, {1441, 3672}, {1465, 3485}, {1706, 3247}, {1724, 3683}, {1785, 1882}, {1962, 3753}, {2887, 3178}
X(3932) lies on these lines: {1, 3589}, {2, 1390}, {8, 344}, {9, 3416}, {10, 37}, {11, 3006}, {12, 313}, {23, 100}, {65, 3710}, {120, 3263}, {141, 984}, {210, 306}, {312, 2886}, {345, 1376}, {346, 2550}, {442, 1089}, {516, 2325}, {518, 3717}, {519, 1279}, {523, 1577}, {524, 1757}, {528, 3685}, {536, 1738}, {726, 1086}, {756, 1211}, {1861, 3693}, {3687, 3740}
X(3933) lies on these lines: {3, 69}, {5, 76}, {11, 3760}, {12, 3761}, {30, 315}, {32, 524}, {39, 141}, {99, 550}, {140, 183}, {187, 3630}, {194, 3314}, {264, 1595}, {304, 337}, {305, 1368}, {316, 3627}, {350, 496}, {394, 441}, {427, 1235}, {495, 1909}, {538, 626}, {549, 1078}, {574, 3631}, {980, 1211}, {1007, 1656}, {1353, 3398}, {1930, 3665}, {3673, 3705}
X(3933) = isotomic conjugate of X(32085)
X(3933) = complement of X(7754)
X(3933) = anticomplement of X(5305)
X(3933) = crossdifference of every pair of points on line X(2489)X(3804)
X(3934) lies on these lines: {2, 39}, {3, 3734}, {5, 141}, {32, 183}, {69, 2548}, {75, 1574}, {83, 385}, {140, 620}, {187, 384}, {230, 736}, {232, 1235}, {262, 3090}, {264, 3199}, {316, 2896}, {325, 1506}, {350, 1500}, {574, 1975}, {726, 3634}, {730, 1125}, {732, 3589}, {1015, 1909}, {1207, 3499}, {1656, 3095}, {2275, 3761}, {2276, 3760}, {3094, 3763}
X(3934) = isotomic conjugate of isogonal conjugate of X(20965)
X(3934) = polar conjugate of isogonal conjugate of X(22062)
X(3934) = centroid of ABCX(76)
X(3934) = Kosnita(X(76),X(2))
X(3934) = complement of X(39)
X(3934) = X(2023)-of-1st-Brocard-triangle
X(3934) = perspector of the medial triangle and the tangential triangle, wrt the medial triangle, of the bicevian conic of X(2) and X(6)
X(3935) lies on these lines: {1, 2}, {44, 765}, {55, 1776}, {63, 3158}, {72, 1005}, {88, 1280}, {100, 518}, {144, 3174}, {149, 908}, {210, 1621}, {238, 3722}, {329, 2900}, {404, 3555}, {522,4724}, {678, 896}, {740, 1109}, {756, 3750}, {758, 3245}, {902, 1757}, {984, 2177}, {1001, 3711}, {1071, 3579}, {2246, 3684}, {3189, 3436}, {3243, 3306}, {3678, 3746}, {3740, 3748}
X(3935) = inverse-in-circumconic-centered-at-X(1) of X(2)X(3936) lies on these lines: {1, 3454}, {2, 6}, {8, 442}, {10, 2650}, {21, 1330}, {31, 3771}, {42, 2887}, {100, 851}, {145, 1834}, {226, 306}, {304, 1228}, {312, 1230}, {320, 2245}, {329, 440}, {346, 1901}, {514, 661}, {518, 3006}, {740, 3120}, {752, 902}, {856, 1809}, {860, 1870}, {1043, 2475}, {2092, 3662}, {2292, 3178}, {3187, 3772}, {3649, 3704}
X(3936) = isogonal conjugate of X(34079)
X(3936) = isotomic conjugate of X(24624)
X(3936) = anticomplement of X(35466)
X(3936) = trilinear pole of line X(4707)X(4736) (the tangent at X(4736) to the inellipse centered at X(10))
X(3937) lies on these lines: {1, 2841}, {3, 1331}, {11, 513}, {25, 1407}, {31, 1401}, {51, 57}, {63, 295}, {100, 2810}, {104, 2818}, {123, 125}, {184, 222}, {214, 2842}, {244, 1357}, {373, 3306}, {511, 3218}, {512, 2611}, {603, 1425}, {1015, 1977}, {1086, 2969}, {1260, 1810}, {1319, 2390}, {1364, 3270}, {1398, 1413}, {1463, 3011}, {1495, 3220}, {1768, 2807}
X(3937) = X(92)-isoconjugate of X(1252)
X(3937) = crossdifference of every pair of points on line X(644)X(1783)
X(3938) lies on these lines: {1, 2}, {31, 518}, {33, 1851}, {37, 2280}, {38, 55}, {63, 902}, {100, 982}, {210, 748}, {238, 3681}, {244, 1376}, {354, 750}, {528, 3782}, {756, 1001}, {896, 3052}, {984, 1621}, {1468, 3555}, {1807, 3478}, {2177, 3666}, {2292, 3295}, {2308, 3751}, {3120, 3434}, {3158, 3677}, {3218, 3550}, {3689, 3752}
X(3939) lies on these lines: {3, 2810}, {9, 294}, {31, 678}, {40, 2835}, {55, 2316}, {59, 677}, {100, 109}, {101, 692}, {190, 522}, {200, 212}, {210, 2328}, {219, 480}, {284, 2311}, {573, 2876}, {643, 645}, {1018, 1783}, {1026, 1332}, {1461, 2283}, {1471, 3243}, {1724, 3189}, {1743, 3174}, {2209, 3169}, {2323, 2340}, {2361, 3689}
X(3939) = isogonal conjugate of X(3676)
X(3939) = trilinear pole of line X(41)X(55)
X(3939) = crossdifference of every pair of points on line X(1086)X(1358)
X(3939) = barycentric product of circumcircle intercepts of line X(8)X(9)
X(3939) = perspector of anticevian triangle of X(101) and unary cofactor triangle of intangents triangle
X(3940) lies on these lines: {1, 210}, {3, 63}, {5, 8}, {30, 329}, {69, 1565}, {144, 376}, {200, 517}, {219, 1807}, {381, 908}, {480, 3428}, {518, 997}, {519, 3452}, {758, 1376}, {936, 942}, {952, 3421}, {956, 3681}, {958, 3678}, {960, 3295}, {995, 3242}, {1064, 2340}, {1159, 3753}, {1265, 3695}, {2098, 3632}, {2099, 3679}
X(3941) lies on these lines: {1, 3286}, {3, 1386}, {6, 2223}, {31, 48}, {36, 1631}, {55, 1100}, {56, 1279}, {100, 3759}, {513, 1423}, {579, 674}, {583, 3779}, {604, 692}, {665, 2498}, {934, 2369}, {1009, 3416}, {1011, 3745}, {1402, 3052}, {1444, 1621}, {1445, 2283}, {1617, 3433}, {2209, 3248}, {2245, 3056}, {2260, 2293}
X(3942) lies on these lines: {1, 1633}, {7, 1953}, {19, 269}, {38, 2877}, {48, 77}, {56, 2097}, {57, 909}, {63, 1332}, {69, 337}, {241, 2183}, {244, 659}, {320, 1959}, {513, 2310}, {651, 2265}, {1086, 1358}, {1108, 1122}, {1364, 3270}, {1418, 2262}, {1419, 2261}, {1422, 1435}, {1443, 2173}, {2294, 3664}, {3271, 3675}
X(3943) lies on these lines: {6, 145}, {8, 45}, {9, 3632}, {10, 37}, {44, 519}, {101, 2758}, {141, 192}, {190, 524}, {306, 3175}, {320, 545}, {523, 661}, {536, 1086}, {1018, 2245}, {1100, 3635}, {1317, 1404}, {1897, 1990}, {2171, 3649}, {2345, 3616}, {3247, 3624}, {3625, 3707}, {3636, 3723}, {3644, 3662}, {3672, 3763}
X(3943) = complement of X(17160)
X(3944) lies on these lines: {1, 4}, {2, 846}, {5, 986}, {11, 982}, {43, 908}, {79, 987}, {115, 3735}, {171, 1836}, {238, 3772}, {312, 2887}, {329, 1757}, {516, 3550}, {726, 3705}, {752, 3769}, {984, 2886}, {1044, 1076}, {1738, 3452}, {1756, 1764}, {2292, 2476}, {3496, 3767}, {3551, 3662}, {3685, 3771}
X(3945) lies on these lines: {1, 7}, {2, 6}, {3, 1014}, {37, 144}, {57, 2269}, {75, 145}, {142, 1449}, {171, 1253}, {226, 1419}, {319, 3617}, {320, 1279}, {344, 3758}, {346, 894}, {354, 3056}, {522, 4724}, {527, 3247}, {941, 980}, {1002, 3779}, {1418, 3666}, {1433, 1440}, {1434, 3522}, {1456, 3485}, {3475, 3745}
X(3945) = intersection of tangents at X(2) and X(7) to Lucas cubic K007
X(3935) = inverse-in-circumconic-centered-at-X(1) of X(2)
X(3945) = {X(2),X(6)}-harmonic conjugate of X(37681)
X(3935) = endo-homothetic center of X(2)- and X(3)-Ehrmann triangles; the homothetic center is X(15139)
X(3946) lies on these lines: {1, 142}, {2, 2321}, {6, 527}, {7, 1419}, {9, 3672}, {37, 3008}, {57, 347}, {81, 553}, {141, 519}, {192, 2325}, {239, 1654}, {284, 1429}, {516, 1386}, {536, 3589}, {740, 1125}, {894, 1266}, {950, 3100}, {1086, 1100}, {1323, 1418}, {2324, 2999}, {3618, 3729}, {3634, 3773}
X(3946) = isogonal conjugate of X(38825)
X(3946) = complement of X(2321)
X(3946) = crosspoint of X(2) and X(1434)
X(3946) = crosssum of X(6) and X(1334)
X(3946) = X(2)-Ceva conjugate of X(38930)
X(3946) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 38825}, {210, 38811}
X(3946) = trilinear product X(i)*X(j) for these {i,j}: {9, 10521}, {81, 4854}, {100, 23729}
X(3946) = trilinear quotient X(i)/X(j) for these (i,j): (1, 38825), (1014, 38811), (4854, 37), (10521, 57), (23729, 513)
X(3946) = barycentric product X(i)*X(j) for these {i,j}: {8, 10521}, {86, 4854}, {190, 23729}
X(3946) = barycentric quotient X(i)/X(j) for these (i,j): (6, 38825), (1412, 38811), (4854, 10), (10521, 7), (23729, 514)
X(3947) lies on these lines: {1, 3091}, {2, 3361}, {7, 1698}, {10, 12}, {57, 3634}, {142, 1329}, {227, 3743}, {354, 3614}, {388, 1125}, {495, 946}, {516, 3085}, {519, 3485}, {551, 1388}, {1089, 1441}, {1478, 3612}, {1770, 3584}, {2099, 3625}, {3090, 3333}, {3340, 3626}, {3476, 3636}, {3600, 3624}
X(3948) lies on these lines: {1, 3765}, {2, 39}, {10, 321}, {37, 313}, {75, 1213}, {86, 3770}, {92, 429}, {190, 2245}, {192, 2092}, {239, 350}, {242, 862}, {257, 312}, {314, 1654}, {329, 2899}, {341, 1834}, {344, 1234}, {514, 661}, {536, 3264}, {714, 3122}, {730, 3009}, {1269, 3739}
X(3948) = isogonal conjugate of X(18268)
X(3948) = isotomic conjugate of X(37128)
X(3949) lies on these lines: {6, 976}, {8, 1953}, {9, 943}, {10, 2294}, {12, 594}, {19, 200}, {37, 42}, {48, 78}, {63, 1796}, {69, 337}, {71, 72}, {100, 1761}, {198, 480}, {201, 2197}, {219, 1807}, {306, 3610}, {319, 1959}, {518, 2260}, {1089, 1826}, {1215, 2345}, {1757, 1778}
X(3950) lies on these lines: {1, 346}, {6, 2325}, {8, 3731}, {9, 519}, {10, 37}, {45, 3625}, {142, 536}, {145, 1743}, {192, 3662}, {226, 3175}, {344, 3008}, {391, 3632}, {573, 3208}, {966, 3626}, {1018, 1400}, {1125, 2345}, {1266, 3644}, {1449, 3635}, {2171, 3671}, {2262, 2802}, {3664, 3729}
X(3950) = complement of X(17151)
X(3951) lies on these lines: {1, 2308}, {3, 63}, {8, 144}, {40, 3681}, {46, 3678}, {69, 3710}, {77, 201}, {329, 3091}, {377, 527}, {518, 3303}, {612, 1046}, {908, 3090}, {936, 3218}, {942, 3305}, {960, 3304}, {976, 1707}, {2093, 3617}, {2292, 3751}, {2475, 3679}, {3419, 3627}
X(3952) lies on these lines: {2, 38}, {8, 80}, {10, 3120}, {37, 3121}, {72, 3701}, {100, 190}, {101, 835}, {110, 645}, {192, 872}, {210, 321}, {312, 3681}, {329, 2835}, {643, 765}, {644, 1783}, {660, 799}, {668, 891}, {726, 899}, {908, 3006}, {1265, 3436}, {3159, 3293}
X(3952) = isogonal conjugate of X(3733)
X(3952) = isotomic conjugate of X(7192)
X(3952) = complement of X(17154)
X(3952) = anticomplement of X(244)
X(3952) = trilinear pole of line X(10)X(37) (the tangent to Kiepert hyperbola at X(10), and the line of the degenerate cross-triangle of Gemini triangles 15 and 16)
X(3952) = X(523)-cross conjugate of X(10)
X(3952) = perspector of ABC and side-triangle of Gemini triangles 17 and 18
X(3952) = barycentric product of vertices of Gemini triangle 17
X(3952) = barycentric product of vertices of Gemini triangle 18
X(3952) = intersection, other than A, B, C, of {ABC, Gemini 17}-circumconic and {ABC, Gemini 18}-circumconic
X(3953) lies on these lines: {1, 3}, {8, 1739}, {10, 244}, {21, 3315}, {38, 1125}, {39, 3726}, {321, 596}, {474, 3242}, {496, 3782}, {518, 3216}, {551, 2292}, {595, 3218}, {614, 1724}, {758, 1201}, {984, 3624}, {1015, 3721}, {1421, 1935}, {1736, 3086}, {3293, 3555}
X(3954) lies on these lines: {1, 6}, {8, 3735}, {10, 762}, {32, 976}, {38, 39}, {76, 321}, {100, 755}, {141, 1930}, {228, 2156}, {257, 668}, {274, 335}, {519, 3727}, {758, 2295}, {1125, 3726}, {1500, 2292}, {1575, 3670}, {1655, 2895}, {1843, 3688}, {2238, 3678}
X(3954) = complement of X(17141)
X(3955) lies on these lines: {1, 987}, {3, 73}, {36, 1401}, {57, 182}, {63, 184}, {72, 1437}, {110, 3219}, {171, 2330}, {219, 3167}, {228, 295}, {293, 1214}, {394, 3781}, {436, 1948}, {511, 2003}, {611, 1460}, {982, 1428}, {1409, 2359}, {1707, 2175}, {3292, 3690}
X(3955) = crosssum of the polar conjugates of PU(6)
X(3956) lies on these lines: {10, 12}, {519, 3740}, {1962, 3293}, {2802, 3679}, {3214, 3743}, {3634, 3742}
X(3957) lies on these lines: {1, 2}, {21, 3555}, {38, 3750}, {55, 3218}, {63, 3243}, {81, 643}, {100, 354}, {149, 226}, {171, 3722}, {518, 1621}, {744, 2667}, {982, 2177}, {1001, 3681}, {1100, 3693}, {3158, 3306}, {3315, 3752}, {3434, 3475}, {3689, 3742}
X(3958) lies on these lines: {1, 1778}, {6, 2292}, {9, 758}, {37, 2650}, {48, 63}, {71, 72}, {191, 284}, {201, 1409}, {219, 3157}, {896, 1333}, {960, 2260}, {1046, 2303}, {1100, 1962}, {1213, 3649}, {1449, 3743}, {1761, 2173}, {1839, 3686}, {1858, 2269}
X(3959) lies on these lines: {1, 1929}, {2, 3727}, {6, 19}, {8, 3721}, {10, 3735}, {37, 3208}, {75, 257}, {85, 1086}, {145, 3726}, {517, 2176}, {982, 2319}, {986, 1107}, {1575, 3061}, {1738, 3094}, {1953, 2277}, {2170, 2275}, {2643, 3764}, {3057, 3290}
X(3959) = polar conjugate of isogonal conjugate of X(23526)
X(3960) lies on these lines: {1, 2254}, {2, 3762}, {3, 2814}, {88, 1022}, {101, 651}, {104, 106}, {214, 3738}, {241, 514}, {513, 1960}, {659, 764}, {667, 3777}, {676, 2826}, {690, 3743}, {812, 1015}, {830, 2530}, {900, 1387}, {1125, 3716}, {2006, 2401}
X(3960) = isotomic conjugate of X(36804)
X(3960) = complement of X(3762)
X(3960) = polar conjugate of isogonal conjugate of X(22379)
X(3960) = trilinear pole of the tangent to the incircle at X(3025)
X(3961) lies on these lines: {1, 2}, {9, 983}, {31, 1757}, {33, 242}, {37, 3684}, {38, 100}, {55, 846}, {63, 3099}, {171, 518}, {210, 238}, {595, 3678}, {668, 1965}, {756, 1621}, {902, 3219}, {982, 1054}, {1279, 3740}, {1763, 3465}, {3666, 3689}
X(3962) lies on these lines: {1, 3683}, {8, 1836}, {10, 12}, {37, 2650}, {40, 3689}, {63, 2646}, {78, 1155}, {144, 145}, {329, 1837}, {354, 960}, {382, 517}, {392, 3636}, {942, 3624}, {1042, 2318}, {1464, 3682}, {1706, 3711}, {3555, 3635}
X(3963) lies on these lines: {2, 1240}, {9, 3765}, {10, 3728}, {37, 313}, {75, 141}, {76, 192}, {190, 3770}, {226, 306}, {495, 3695}, {536, 1269}, {668, 1654}, {730, 2309}, {732, 894}, {1018, 3729}, {1089, 3178}, {1920, 1926}, {3264, 3739}
X(3964) lies on these lines: {3, 69}, {25, 317}, {95, 183}, {99, 1294}, {159, 1634}, {253, 2071}, {264, 1105}, {326, 1259}, {340, 3515}, {394, 577}, {491, 1584}, {492, 1583}, {524, 1609}, {648, 1033}, {1073, 2063}, {1270, 1599}, {1271, 1600}
X(3964) = isogonal conjugate of X(6524)
X(3964) = isotomic conjugate of X(1093)
X(3964) = X(92)-isoconjugate of X(2207)
X(3965) lies on these lines: {6, 78}, {8, 37}, {9, 55}, {44, 2220}, {69, 241}, {72, 573}, {220, 3692}, {341, 346}, {391, 1212}, {518, 1400}, {872, 2340}, {960, 2269}, {1098, 1792}, {1211, 2092}, {1376, 2285}, {3057, 3169}, {3290, 3705}
X(3966) lies on these lines: {1, 1211}, {2, 1386}, {8, 210}, {9, 3703}, {38, 3764}, {55, 3687}, {69, 354}, {75, 1836}, {141, 614}, {219, 3686}, {306, 1001}, {333, 2194}, {345, 3683}, {391, 2348}, {958, 1036}, {3434, 3696}, {3715, 3717}
X(3967) lies on these lines: {12, 3710}, {37, 714}, {42, 3175}, {43, 536}, {65, 3701}, {72, 1089}, {75, 3740}, {210, 321}, {312, 518}, {329, 3416}, {726, 3752}, {908, 3703}, {1376, 3729}, {1901, 3610}, {2886, 3717}, {3681, 3706}
X(3968) lies on these lines: {2, 2802}, {10, 12}, {517, 547}, {519, 3742}
X(3969) lies on these lines: {2, 594}, {8, 405}, {10, 1962}, {42, 3773}, {63, 544}, {81, 2295}, {100, 199}, {190, 2895}, {226, 306}, {319, 3219}, {345, 1150}, {502, 1089}, {961, 3476}, {1500, 3661}, {3006, 3706}, {3681, 3690}
X(3970) lies on these lines: {1, 6}, {35, 3509}, {39, 3726}, {55, 1759}, {65, 1018}, {79, 2795}, {321, 1930}, {758, 1334}, {942, 3693}, {1500, 3721}, {2170, 3244}, {2171, 3671}, {2276, 3670}, {2294, 2321}, {3216, 3290}, {3496, 3746}
X(3971) lies on these lines: {1, 979}, {2, 726}, {10, 321}, {37, 714}, {43, 192}, {171, 190}, {210, 740}, {312, 984}, {354, 537}, {519, 3681}, {536, 3740}, {894, 1961}, {1211, 3773}, {1757, 1999}, {2901, 3678}, {3263, 3663}
X(3971) = incentral-to-ABC barycentric image of X(2)
X(3971) = complement of X(17155)
X(3972) = eigencenter of 3rd Brocard triangle (Peter Moses, January 24, 2012); see X(3734)
X(3972) lies on these lines: {2, 187}, {3, 83}, {6, 99}, {32, 76}, {39, 3552}, {69, 1285}, {112, 264}, {183, 1384}, {251, 305}, {350, 609}, {376, 3618}, {574, 3329}, {691, 1316}, {754, 3314}, {1078, 3053}
X(3972) = perspector of ABC and circumsymmedial triangle of 1st Brocard triangle
X(3973) lies on these lines: {1, 6}, {144, 3008}, {165, 2348}, {173, 363}, {200, 902}, {346, 3632}, {391, 3679}, {484, 2270}, {519, 3161}, {572, 3217}, {610, 2265}, {2163, 2297}, {2345, 3707}, {2347, 3730}, {2999, 3219}
X(3973) = isogonal conjugate of X(36603)
X(3973) = crosssum of X(244) and X(4394)
X(3973) = crosspoint of X(765) and X(27834)
X(3974) lies on these lines: {2, 1390}, {4, 1089}, {8, 210}, {33, 200}, {55, 346}, {321, 2550}, {329, 3416}, {344, 3757}, {391, 3715}, {612, 2345}, {756, 966}, {1219, 3304}, {3085, 3695}, {3161, 3683}, {3474, 3729}
X(3975) lies on these lines: {2, 330}, {8, 210}, {9, 3596}, {75, 966}, {190, 2183}, {239, 350}, {314, 3686}, {333, 3691}, {645, 2323}, {646, 2325}, {730, 2664}, {1655, 3666}, {1999, 3780}, {2340, 3699}, {3739, 3770}
X(3976) lies on these lines: {1, 3}, {8, 244}, {38, 3616}, {43, 3555}, {256, 3296}, {518, 978}, {749, 984}, {1015, 3061}, {1046, 1191}, {1329, 3756}, {1393, 3476}, {1739, 3632}, {2275, 3726}, {2292, 3622}, {2975, 3315}
X(3977) lies on these lines: {2, 2415}, {3, 3710}, {63, 69}, {100, 2751}, {110, 2760}, {190, 908}, {344, 3306}, {441, 525}, {516, 3006}, {518, 3712}, {519, 902}, {726, 3011}, {1150, 2321}, {3219, 3687}, {3589, 3666}
X(3977) = isogonal conjugate of X(8752)
X(3977) = isotomic conjugate of X(6336)
X(3977) = crossdifference of every pair of points on line X(25)X(8643)
X(3978) lies on these lines: {2, 39}, {6, 706}, {75, 256}, {83, 1207}, {99, 237}, {290, 325}, {308, 3589}, {315, 1899}, {316, 512}, {524, 670}, {561, 3765}, {671, 886}, {694, 698}, {702, 1084}, {1215, 1237}
X(3978) = isogonal conjugate of X(9468)
X(3978) = isotomic conjugate of X(694)
X(3978) = complement of polar conjugate of isogonal conjugate of X(23174)
X(3978) = complement of Steiner-circumellipse-inverse of X(39)
X(3978) = anticomplement of X(3229)
X(3978) = perspector of conic {{A,B,C,PU(11)}}
X(3978) = trilinear pole of line PU(1) of 1st anti-Brocard triangle
X(3978) = trilinear pole of PU(133) (line X(804)X(5976))
X(3978) = X(694)-of-1st-anti-Brocard-triangle
X(3979) lies on these lines: {1, 2}, {81, 3722}, {238, 3748}, {354, 1054}, {518, 846}, {1046, 3746}, {1051, 1386}, {1621, 1757}
X(3980) lies on these lines: {1, 3210}, {2, 846}, {10, 46}, {43, 894}, {57, 3741}, {75, 171}, {192, 1961}, {321, 750}, {612, 726}, {740, 940}, {968, 1125}, {986, 1010}, {1215, 1376}, {2345, 3509}, {3550, 3757}
X(3980) = X(612)-of-1st-Brocard-triangle
X(3981) lies on these lines: {2, 694}, {4, 695}, {6, 25}, {22, 1691}, {23, 1501}, {305, 698}, {386, 2653}, {493, 1584}, {494, 1583}, {511, 1196}, {1350, 1611}, {1993, 2056}, {2979, 3231}, {3051, 3060}, {3095, 3117}
X(3981) = X(305)-of-1st-Brocard-triangle
X(3981) = 1st-Brocard-isotomic conjugate of X(9306)
X(3982) lies on these lines: {1, 3529}, {2, 7}, {65, 3626}, {382, 950}, {388, 3632}, {516, 3748}, {546, 942}, {554, 3639}, {1081, 3638}, {1319, 3636}, {1374, 1659}, {2099, 3244}, {3487, 3528}, {3664, 3782}
X(3983) lies on these lines: {1, 3711}, {8, 3740}, {10, 12}, {37, 3214}, {40, 3715}, {354, 1698}, {392, 3626}, {405, 3689}, {518, 3619}, {936, 1319}, {960, 3617}, {3057, 3679}, {3555, 3634}, {3696, 3701}
X(3984) lies on these lines: {1, 748}, {3, 63}, {8, 908}, {306, 1265}, {329, 3146}, {518, 3304}, {519, 2478}, {546, 3419}, {936, 3306}, {960, 3303}, {2476, 3679}, {3219, 3601}, {3243, 3622}, {3340, 3617}
X(3985) lies on these lines: {9, 312}, {37, 714}, {190, 3509}, {210, 2321}, {341, 3208}, {522, 650}, {537, 3726}, {726, 3290}, {740, 2238}, {1089, 3294}, {1334, 3701}, {3684, 3685}, {3686, 3706}, {3691, 3702}
X(3986) lies on these lines: {1, 391}, {2, 2415}, {6, 551}, {9, 1125}, {10, 37}, {346, 1698}, {405, 1696}, {519, 966}, {1100, 3707}, {1400, 3294}, {1449, 3636}, {1743, 3616}, {2345, 3634}, {3244, 3686}
X(3986) = complement of X(25590)
X(3987) lies on these lines:
{1, 474}, {8, 3670}, {10, 321}, {38, 3626}, {40, 1724}, {42, 3754},
{65, 3293}, {244, 3244}, {517, 3216}, {758, 3214}, {982, 3632}, {986,
3679}, {1201, 2802}, {1574, 3727}
X(3987) = crossdifference of every pair of points on line X(644)X(1783)
X(3988) lies on these lines: {10, 12}, {518, 3636}, {960, 3635}, {3632, 3681}
X(3989) lies on these lines: {1, 2308}, {2, 726}, {37, 38}, {42, 984}, {45, 748}, {165, 612}, {517, 2292}, {518, 1962}, {614, 3731}, {756, 899}, {846, 902}, {896, 3745}, {1051, 1757}, {1961, 3218}
X(3990) lies on these lines: {1, 6}, {48, 184}, {71, 73}, {255, 577}, {275, 321}, {287, 336}, {326, 394}, {517, 1841}, {604, 2198}, {836, 3682}, {919, 2749}, {1331, 2327}, {2269, 2288}, {3197, 3198}
X(3990) = X(92)-isoconjugate of X(28)
X(3991) lies on these lines: {1, 728}, {9, 3295}, {10, 37}, {44, 2241}, {65, 1018}, {72, 1334}, {192, 3673}, {210, 3294}, {517, 3208}, {518, 3730}, {519, 1212}, {672, 3555}, {942, 3501}, {3509, 3579}
X(3992) lies on these lines: {1, 341}, {10, 321}, {100, 2758}, {190, 484}, {312, 3679}, {523, 1577}, {726, 1739}, {1111, 3263}, {1479, 2899}, {1737, 3717}, {2901, 3214}, {3293, 3725}, {3626, 3702}, {3697, 3714}
X(3993) lies on these lines: {1, 87}, {10, 37}, {75, 1125}, {226, 1365}, {321, 1962}, {335, 2796}, {518, 3244}, {519, 751}, {536, 551}, {846, 1999}, {1215, 3175}, {1278, 3616}, {2667, 3159}, {3636, 3644}
X(3994) lies on these lines: {10, 321}, {37, 2229}, {38, 312}, {42, 3175}, {190, 896}, {244, 726}, {523, 661}, {536, 899}, {750, 3729}, {1215, 1962}, {1266, 3263}, {2325, 3011}, {3632, 3681}, {3685, 3722}
X(3995) lies on these lines: {1, 3159}, {2, 37}, {8, 2901}, {9, 3187}, {72, 145}, {81, 190}, {86, 1255}, {239, 3294}, {726, 3720}, {740, 756}, {1089, 3743}, {1215, 1962}, {1655, 2895}, {1999, 3219}
X(3995) = complement of X(4359)X(3996) lies on these lines: {8, 21}, {10, 3750}, {145, 940}, {171, 519}, {190, 3681}, {200, 312}, {210, 3685}, {239, 3744}, {1222, 3621}, {2321, 3684}, {3210, 3242}, {3550, 3632}, {3689, 3706}, {3696, 3757}
X(3997) lies on these lines: {1, 672}, {6, 519}, {10, 213}, {37, 758}, {42, 1018}, {44, 1573}, {58, 2329}, {81, 644}, {101, 171}, {386, 3501}, {551, 3230}, {1125, 2176}, {3625, 3780}
X(3998) lies on these lines: {2, 37}, {3, 63}, {81, 1257}, {100, 1297}, {213, 2221}, {306, 307}, {326, 394}, {518, 2352}, {1073, 3692}, {1108, 3187}, {1210, 2901}, {1331, 1801}, {2287, 3219}
X(3998) = isogonal conjugate of X(5317)
X(3999) lies on these lines: {1, 3}, {38, 3742}, {44, 3290}, {88, 1280}, {244, 518}, {896, 3246}, {908, 3756}, {1054, 3689}, {1279, 3218}, {3240, 3752}, {3242, 3306}, {3667, 3676}, {3693, 3726}
X(3999) = incircle-inverse of X(34583)
As described in the preamble to X(3758), the notation T(-a, c) represents the central triangle whose A-vertex is the point -a : c : b. This triangle and seven others are discussed just before X(4357).
X(4000) is the center of the inellipse that is the barycentric square of the Gergonne line. The Brianchon point (perspector) of the inellipse is X(279). (Randy Hutson, October 15, 2018)
X(4000) lies on these lines: {1, 142}, {2, 37}, {3, 1612}, {4, 990}, {6, 7}, {8, 141}, {9, 3008}, {10, 4353}, {19, 57}, {20, 1104}, {27, 2221}, {31, 3474}, {42, 3475}, {44, 144}, {48, 1429}, {69, 239}, {77, 3554}, {105, 1486}, {145, 3834}, {172, 4209}, {193, 320}, {226, 2999}, {241, 347}, {273, 393}, {279, 1418}, {319, 3620}, {329, 3782}, {348, 2275}, {386, 2140}, {387, 942}, {388, 4327}, {390, 1279}, {497, 614}, {499, 1733}, {518, 4310}, {527, 1743}, {594, 3763}, {894, 3618}, {910, 3598}, {938, 1834}, {962, 1191}, {1100, 3945}, {1107, 4352}, {1125, 4356}, {1193, 3485}, {1210, 1861}, {1266, 3729}, {1386, 4307}, {1423, 2183}, {1449, 3664}, {1453, 1890}, {1471, 4331}, {1473, 1851}, {1618, 2175}, {1714, 3670}, {1722, 2551}, {1760, 3218}, {3086, 4008}, {3212, 3959}, {3247, 4021}, {3486, 3924}, {3619, 3661}, {3875, 3912}
X(4000) = isogonal conjugate of X(7123)
X(4000) = complement of X(346)
X(4000) = {X(2),X(75)}-harmonic conjugate of X(2345)
X(4000) = isotomic conjugate of X(30701)
X(4000) = anticomplement of X(17279)
X(4001) lies on these lines: {2, 1743}, {8, 2093}, {63, 69}, {222, 307}, {226, 1150}, {320, 333}, {321, 527}, {524, 3666}, {553, 3578}, {1125, 2308}, {1269, 1839}, {2895, 3218}, {3187, 3663}
X(4001) = isotomic conjugate of isogonal conjugate of X(22054)
X(4002) lies on these lines: {10, 12}, {354, 3626}, {392, 1698}, {405, 1706}, {474, 1420}, {517, 3090}, {942, 3617}, {956, 3361}, {1376, 3612}, {3057, 3634}, {3555, 3679}, {3632, 3742}
X(4003) lies on these lines: {1, 3}, {11, 3663}, {37, 244}, {38, 210}, {45, 3290}, {88, 1390}, {518, 3240}, {596, 3634}, {614, 3683}, {1386, 3218}, {3210, 3706}, {3242, 3689}
X(4004) lies on these lines: {10, 12}, {78, 1159}, {145, 942}, {354, 3635}, {392, 3624}, {405, 2093}, {474, 3340}, {517, 631}, {956, 3339}, {1482, 3306}, {3057, 3636}, {3555, 3632}
X(4005) lies on these lines: {1, 3715}, {10, 12}, {40, 3711}, {44, 976}, {145, 960}, {354, 3624}, {392, 3635}, {518, 3616}, {1898, 3059}, {2318, 2594}, {3057, 3632}, {3555, 3636}
X(4006) lies on these lines: {9, 3746}, {25, 200}, {37, 762}, {72, 1018}, {210, 3294}, {442, 594}, {1089, 1826}, {1334, 3678}, {1574, 3726}, {2170, 3625}, {3061, 3632}, {3681, 3730}
X(4007) lies on these lines: {1, 594}, {6, 3632}, {8, 9}, {10, 3247}, {37, 3679}, {319, 3729}, {519, 1449}, {966, 3626}, {1100, 3633}, {1266, 3620}, {2323, 3713}, {3624, 3723}
X(4007) = complement of X(4460)
X(4008) lies on these lines: {1, 75}, {19, 158}, {31, 92}, {82, 91}, {239, 613}, {240, 774}, {560, 1955}, {611, 894}, {920, 1760}, {1210, 1738}, {2345, 3085}, {2550, 3419}
Let A'B'C' be the Artzt triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(4008). (Randy Hutson, April 9, 2016)
X(4009) lies on these lines: {8, 210}, {11, 3717}, {43, 3175}, {190, 1155}, {321, 3740}, {522, 650}, {536, 899}, {986, 1698}, {1125, 1215}, {1329, 3710}, {3452, 3703}, {3685, 3689}
X(4010) lies on these lines: {1, 2787}, {11, 244}, {149, 3448}, {320, 350}, {512, 1577}, {522, 1491}, {523, 661}, {659, 812}, {663, 814}, {804, 3027}, {891, 3762}, {1639, 2977}
X(4010) = isotomic conjugate of X(4589)
X(4010) = crossdifference of every pair of points on line X(58)X(101)
X(4011) lies on these lines: {1, 979}, {2, 846}, {9, 3741}, {10, 1479}, {43, 3685}, {190, 982}, {238, 312}, {321, 748}, {614, 726}, {908, 3771}, {997, 3465}, {1001, 1215}
X(4011) = X(614)-of-1st-Brocard-triangle
Let DEF be the extouch triangle of ABC. Let Ha be the hyperbola with foci E and F, passing through A, and define Hb and Hc cyclically. These three hyperbolas have two common points, U and V. Let Pa be the pole of the line UV with respect to Ha, and define Pb and Pc cyclically. Then DEF and PaPBPc are perspective at X(4012). (Angel Montesdeoca, September 23, 2018)
X(4012) lies on these lines: {7, 8}, {33, 200}, {346, 480}, {2823, 3421}
X(4012) = X(8)-Ceva conjugate of X(497)
X(4013) lies on these lines: {10, 3120}, {12, 1365}, {80, 519}, {88, 1224}, {106, 1125}, {115, 594}, {121, 1086}, {502, 3178}, {596, 1329}, {901, 2372}, {903, 1268}, {1089, 1109}
X(4014) lies on these lines: {7, 2481}, {11, 1357}, {65, 2801}, {513, 1086}, {516, 1463}, {1015, 3123}, {1356, 3026}, {1358, 2820}, {1364, 1365}, {1401, 1836}, {2223, 3000}, {2310, 3020}
X(4014) = X(648)-of-intouch-triangle
X(4014) = trilinear pole wrt intouch triangle of line X(1)X(3)
X(4015) lies on these lines: {10, 12}, {100, 3065}, {392, 3625}, {405, 3711}, {518, 3634}, {756, 3293}, {762, 2238}, {936, 1476}, {960, 2802}, {1125, 3740}, {1126, 1961}, {1698, 3681}
X(4016) lies on these lines: {1, 1333}, {6, 758}, {37, 65}, {38, 1755}, {257, 3770}, {740, 3416}, {902, 1962}, {1100, 2650}, {1213, 3125}, {2220, 3496}, {2305, 3743}, {2643, 3728}
X(4017) lies on these lines: {56, 3733}, {73, 3657}, {109, 1290}, {244, 1365}, {513, 663}, {522, 693}, {523, 656}, {647, 661}, {798, 1400}, {934, 2701}, {3020, 3123}
X(4017) = isogonal conjugate of X(643)
X(4017) = crossdifference of every pair of points on line X(9)X(21)
X(4018) lies on these lines: {1, 3052}, {10, 12}, {20, 145}, {354, 3636}, {392, 942}, {474, 3339}, {518, 3632}, {956, 3340}, {960, 3624}, {1385, 3218}, {3057, 3635}
X(4018) = anticomplement of anticomplement of X(31794)
X(4019) lies on these lines: {46, 1089}, {63, 3718}, {69, 337}, {171, 385}, {190, 1761}, {192, 986}, {306, 307}, {321, 1400}, {740, 3778}, {813, 2868}, {1237, 1840}
X(4020) lies on these lines: {1, 1755}, {3, 295}, {39, 1401}, {48, 255}, {63, 304}, {799, 1925}, {922, 1917}, {1475, 2225}, {1496, 1973}, {1923, 1964}, {1930, 3404}
X(4020) = X(82)-isoconjugate of X(92)
X(4021) lies on these lines: {1, 7}, {37, 3008}, {69, 3244}, {75, 1125}, {86, 1266}, {319, 519}, {527, 1100}, {1086, 3723}, {1268, 3634}, {1387, 2783}, {2325, 3589}
X(4022) lies on these lines: {1, 1918}, {37, 38}, {69, 3764}, {75, 982}, {141, 2228}, {244, 3728}, {256, 320}, {518, 872}, {740, 3670}, {749, 984}, {2667, 3666}
X(4023) lies on these lines: {8, 11}, {9, 3712}, {43, 1211}, {141, 899}, {210, 3687}, {306, 3740}, {345, 3715}, {524, 750}, {908, 3696}, {1150, 3035}, {3452, 3706}
X(4024) lies on the Yff parabola and these lines: {37, 650}, {42, 2395}, {101, 476}, {190, 892}, {321, 693}, {522, 649}, {523, 661}, {657, 3064}, {685, 1897}, {784, 3250}, {850, 1577}
X(4024) = isogonal conjugate of X(4556)
X(4024) = isotomic conjugate of X(4610)
X(4024) = trilinear pole of line X(115)X(2643)
X(4024) = barycentric product X(i)*X(j) for these {i,j}: {10,523}, {514,594}
X(4025) lies on these lines: {2, 2400}, {63, 652}, {239, 514}, {441, 525}, {513, 3004}, {521,4131}, {522, 693}, {650, 918}, {658, 1897}, {812, 3776}, {934, 2765}, {1638, 3700}
X(4025) = isogonal conjugate of X(8750)
X(4025) = isotomic conjugate of X(1897)
X(4025) = anticomplement of X(3239)
X(4025) = pole of Soddy line wrt Steiner circumellipse
X(4025) = crossdifference of every pair of points on line X(25)X(41)
X(4026) lies on these lines: {1, 141}, {2, 11}, {10, 37}, {12, 1284}, {42, 1211}, {65, 307}, {238, 3589}, {405, 1486}, {519, 3775}, {1125, 1279}, {1738, 3739}
X(4026) = barycentric product X(4366)*X(6645)
X(4027) lies on these lines: {2, 98}, {6, 1916}, {32, 99}, {83, 115}, {239, 1281}, {384, 2782}, {385, 732}, {620, 1078}, {733, 1084}, {2023, 3329}, {3044, 3203}
X(4027) = barycentric square of X(385)
X(4027) = 6th-Brocard-to-ABC similarity image of X(384)
X(4027) = homothetic center of 1st and 6th anti-Brocard triangles
X(4027) = perspector of ABC and 1st Brocard triangle of 6th anti-Brocard triangle
X(4027) = perspector of ABC and cross-triangle of ABC and 1st anti-Brocard triangle
X(4027) = endo-homothetic center of 1st and 6th Brocard triangles
X(4027) = X(10997)-of-1st-Brocard-triangle
X(4027) = X(384)-of-1st-anti-Brocard-triangle
X(4028) lies on these lines: {1, 2}, {101, 2374}, {171, 332}, {193, 1707}, {226, 740}, {345, 3751}, {430, 1867}, {1215, 2321}, {1869, 3189}, {2129, 2333}, {2887, 3755}
X(4028) = complement of X(17156)
X(4029) lies on these lines: {1, 2325}, {6, 3635}, {9, 145}, {10, 37}, {44, 3244}, {45, 519}, {142, 192}, {346, 3247}, {1449, 3161}, {2345, 3624}, {3632, 3686}
X(4030) lies on these lines: {8, 21}, {10, 3744}, {312, 3058}, {321, 528}, {519, 3666}, {594, 1914}, {3679, 3749}, {3683, 3717}, {3686, 3693}, {3687, 3689}, {3695, 3746}
X(4031) lies on these lines: {1, 3528}, {2, 7}, {56, 3636}, {65, 1317}, {546, 1210}, {550, 942}, {946, 1768}, {950, 3529}, {1000, 2093}, {3339, 3632}
X(4032) lies on these lines: {7, 192}, {37, 226}, {57, 75}, {65, 740}, {85, 1221}, {201, 388}, {304, 3729}, {536, 553}, {894, 2329}, {1400, 1441}
X(4033) is the intersection, other than the vertices of the Gemini triangle 17, of the {ABC, Gemini 17}-circumconic and {Gemini 17, Gemini 18}-circumconic. (Randy Hutson, November 30, 2018)
X(4033) lies on these lines: {10, 3122}, {75, 141}, {100, 835}, {101, 839}, {190, 646}, {313, 2321}, {341, 3695}, {645, 1016}, {1897, 3699}, {2295, 3758}
X(4033) = isotomic conjugate of X(1019)
X(4033) = trilinear pole of line X(10)X(321)
X(4034) lies on these lines: {1, 1213}, {6, 3679}, {8, 9}, {10, 1449}, {37, 3632}, {519, 966}, {553, 3646}, {594, 1743}, {1100, 1698}, {2345, 3626}
X(4035) lies on these lines: {2, 1449}, {142, 3687}, {226, 306}, {329, 2325}, {345, 527}, {519, 3772}, {1427, 3694}, {2887, 3755}, {3061, 3452}, {3671, 3704}
X(4036) lies on these lines: {10, 522}, {37, 2395}, {100, 476}, {313, 3261}, {424, 2501}, {513, 2517}, {523, 1577}, {668, 892}, {685, 692}, {2787, 3733}
X(4037) lies on these lines: {2, 37}, {213, 2901}, {230, 3712}, {523, 661}, {594, 756}, {726, 3726}, {740, 2238}, {1089, 1500}, {1107, 3702}, {1914, 3685}
X(4038) lies on these lines: {1, 3}, {2, 3775}, {81, 238}, {86, 3741}, {333, 1125}, {518, 1961}, {1100, 3684}, {1126, 3634}, {1206, 3231}, {1962, 3218}
X(4039) lies on these lines: {1, 2}, {31, 3765}, {171, 1909}, {313, 983}, {385, 1580}, {714, 2245}, {730, 2223}, {740, 1284}, {846, 1655}, {1215, 2295}
X(4040) lies on these lines: {1, 514}, {36, 238}, {41, 2141}, {43, 4893}, {101, 9323}, {512, 659}, {522, 3465}, {649, 2664}, {650, 1734}, {661, 830}, {1491, 3216}, {1577, 3716}
X(4040) = reflection of X(1) in X(663)
X(4040) = antipode of X(1) in circle {{X(1),X(15),X(16)}} (or V(X(1)))
X(4041) lies on these lines: {10, 1577}, {42, 810}, {100, 2701}, {512, 661}, {514, 1734}, {522, 3717}, {523, 656}, {650, 663}, {667, 1635}, {891, 2530}
X(4041) = isogonal conjugate of X(1414)
X(4041) = isotomic conjugate of X(4625)
X(4041) = crossdifference of every pair of points on line X(57)X(77)
X(4042) lies on these lines: {8, 21}, {9, 3706}, {10, 940}, {63, 3696}, {171, 3679}, {219, 3686}, {312, 3715}, {391, 497}, {1150, 1376}, {3632, 3750}
X(4043) lies on these lines: {2, 37}, {190, 314}, {213, 3759}, {313, 2321}, {518, 3702}, {740, 872}, {984, 3159}, {1043, 3191}, {1215, 2667}, {3696, 3701}
X(4044) lies on these lines: {2, 3760}, {10, 321}, {76, 85}, {142, 1269}, {306, 1230}, {307, 1229}, {313, 2321}, {519, 3765}, {1400, 3729}, {1500, 3175}
X(4045) lies on these lines: {2, 99}, {6, 754}, {30, 3589}, {39, 325}, {141, 538}, {182, 2794}, {194, 3096}, {316, 3329}, {597, 2030}, {2679, 3111}
X(4045) = X(141)-of-1st-Brocard-triangle
X(4045) = 1st-Brocard-isogonal conjugate of X(10328)
X(4045) = 1st-Brocard-isotomic conjugate of X(24273)
X(4046) lies on these lines: {8, 21}, {11, 3687}, {42, 594}, {210, 2321}, {306, 3696}, {346, 3715}, {519, 3745}, {740, 1211}, {1213, 1962}, {3683, 3686}
X(4047) lies on these lines: {9, 65}, {10, 1901}, {37, 758}, {46, 965}, {63, 77}, {71, 72}, {144, 1441}, {392, 2260}, {516, 3686}, {579, 960}
X(4048) lies on these lines: {3, 66}, {6, 194}, {32, 732}, {69, 2076}, {76, 1691}, {99, 737}, {182, 2782}, {206, 3492}, {305, 1915}, {524, 1003}
X(4048) = 1st-Brocard-isogonal conjugate of X(3094)
X(4049) lies on the Kiepert hyperbola and these lines: {2, 514}, {4, 2457}, {10, 523}, {76, 3261}, {98, 106}, {321, 1577}, {671, 903}, {901, 2690}, {1797, 2986}
X(4049) = orthocenter of X(2)X(4)X(10)
X(4049) = barycentric product X(106)*X(850)
X(4049) = barycentric quotient X(106)/X(110)
X(4050) lies on these lines: {1, 1575}, {8, 9}, {40, 2784}, {519, 3501}, {672, 3621}, {1018, 3632}, {1449, 2295}, {2092, 3247}, {3625, 3730}
X(4051) lies on these lines: {1, 2271}, {8, 2170}, {9, 3057}, {956, 3496}, {982, 2319}, {984, 3727}, {1212, 3208}, {2082, 2329}, {2802, 3730}
X(4052) lies on these lines: {2, 2415}, {4, 519}, {30, 3429}, {98, 1293}, {226, 3175}, {262, 726}, {516, 3424}, {536, 2051}, {551, 3445}
X(4053) lies on these lines: {1, 6}, {12, 594}, {321, 3262}, {523, 661}, {524, 1959}, {758, 2245}, {1030, 1761}, {1213, 2294}, {1259, 2178}
X(4054) lies on these lines: {2, 2415}, {10, 3120}, {75, 908}, {226, 306}, {312, 1269}, {329, 391}, {442, 3710}, {527, 1150}, {3649, 3714}
X(4055) lies on these lines: {6, 31}, {10, 275}, {58, 1794}, {184, 2200}, {201, 2650}, {228, 1409}, {255, 394}, {287, 293}, {516, 1860}
X(4055) = X(27)-isoconjugate of X(92)
X(4056) lies on these lines: {4, 1111}, {7, 79}, {30, 3665}, {69, 1089}, {80, 3212}, {85, 3585}, {315, 1930}, {320, 3760}, {3583, 3673}
X(4057) lies on these lines: {3, 3667}, {6, 1919}, {23, 385}, {36, 238}, {522, 1324}, {663, 834}, {884, 3415}, {1960, 2605}, {2483, 3709}
X(4057) = isogonal conjugate of X(8050)
X(4057) = polar conjugate of isotomic conjugate of X(22154)
X(4057) = pole, with respect to circumcircle, of the Nagel line
X(4057) = crossdifference of every pair of points on line X(37)X(39)
X(4058) lies on these lines: {6, 3625}, {8, 1743}, {9, 3626}, {10, 37}, {346, 3679}, {519, 1449}, {1278, 3661}, {3247, 3634}, {3617, 3731}
X(4059) lies on these lines: {7, 8}, {226, 241}, {279, 3485}, {354, 3673}, {942, 1111}, {950, 3664}, {1358, 2795}, {1434, 1447}, {3691, 3739}
X(4060) lies on these lines: {8, 9}, {37, 3626}, {319, 527}, {519, 594}, {1449, 3621}, {2345, 3632}, {3247, 3617}, {3634, 3723}
X(4061) lies on these lines: {1, 2}, {55, 3686}, {209, 3059}, {210, 2321}, {226, 3696}, {1211, 3755}, {2325, 3715}, {3452, 3706}, {3683, 3707}
X(4061) = excentral-to-ABC barycentric image of X(940)
X(4062) is the point of intersection of the line X(1)X(10) and the trilinear polar of X(10); cf. X(1323). (Randy Hutson, June 7, 2019)
X(4062) lies on these lines: {1, 2}, {101, 2770}, {523, 661}, {524, 896}, {740, 3120}, {846, 2895}, {1211, 1962}, {2177, 3416}, {2650, 3704}
X(4063) lies on these lines: {1, 667}, {40, 3309}, {57, 1022}, {239, 514}, {484, 513}, {512, 659}, {764, 3336}, {798, 812}, {834, 3737}
X(4063) = pole, with respect to Bevan circle, of line X(1)X(6)
X(4064) lies on these lines: {10, 2394}, {37, 2509}, {71, 879}, {72, 521}, {101, 935}, {190, 2966}, {522, 3465}, {523, 661}, {525, 656}
X(4064) = trilinear pole of line X(125)X(3708)
X(4064) = crossdifference of every pair of points on line X(58)X(1474)
X(4065) lies on these lines: {1, 596}, {10, 37}, {42, 3159}, {190, 1126}, {519, 2292}, {726, 2667}, {758, 3057}, {1125, 1962}, {2650, 3635}
X(4065) = reflection of X(4647) in X(1125)
X(4065) = complement of X(4647) wrt incentral triangle
X(4065) = anticomplement of X(1125) wrt incentral triangle
X(4065) = X(1)-Ceva conjugate of X(1125)
X(4065) = QA-P20 (Reflection of QA-P5 in QA-P1) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/48-qa-p20.html)
X(4066) lies on these lines: {8, 3583}, {10, 321}, {75, 3634}, {312, 1125}, {758, 3714}, {1215, 2901}, {3175, 3743}, {3244, 3702}, {3454, 3773}
X(4067) lies on these lines: {1, 2308}, {8, 3585}, {10, 12}, {63, 3612}, {517, 3625}, {518, 3244}, {551, 960}, {997, 3361}, {3626, 3681}
X(4068) lies on these lines: {1, 3286}, {6, 2667}, {45, 3728}, {55, 199}, {523, 885}, {740, 1001}, {1486, 3295}, {2223, 3723}, {2292, 3242}
X(4069) lies on these lines: {1, 597}, {9, 3056}, {190, 1026}, {645, 3737}, {646, 3699}, {692, 1023}, {1296, 2748}, {2325, 2340}
X(4070) lies on these lines: {8, 41}, {320, 3509}, {522, 650}, {752, 2243}, {1125, 3290}, {2246, 3006}, {2321, 3689}, {2348, 3707}
X(4071) lies on these lines: {10, 213}, {37, 744}, {226, 306}, {334, 350}, {594, 1215}, {672, 3006}, {1281, 3509}, {1500, 3178}
X(4072) lies on these lines: {9, 3625}, {10, 37}, {346, 519}, {551, 2345}, {1449, 3244}, {3161, 3632}, {3626, 3731}, {3644, 3663}
X(4073) lies on these lines: {8, 192}, {9, 55}, {78, 238}, {346, 2310}, {518, 1423}, {982, 2887}, {3056, 3061}, {3190, 3725}
X(4074) lies on these lines: {2, 694}, {6, 305}, {76, 1613}, {141, 427}, {384, 1915}, {698, 1194}, {732, 3051}, {1799, 2076}
X(4074) = X(1194)-of-1st-Brocard-triangle
X(4075) lies on these lines: {2, 596}, {10, 321}, {12, 1365}, {210, 2901}, {519, 960}, {726, 3634}, {1125, 1215}, {3175, 3697}
X(4075) = complement of X(596)
X(4075) = centroid of X(10) plus the vertices of the anticevian triangle of X(10)
X(4076) lies on these lines: {99, 2748}, {190, 3667}, {238, 519}, {522, 3699}, {644, 3063}, {645, 3737}, {1447, 3263}, {2325, 3684}
X(4076) = perspector of ABC and the reflection of the intouch triangle in the Nagel line
X(4076) = isogonal conjugate of X(1357)
X(4076) = isotomic conjugate of X(1358)
X(4077) lies on these lines: {109, 2864}, {226, 661}, {514, 3064}, {522, 693}, {525, 1577}, {934, 2689}, {1365, 3323}, {1367, 3326}
X(4077) = isotomic conjugate of X(643)
X(4077) = trilinear product of Kiepert hyperbola intercepts of Gergonne line
X(4078) lies on these lines: {1, 344}, {10, 37}, {45, 3416}, {142, 726}, {192, 1738}, {306, 756}, {519, 1001}, {1279, 3244}
X(4079) lies on these lines: {37, 513}, {101, 691}, {213, 3063}, {512, 798}, {523, 661}, {663, 1919}, {786, 3766}, {1918, 2422}
X(4079) = isogonal conjugate of X(4610)
Let A14B14C14 be Gemini triangle 14. Let A' be the perspector of conic {{A,B,C,B14,C14}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(4080). (Randy Hutson, January 15, 2019)
X(4080) lies on these lines: {2, 45}, {4, 145}, {10, 3120}, {76, 1978}, {98, 901}, {106, 835}, {329, 1751}, {908, 1266}
X(4080) = isogonal conjugate of X(3285)X(4081) lies on these lines: {8, 190}, {11, 123}, {12, 318}, {56, 280}, {346, 480}, {653, 1360}, {1146, 2310}, {2321, 3059}
X(4081) = trilinear pole wrt extouch triangle of Nagel lineX(4082) lies on these lines: {10, 321}, {55, 2325}, {200, 346}, {210, 2321}, {312, 3717}, {3175, 3755}, {3452, 3703}, {3686, 3715}
X(4083) lies on these lines: {1, 667}, {30, 511}, {65, 876}, {650, 3250}, {659, 663}, {693, 2533}, {1734, 2530}, {2254, 3777}
X(4084) lies on these lines: {1, 89}, {10, 12}, {517, 550}, {518, 3625}, {551, 942}, {958, 1159}, {997, 3339}, {2802, 3555}
X(4085) lies on these lines: {2, 2177}, {6, 752}, {8, 3775}, {10, 37}, {42, 2887}, {141, 519}, {528, 3589}, {537, 3663}
X(4085) = complement of X(32941)
X(4086) lies on these lines: {10, 656}, {100, 2689}, {513, 3762}, {514, 2517}, {522, 3717}, {523, 1577}, {3064, 3239}
X(4086) = isotomic conjugate of X(1414)X(4087) lies on these lines: {75, 982}, {312, 2321}, {314, 3706}, {350, 740}, {517, 668}, {646, 3693}, {1978, 3263}
X(4088) lies on these lines:
{8, 2785}, {523, 661}, {676, 1639}, {918, 2254}, {1109, 2632}, {1282,
2786}, {1635, 2977}
X(4088) = anticomplement of X(4458)
X(4089) lies on these lines: {1, 7}, {11, 1111}, {495, 3665}, {527, 1023}, {1022, 2401}, {1086, 2087}, {3323, 3328}
X(4090) lies on these lines: {10, 12}, {43, 726}, {171, 3699}, {312, 519}, {537, 3752}, {3625, 3706}, {3681, 3741}
X(4091) lies on these lines: {101, 1262}, {110, 2727}, {239, 514}, {652, 905}, {654, 3669}, {663, 928}, {1734, 2812}
X(4091) = isogonal conjugate of isotomic conjugate of X(30805)
X(4091) = crossdifference of every pair of points on line X(33)X(42)
X(4092) lies on these lines:
{8, 645}, {10, 2652}, {115, 2643}, {125, 1109}, {281, 2175}, {542,
2607}, {1146, 3271}
X(4093) lies on these lines: {37, 42}, {38, 1964}, {350, 740}, {668, 718}, {688, 3005}, {1914, 2210}, {3009, 3726}
X(4094) lies on the incentral inellipse and the inellipse centered at X(21254) and on these lines: {31, 643}, {37, 2054}, {42, 2107}, {238, 239}, {244, 1962}, {1100, 2309}, {2269, 2310}
X(4094) = antipode in the incentral inellipse of X(2643)
X(4094) = reflection of X(75) in X(21254)
X(4094) = reflection of X(2643) in X(37)
X(4094) = trilinear square of X(2238)
X(4095) lies on these lines: {8, 2170}, {9, 341}, {10, 37}, {312, 3208}, {1018, 1089}, {1215, 2295}, {1334, 3701}
X(4096) lies on these lines: {2, 38}, {210, 740}, {341, 1089}, {519, 960}, {726, 3740}, {846, 3699}
X(4097) lies on these lines: {10, 1001}, {42, 1449}, {55, 3686}, {71, 3174}, {100, 1014}, {3688, 3689}
Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(4098) and X(24150) are equal.
X(4098) lies on these lines: {1, 3161}, {10, 37}, {346, 1125}, {391, 519}, {551, 3247}, {1743, 3635}
X(4099) lies on these lines: {192, 1930}, {740, 3294}, {1089, 1500}, {1224, 2345}, {1334, 2901}, {2171, 3671}
X(4100) lies on these lines: {19, 775}, {48, 255}, {63, 2148}, {163, 610}, {283, 2302}, {1820, 3708}
X(4100) = isogonal conjugate of X(6521)
X(4101) lies on these lines: {8, 226}, {10, 2650}, {69, 73}, {72, 306}, {391, 1449}, {3649, 3696}
X(4102) lies on these lines: {2, 594}, {8, 3058}, {190, 3578}, {257, 3175}, {333, 2321}, {519, 1126}
X(4103) lies on these lines: {10, 762}, {101, 3699}, {115, 594}, {514, 668}, {596, 1574}, {644, 3239}
X(4104) lies on these lines: {2, 3751}, {10, 12}, {141, 3740}, {306, 756}, {984, 3687}, {3452, 3741}
X(4105) lies on these lines: {55, 652}, {109, 677}, {200, 3239}, {521, 2254}, {649, 926}, {650, 663}
X(4105) = isogonal conjugate of X(4626)
X(4105) = crossdifference of every pair of points on line X(57)X(279)
X(4106) lies on these lines: {320, 350}, {514, 3700}, {522, 2526}, {650, 812}, {2786, 3776}, {3667, 3676}
X(4106) = complement of X(4380)
X(4106) = anticomplement of X(4394)
X(4107) lies on these lines: {239, 514}, {786, 2483}, {802, 3063}, {1027, 2665}, {1054, 2789}, {1919, 3261}
X(4107) = X(649)-of-1st-Brocard-triangle
X(4107) = 1st-Brocard-isogonal conjugate of X(24281)
X(4108) lies on these lines: {2, 512}, {98, 2770}, {351, 523}, {476, 1302}, {669, 804}, {2373, 2857}
X(4108) = X(351)-of-1st-anti-Brocard-triangle
X(4109) lies on these lines: {10, 213}, {37, 3178}, {226, 1231}, {1089, 1826}, {1330, 3509}, {3006, 3691}
X(4110) lies on these lines: {8, 3056}, {9, 646}, {75, 141}, {312, 2321}, {341, 3704}, {668, 3729}
X(4111) lies on these lines: {8, 314}, {210, 2321}, {2092, 3728}, {3271, 3686}, {3679, 3779}, {3704, 3717}
X(4112) lies on these lines: {6, 714}, {31, 730}, {76, 1580}, {313, 560}, {1582, 3596}, {2210, 3765}
X(4112) = X(31)-of-1st-Brocard-triangle
X(4112) = 1st-Brocard-isogonal conjugate of X(3735)
X(4113) lies on these lines: {8, 210}, {333, 3689}, {1215, 3626}, {3681, 3696}, {3686, 3693}
X(4114) lies on these lines: {2, 7}, {65, 3625}, {942, 3627}, {1319, 3671}, {2099, 3635}
X(4115) lies on these lines: {10, 115}, {37, 537}, {72, 2809}, {99, 101}, {213, 3159}
X(4116) lies on these lines: {1, 76}, {32, 904}, {214, 995}, {766, 3116}, {1911, 2242}
X(4117) lies on these lines: {1, 799}, {31, 1927}, {42, 2107}, {561, 3223}, {669, 1977}
X(4118) lies on these lines: {1, 82}, {38, 1755}, {75, 1581}, {760, 1918}, {1959, 1964}
X(4118) = isogonal conjugate of isotomic conjugate of X(20627)
X(4119) lies on these lines: {8, 41}, {10, 3290}, {594, 1575}, {2321, 3693}, {2348, 3686}
X(4120) lies on these lines: {2, 2786}, {115, 125}, {523, 661}, {649, 3239}, {900, 1635}
X(4121) lies on these lines: {51, 325}, {69, 184}, {125, 305}, {141, 1194}, {626, 3118}
X(4122) lies on these lines: {522, 659}, {523, 661}, {525, 2533}, {824, 1491}, {826, 1089}
X(4123) lies on these lines: {1, 977}, {22, 1760}, {29, 33}, {345, 3100}, {643, 3719}
X(4124) lies on these lines: {8, 210}, {11, 1146}, {239, 3573}, {242, 1428}, {514, 3675}
X(4125) lies on these lines: {10, 321}, {312, 519}, {341, 3626}, {3625, 3702}, {3678, 3714}
X(4126) lies on these lines: {8, 3058}, {10, 3782}, {200, 3712}, {210, 3687}, {345, 3711}
X(4127) lies on these lines: {10, 12}, {518, 3635}, {960, 3636}, {2802, 3632}
X(4128) lies on these lines: {1, 99}, {512, 1015}, {804, 3023}, {1356, 3122}, {2170, 2643}
X(4129) lies on these lines: {2, 1019}, {10, 512}, {514, 661}, {656, 3667}, {830, 3716}, {3814, 3836}
X(4129) = isotomic conjugate of X(37205)
X(4129) = complement of X(1019)
X(4129) = pole wrt excircles-radical-circle of Brocard axis
X(4130) lies on these lines: {9, 521}, {37, 2509}, {522, 650}, {644, 765}, {918, 3669}
X(4130) = isogonal conjugate of X(4617)
X(4130) = isotomic conjugate of X(36838)
X(4130) = crossdifference of every pair of points on line X(56)X(269)
X(4131) lies on these lines: {99, 2719}, {100, 677}, {320, 350}, {520, 3265}, {521, 4025}, {3676, 3738}
X(4131) = anticomplement of X(14298)
X(4132) lies on these lines: {1, 3733}, {30, 511}, {37, 798}, {649, 3726}, {3050, 3063}
X(4132) = isogonal conjugate of X(34594)
X(4132) = crosspoint of X(i) and X(j) for these {i,j}: {1, 3952}, {668, 1255}, {1018, 18098}
X(4132) = crosssum of X(i) and X(j) for these {i,j}: {1, 3733}, {513, 3670}, {523, 4187}, {667, 1100}, {905, 18733}, {1019, 16696}, {3737, 18178}
X(4132) = crossdifference of every pair of points on line X(6)X(474)
X(4133) lies on these lines: {10, 37}, {306, 3120}, {519, 1992}
X(4134) lies on these lines: {8, 3583}, {10, 12}, {518, 551}, {519, 3681}, {960, 3244}
X(4135) lies on these lines: {10, 321}, {312, 726}, {346, 3771}, {1215, 3175}, {3625, 3681}
X(4136) lies on these lines: {8, 41}, {10, 37}, {626, 712}, {2887, 3721}, {3061, 3705}
X(4137) lies on these lines: {1, 2206}, {38, 1755}, {517, 2292}, {1962, 3744}, {2294, 3720}
X(4138) lies on these lines: {2, 1707}, {10, 12}, {306, 3120}, {516, 3771}
X(4139) lies on these lines: {30, 511}, {647, 1635}, {1643, 2489}, {3063, 3288}
X(4139) = isogonal conjugate of X(8690)
X(4139) = crossdifference of every pair of points on line X(6)X(404)
X(4140) lies on these lines: {37, 2395}, {522, 2321}, {523, 661}, {804, 2533}
X(4141) lies on these lines: {2, 726}, {519, 902}, {1647, 2325}, {2796, 3006}
X(4142) lies on these lines: {240, 522}, {514, 659}, {525, 3716}, {663, 2785}
X(4143) lies on these lines: {69, 2419}, {99, 2764}, {520, 3265}, {525, 3267}
X(4143) = isotomic conjugate of X(6529)
X(4144) lies on these lines: {10, 213}, {44, 3006}, {523, 661}, {752, 2243}
X(4145) lies on these lines: {30, 511}, {244, 2611}, {647, 2516}, {659, 3722}
Let A'B'C' be the excentral triangle. Let Oa be the circle centered at A with radius |B'C'|, and define Ob and Oc cyclically. X(4146) is the trilinear pole of the Monge line of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)
X(4146) lies on these lines: {7, 2091}, {85, 178}, {174, 556}, {188, 555}
X(4146) = isotomic conjugate of X(188)
X(4146) = cevapoint of X(174) and X(188)X(4147) lies on these lines: {8, 663}, {10, 514}, {522, 3717}, {1734, 3762}
X(4147) = complement of X(4449)
X(4148) lies on these lines: {812, 3766}, {966, 1769}, {1146, 2968}, {3686, 3738}
X(4149) lies on these lines: {1, 141}, {9, 2175}, {78, 1229}, {1631, 1759}
X(4150) lies on these lines: {10, 82}, {92, 264}, {315, 1760}, {344, 2478}
X(4151) lies on these lines: {10, 1577}, {30, 511}, {596, 876}, {693, 1734}
X(4151) = crossdifference of every pair of points on line X(6)X(2350)
X(4152) lies on these lines: {8, 11}, {210, 3271}, {883, 3321}, {2325, 3689}
X(4153) lies on these lines: {10, 213}, {37, 3454}, {306, 1230}, {626, 742}
X(4154) lies on these lines: {238, 239}, {1045, 1178}, {1580, 1966}
X(4154) = X(3747)-of-1st-Brocard-triangle
X(4155) lies on these lines: {30, 511}, {351, 1635}, {2642, 2643}
X(4155) = ideal point of PU(i) for these i: 79, 143
X(4155) = crossdifference of every pair of points on line X(6)X(662)
X(4155) = isogonal conjugate of X(36066)
X(4156) lies on these lines: {10, 82}, {523, 661}, {754, 2244}
X(4157) lies on these lines: {8, 2175}, {522, 650}, {754, 2244}
X(4158) lies on these lines: {72, 306}, {836, 3682}, {1018, 1490}
X(4159) lies on these lines: {184, 3734}, {384, 3117}, {736, 1501}
X(4159) = X(1501)-of-1st-Brocard-triangle
X(4160) lies on these lines: {1, 661}, {30, 511}, {1022, 1390}
X(4160) = isogonal conjugate of X(8691)
X(4160) = isotomic conjugate of X(35181)
X(4160) = X(2)-Ceva conjugate of X(35135)
X(4160) = crossdifference of every pair of points on line X(6)X(896)
X(4161) lies on these lines: {1, 76}, {904, 1914}, {1201, 1279}
X(4162) lies on these lines: {1, 3309}, {55, 667}, {650, 663}, {513,5048}
X(4162) = crossdifference of every pair of points on line X(57)X(1122)
X(4163) lies on these lines: {514, 2526}, {522, 3717}, {1016, 3699}
X(4163) = isogonal conjugate of X(6614)
X(4163) = isotomic conjugate of X(4626)
X(4164) lies on these lines: {36, 238}, {182, 2788}, {693, 1980}
X(4164) = X(667)-of-1st-Brocard-triangle
X(4165) lies on these lines: {8, 41}, {1146, 3703}, {2170, 3705}
X(4166) lies on these lines: {1, 2068}, {43, 2069}, {165, 364}
X(4167) lies on these lines: {8, 41}, {1146, 3704}, {2264, 3686}
X(4168) lies on these lines: {8, 41}, {2264, 2321}
X(4169) lies on these lines: {10, 37}, {519, 2087}
X(4170) lies on these lines: {512, 1577}, {900, 2530}
X(4171) lies on these lines: {37, 656}, {523, 661}
X(4171) = isogonal conjugate of X(4637)X(4172) lies on these lines: {76, 1933}, {560, 734}
X(4172) = X(560)-of-1st-Brocard-triangle
X(4173) lies on these lines: {3, 1808}, {39, 2387}
X(4173) = isotomic conjugate of isogonal conjugate of X(23209)
X(4173) = isotomic conjugate of polar conjugate of X(8265)
X(4173) = X(19)-isoconjugate of X(38830)
X(4174) lies on this line: {10, 3407}
X(4175) lies on this line: {125, 305}
X(4176) lies on this line: {69, 305}
X(4176) = isotomic conjugate of X(6524)
X(4176) = polar conjugate of X(36434)
X(4177) lies on this line: {10, 2205}
X(4178) lies on this line: {8, 2175}
X(4179) lies on this line: {365, 366}
X(4180) lies on this line: {1, 366}
X(4181) lies on this line: {2, 366}
X(4182) lies on this line: {365, 366}
Points on the Euler Line: X(4183)-X(4250)
Many points are found as points of intersection of the Euler line and other central lines.
As a point on the Euler line, X(4183) has Shinagawa coefficients ($aSA$F+abcF, $aSA2$-$aSA$(E+F) +$a$S2).
X(4183) lies on these lines: {1, 204}, {2, 3}, {9, 33}, {42, 1783}, {55, 281}, {81, 162}, {92, 1621}, {107, 972}, {108, 226}, {191, 1844}, {200, 1802}, {240, 846}, {278, 1001}, {282, 284}, {283, 2000}, {285, 1437}, {346, 1260}, {943, 1896}, {968, 1096}, {1430, 3720}, {1474, 2267}, {1490, 2360}, {1713, 1778}, {1728, 1780}, {1859, 3683}, {1864, 2194}, {2287, 2326}, {2906, 3193}
X(4183) = isogonal conjugate of X(1439)
X(4183) = polar conjugate of X(1446)
As a point on the Euler line, X(4184) has Shinagawa coefficients (2$bc$ + E, -2$bc$)
X(4184) lies on these lines: {2, 3}, {31, 3736}, {35, 42}, {36, 3720}, {55, 81}, {60, 1780}, {86, 1621}, {99, 310}, {100, 333}, {103, 110}, {228, 3219}, {283, 947}, {284, 672}, {476, 2688}, {573, 3060}, {846, 3724}, {917, 925}, {991, 1779}, {1014, 1617}, {1030, 2238}, {1043, 2975}, {1326, 2206}, {1333, 2276}, {1412, 2078}, {1444, 3433}, {1623, 1634}
X(4184) = isogonal conjugate of the isotomic conjugate of X(33297)
As a point on the Euler line, X(4185) has Shinagawa coefficients ($a$F, -$a$(E + F) - 2abc)
X(4185) lies on these lines: {1, 1824}, {2, 3}, {6, 19}, {9, 1868}, {12, 197}, {33, 1900}, {46, 1707}, {55, 1869}, {56, 225}, {57, 1426}, {275, 3597}, {278, 961}, {672, 2333}, {958, 1867}, {999, 1068}, {1155, 1878}, {1452, 1454}, {1474, 2278}, {1610, 3485}, {1753, 1902}, {1825, 2099}, {1836, 3556}, {1861, 1891}, {1968, 2204}, {3167, 3193}
X(4185) = isogonal conjugate of X(34259)
X(4185) = polar conjugate of X(34258)
X(4185) = crosssum of X(219) and X(7085)
X(4185) = crossifference of every pair of points on line X(521)X(647)
X(4185) = X(63)-isoconjugate of X(941)
As a point on the Euler line, X(4186) has Shinagawa coefficients ($a$F, -$a$(E + F) + 2abc)
X(4186) lies on these lines: {1, 1828}, {2, 3}, {12, 1486}, {19, 45}, {33, 1829}, {34, 1319}, {55, 1842}, {56, 1877}, {100, 2899}, {108, 1398}, {208, 1876}, {242, 318}, {513, 1406}, {607, 2201}, {1068, 2969}, {1145, 1862}, {1210, 1473}, {1351, 3193}, {1452, 1887}, {1633, 1788}, {1837, 3556}, {1866, 2099}, {3214, 3217}
As a point on the Euler line, X(4187) has Shinagawa coefficients (abc$a$ - S2, -S2)
X(4187) lies on these lines: {1, 1329}, {2, 3}, {8, 496}, {10, 11}, {12, 1125}, {35, 3035}, {72, 1210}, {119, 1385}, {495, 3616}, {498, 1001}, {499, 958}, {908, 942}, {936, 3419}, {946, 3753}, {956, 2551}, {960, 1737}, {997, 1837}, {999, 3436}, {1376, 1479}, {1698, 1706}, {1834, 3216}, {3679, 3680}
X(4187) = complement of X(404)
As a point on the Euler line, X(4188) has Shinagawa coefficients (abc$a$ - 4S2, 4S2)
X(4188) lies on these lines: {1, 1392}, {2, 3}, {8, 36}, {35, 3616}, {55, 3622}, {56, 100}, {78, 3218}, {149, 3086}, {346, 2178}, {936, 3219}, {962, 2077}, {970, 2979}, {999, 3623}, {1055, 3501}, {1201, 3550}, {1376, 2975}, {1420, 3680}, {1468, 3240}, {1470, 3600}, {3306, 3601}
X(4188) = {X(2),X(3)}-harmonic conjugate of X(4189)
X(4188) = {X(3),X(631)}-harmonic conjugate of X(37106)
X(4188) = {X(3),X(6876)}-harmonic conjugate of X(37105)
As a point on the Euler line, X(4189) has Shinagawa coefficients (abc$a$ + 4S2, -4S2)
X(4189) lies on these lines: {1, 89}, {2, 3}, {8, 35}, {36, 3616}, {55, 145}, {56, 1621}, {63, 3601}, {78, 3219}, {100, 958}, {956, 3621}, {966, 1030}, {970, 3060}, {1043, 1150}, {1125, 1770}, {1158, 3576}, {1727, 3612}, {3241, 3746}, {3295, 3623}
X(4189) = isogonal conjugate of isotomic conjugate of X(34282)
X(4189) = {X(2),X(3)}-harmonic conjugate of X(4188)
X(4189) = {X(3),X(376)}-harmonic conjugate of X(37105)
X(4189) = {X(3),X(6875)}-harmonic conjugate of X(37106)
As a point on the Euler line, X(4190) has Shinagawa coefficients (abc$a$ - 2S2, 4S2)
X(4190) lies on these lines: {2, 3}, {7, 224}, {8, 46}, {56, 3434}, {65, 145}, {100, 388}, {390, 2646}, {391, 2245}, {528, 3304}, {950, 3306}, {997, 1770}, {1155, 3617}, {1320, 3296}, {1376, 3436}, {2550, 2975}, {3612, 3616}
X(4190) = anticomplement of X(2478)
As a point on the Euler line, X(4191) has Shinagawa coefficients ($bc$ - E, -$bc$)
X(4191) lies on these lines: {2, 3}, {6, 2350}, {36, 43}, {42, 56}, {51, 991}, {55, 750}, {57, 228}, {182, 1790}, {183, 310}, {198, 672}, {582, 1437}, {614, 2223}, {1155, 3185}, {1403, 3724}, {2178, 2276}, {2352, 3752}
As a point on the Euler line, X(4192) has Shinagawa coefficients ($bcSBSC$ + ES2, -$bc$S2)
X(4192) lies on these lines: {2, 3}, {40, 43}, {42, 517}, {165, 2108}, {182, 1754}, {228, 908}, {511, 1764}, {515, 3741}, {516, 2051}, {573, 2238}, {899, 3579}, {1385, 3720}, {1766, 2276}
X(4192) = anticomplement of X(37365)
As a point on the Euler line, X(4193) has Shinagawa coefficients (abc$a$ - 2S2, -S2).
X(4193) lies on these lines: {2, 3}, {8, 11}, {12, 1388}, {100, 1479}, {119, 944}, {145, 496}, {388, 1476}, {495, 3622}, {498, 1621}, {499, 2975}, {908, 1210}, {3086, 3436}, {3701, 3705}
As a point on the Euler line, X(4194) has Shinagawa coefficients ($a$F, 2abc).
X(4194) lies on these lines: {2, 3}, {7, 208}, {8, 33}, {34, 3616}, {108, 3600}, {193, 3193}, {281, 318}, {391, 1172}, {393, 941}, {968, 1785}, {1788, 1887}, {1870, 3622}, {1875, 3485}
As a point on the Euler line, X(4195) has Shinagawa coefficients ((E + F)2 + $bc$(E + F) - S2, 2S2).
X(4195) lies on these lines: {1, 87}, {2, 3}, {6, 1043}, {8, 31}, {10, 3550}, {55, 1220}, {75, 1104}, {86, 1975}, {239, 1453}, {346, 2298}, {2322, 3172}, {3714, 3769}
X(4195) = anticomplement of X(16062)
As a point on the Euler line, X(4196) has Shinagawa coefficients (F, -$bc$).
X(4196) lies on these lines: {2, 3}, {19, 672}, {33, 3720}, {34, 42}, {184, 3332}, {225, 1435}, {264, 310}, {278, 1002}, {393, 2350}, {1730, 1779}, {1836, 2182}, {1841, 2276}
X(4196) = polar conjugate of X(32022)
As a point on the Euler line, X(4197) has Shinagawa coefficients (3abc$a$ + 2S2, 2S2).
X(4197) lies on these lines: {2, 3}, {7, 12}, {8, 3475}, {10, 3681}, {63, 1698}, {76, 3263}, {81, 1714}, {331, 1441}, {495, 3617}, {2346, 2550}, {2886, 3616}
X(4197) = homothetic center of 4th Euler triangle and 1st Conway triangle
As a point on the Euler line, X(4198) has Shinagawa coefficients ($a$F, -2$a$(E + F) - 2abc).
X(4198) lies on these lines: {2, 3}, {7, 34}, {8, 19}, {193, 1829}, {278, 3600}, {393, 1169}, {497, 1852}, {944, 1871}, {1848, 3616}, {1859, 3486}, {1888, 3474}
X(4198) = anticomplement of X(37179)
As a point on the Euler line, X(4199) has Shinagawa coefficients (abc$a$(E + F) + ($bc$ + 2E)S2, -$bc$S2).
X(4199) lies on these lines: {2, 3}, {9, 43}, {42, 72}, {55, 1211}, {226, 1284}, {392, 1064}, {950, 3741}, {958, 1834}, {1213, 1376}, {3454, 3771}
As a point on the Euler line, X(4200) has Shinagawa coefficients ($a$F, -2abc).
X(4200) lies on these lines: {2, 3}, {8, 34}, {33, 3616}, {145, 1870}, {278, 318}, {962, 1753}, {1785, 3086}, {1788, 1875}, {1841, 2345}, {1887, 3485}
As a point on the Euler line, X(4201) has Shinagawa coefficients ((E+F)2 + $bc$(E + F) + S2, -2S2).
X(4201) lies on these lines: {1, 2896}, {2, 3}, {8, 38}, {141, 1043}, {194, 1654}, {386, 1330}, {388, 1403}, {988, 3705}
X(4201) = complement of X(13740)
As a point on the Euler line, X(4202) has Shinagawa coefficients (2(E + F)2 + 2$bc$(E + F) - abc$a$, -2S2).
X(4202) lies on these lines: {2, 3}, {8, 141}, {10, 38}, {121, 1054}, {1150, 1714}, {1193, 2887}, {3216, 3454}, {3663, 3710}
As a point on the Euler line, X(4203) has Shinagawa coefficients ($bc$(E + F)2 - $bc(SA)2$ - (2$bc$ + E)S2, 2$bc$S2).
X(4203) lies on these lines: {2, 3}, {31, 43}, {171, 2309}, {312, 2352}, {1402, 3685}, {1580, 3494}, {2276, 2298}, {2975, 3741}
As a point on the Euler line, X(4204) has Shinagawa coefficients (2$bc$(E + F)2 - 2$bc(SA)2$ + $bcSBSC$ + 4ES2, -$bc$S2).
X(4204) lies on these lines: {2, 3}, {37, 42}, {55, 1213}, {968, 2092}, {1001, 1211}, {1104, 3720}, {2245, 3683}, {3294, 3690}
As a point on the Euler line, X(4205) has Shinagawa coefficients (2(E + F)2 + 4$bc$(E + F) - 2$bcSA$ + S2, -S2).
X(4205) lies on these lines: {1, 1211}, {2, 3}, {10, 37}, {12, 1402}, {86, 1330}, {387, 966}, {1104, 1125}, {1698, 3712}
As a point on the Euler line, X(4206) has Shinagawa coefficients ([$a$(E + F) + abc]F, -[(E + F)2 + S2]$a$ + $a(SA)2$ + 2abc(E + F) ).
X(4206) lies on these lines: {2, 3}, {19, 31}, {33, 1474}, {1172, 1824}, {1396, 1876}, {1766, 2328}, {1829, 3194}, {2182, 2194}
As a point on the Euler line, X(4207) has Shinagawa coefficients (F, $bc$).
X(4207) lies on these lines: {2, 3}, {33, 42}, {34, 3720}, {193, 2905}, {278, 1893}, {281, 1824}, {1848, 1851}, {1899, 3332}
As a point on the Euler line, X(4208) has Shinagawa coefficients (2abc$a$ + S2, 2S2).
X(4208) lies on these lines: {2, 3}, {7, 10}, {142, 938}, {391, 1330}, {1441, 1847}, {2550, 2894}, {3617, 3753}
As a point on the Euler line, X(4209) has Shinagawa coefficients ((E + F)2 - $bcSA$ - S2, 2S2).
X(4209) lies on these lines: {2, 3}, {6, 1434}, {7, 41}, {56, 673}, {85, 910}, {169, 3177}, {279, 294}
As a point on the Euler line, X(4210) has Shinagawa coefficients (E - 2$bc$, 2$bc$).
X(4210) lies on these lines: {2, 3}, {35, 3720}, {36, 42}, {228, 3218}, {310, 1078}, {573, 2979}, {991, 3060}
As a point on the Euler line, X(4211) has Shinagawa coefficients ([$a$(E + F) - abc]F, -$a$(E + F)2 + abc$bc$).
X(4211) lies on these lines: {2, 3}, {19, 38}, {1106, 1435}, {1396, 1462}, {1473, 1851}, {1474, 2191}, {2201, 3720}
As a point on the Euler line, X(4212) has Shinagawa coefficients (2F, -$bc$).
X(4212) lies on these lines: {2, 3}, {34, 43}, {42, 1870}, {278, 291}, {310, 1235}, {1575, 1841}, {1861, 3741}
X(4212) = polar conjugate of isogonal conjugate of X(33863)
As a point on the Euler line, X(4213) has Shinagawa coefficients (2F, $bc$).
X(4213) lies on these lines: {2, 3}, {33, 43}, {242, 1848}, {278, 1874}, {1172, 2238}, {1654, 2905}, {1870, 3720}
X(4213) = pole wrt polar circle of trilinear polar of X(6625) (line X(523)X(2487))
X(4213) = polar conjugate of X(6625)
As a point on the Euler line, X(4214) has Shinagawa coefficients ($a$F, -$a$(E + F) - 4abc).
X(4214) lies on these lines: {1, 1900}, {2, 3}, {19, 44}, {34, 1824}, {225, 1398}, {484, 1878}, {1869, 1877}
As a point on the Euler line, X(4215) has Shinagawa coefficients ($a$(E + F)F + $a(SA)2$ - $aSA$F + abc(E + 2F) + 2$a$S2, -$a$(E + F)2 - 2abc(E + F) - 2$a$S2).
X(4215) lies on these lines: {2, 3}, {31, 48}, {228, 284}, {1473, 3286}, {2193, 2203}
As a point on the Euler line, X(4216) has Shinagawa coefficients ($a$(E + 2F) + 2$aSA$, -2$a$(E + F) - 2$aSA$).
X(4216) lies on these lines: {2, 3}, {31, 36}, {160, 1631}, {2178, 2298}
As a point on the Euler line, X(4217) has Shinagawa coefficients (2(E + F)2 + 2$bc$(E + F) - abc$a$ - 2S2,6S2).
X(4217) lies on these lines: {2, 3}, {8, 44}
As a point on the Euler line, X(4218) has Shinagawa coefficients ($a$(3E + 2F) - 2$aSA$, -2$a$(E + F) + 2$aSA$).
X(4218) lies on these lines: {2, 3}, {36, 38}
As a point on the Euler line, X(4219) has Shinagawa coefficients (($aSA$ - abc)F, $aSBSC$).
X(4219) lies on these lines: {1, 951}, {2, 3}, {19, 165}, {33, 57}, {34, 3601}, {35, 1838}, {42, 1430}, {43, 1783}, {48, 2947}, {55, 278}, {74, 1243}, {92, 100}, {104, 3418}, {185, 2906}, {204, 2999}, {281, 1376}, {386, 3194}, {516, 1848}, {579, 1172}, {610, 1750}, {942, 1902}, {991, 1396}, {1119, 3672}, {1155, 1859}, {1633, 1709}, {1844, 3336}, {1871, 3579}, {1888, 2646}, {2688, 2766}
As a point on the Euler line, X(4220) has Shinagawa coefficients ($a$(E + F) + abc, -$a$(E + F)).
X(4220) lies on these lines: {1, 3430}, {2, 3}, {9, 2312}, {40, 612}, {41, 43}, {55, 1284}, {81, 511}, {98, 100}, {105, 1283}, {108, 1214}, {111, 1292}, {165, 846}, {171, 256}, {197, 2550}, {198, 1376}, {208, 1038}, {230, 1030}, {325, 1444}, {573, 1754}, {614, 3576}, {842, 1290}, {940, 1350}, {1211, 1503}, {1824, 3101}, {2691, 2770}, {2697, 2766}, {2895, 3564}, {2975, 3705}
X(4220) = complement of X(37456)
X(4220) = anticomplement of X(37360)
X(4220) = circumcircle-inverse of X(37959)
X(4220) = Euler line intercept, other than X(28), of circle {X(28),PU(4)}
X(4220) = Thomson-isogonal conjugate of X(392)
As a point on the Euler line, X(4221) has Shinagawa coefficients ($a$(E + F) + 2abc, -$a$(E + F) - 3abc).
X(4221) lies on these lines: {1, 1412}, {2, 3}, {40, 58}, {81, 517}, {99, 104}, {105, 1296}, {112, 1295}, {573, 1778}, {691, 2687}, {759, 1292}, {915, 3565}, {935, 2694}, {944, 1043}, {990, 3576}, {999, 1014}, {1064, 3736}, {1408, 3057}, {1766, 2303}, {2696, 2752}, {3286, 3428}
X(4221) = circumcircle-inverse of X(37960)
As a point on the Euler line, X(4222) has Shinagawa coefficients ($a$F, -$a$(E + F) + abc).
The trilinear polar of X(4222) meets the line at infinity at X(4132). (Randy Hutson, June 7, 2019)
X(4222) lies on these lines: {2, 3}, {19, 3731}, {33, 1697}, {34, 106}, {46, 1633}, {208, 1467}, {225, 2078}, {1068, 1851}, {1210, 3220}, {1486, 3085}, {1629, 1896}, {1736, 1782}, {1783, 1973}, {1785, 1842}, {1828, 1870}, {1843, 2906}, {1900, 2355}, {2201, 2333}, {2758, 2766}, {3060, 3193}
As a point on the Euler line, X(4223) has Shinagawa coefficients ((E - 2F)S2 + abc$aSA$, 2(E + F)S2).
X(4223) lies on these lines: {1, 41}, {2, 3}, {56, 948}, {142, 3220}, {172, 1104}, {198, 1001}, {228, 1621}, {229, 1931}, {238, 1400}, {242, 1441}, {614, 1453}, {1279, 2220}, {1446, 1447}, {1486, 2550}, {1730, 2328}, {1827, 3100}, {2355, 3101}, {2690, 2752}
As a point on the Euler line, X(4224) has Shinagawa coefficients ((E + 2F)S2 - abc$aSA$, -2(E + F)S2).
X(4224) lies on these lines: {1, 2187}, {2, 3}, {7, 1473}, {19, 1040}, {31, 57}, {81, 184}, {154, 940}, {226, 3220}, {497, 1486}, {610, 2268}, {612, 3601}, {1610, 2646}, {1633, 1836}, {1764, 2328}, {1824, 3100}, {2689, 2752}, {2975, 3757}, {3485, 3556}
As a point on the Euler line, X(4225) has Shinagawa coefficients ($a$(E + 2F) + 2$aSA$ + 2abc, -2$a$(E + F) - 2$aSA$ - 2abc).
X(4225) lies on these lines: {1, 994}, {2, 3}, {36, 58}, {56, 81}, {99, 1311}, {100, 1043}, {102, 110}, {198, 2287}, {284, 1400}, {333, 1610}, {476, 2695}, {581, 3060}, {1333, 2277}, {1437, 3417}, {1444, 3435}, {2178, 2303}
As a point on the Euler line, X(4226) has Shinagawa coefficients ((E + F)(E + 4F) - 4S2, -2(E + F)2 + 8S2).
X(4226) lies on these lines: {2, 3}, {99, 110}, {107, 3565}, {112, 925}, {476, 691}, {523, 2407}, {685, 877}, {827, 930}, {879, 2966}, {1287, 1291}, {1296, 1302}, {1640, 2420}, {1649, 3233}
X(4226) = isogonal conjugate of X(35364)
X(4226) = cevapoint of X(i) and X(j) for these {i,j}: {511, 6132}, {523, 6036}
X(4226) = crosspoint of X(99) and X(2966)
X(4226) = crosssum of X(512) and X(3569)
X(4226) = trilinear pole of line X(114)X(230)
X(4226) = crossdifference of every pair of points on line X(647)X(3124)
X(4226) = antigonal conjugate of X(34174)
X(4226) = trilinear quotient X(i)/X(j) for these (i,j): (230, 661), (1733, 523), (3564, 656), (8772, 512)
As a point on the Euler line, X(4227) has Shinagawa coefficients (2FS2, (E - 2F)S2 + abc$a$(E + F)).
X(4227) lies on these lines: {2, 3}, {19, 993}, {35, 1891}, {36, 1848}, {99, 286}, {104, 112}, {935, 2687}, {1014, 1565}, {1158, 1780}, {1289, 1295}, {1396, 1870}, {1474, 3576}, {1748, 2975}
X(4227) = circumcircle-inverse of X(37961)
As a point on the Euler line, X(4228) has Shinagawa coefficients (2FS2 - abc$aSA$, -2(E + F)S2).
X(4228) lies on these lines: {2, 3}, {58, 614}, {81, 105}, {104, 1302}, {476, 2752}, {583, 1778}, {1014, 3598}, {1333, 3290}, {1486, 3434}, {1621, 3185}, {1775, 1780}, {1792, 3006}, {1859, 3100}
As a point on the Euler line, X(4229) has Shinagawa coefficients (E + F + 2$bc$, -E - F - 3$bc$).
X(4229) lies on these lines: {1, 1434}, {2, 3}, {86, 516}, {99, 103}, {165, 333}, {390, 1014}, {675, 1296}, {691, 2688}, {740, 2938}, {917, 3565}, {1742, 3736}
As a point on the Euler line, X(4230) has Shinagawa coefficients ((E + F)(E - 2F)F - 2FS2, 2(E + F)2F - (E - 2F)S2).
X(4230) lies on these lines: {2, 3}, {99, 107}, {110, 112}, {476, 935}, {648, 1634}, {691, 1304}, {827, 933}, {877, 2396}, {925, 1289}, {1286, 1288}, {1301, 3565}
X(4230) = isogonal conjugate of X(879)
X(4230) = trilinear pole of line X(232)X(511)
As a point on the Euler line, X(4231) has Shinagawa coefficients (($aSA$ - abc)F, - $aSA$(E + F)).
X(4231) lies on these lines: {2, 3}, {33, 240}, {98, 108}, {100, 1824}, {230, 1865}, {232, 1172}, {511, 1812}, {842, 2766}, {997, 3430}, {1292, 2374}, {1716, 2212}
X(4231) = Euler line intercept, other than X(21), of circle {X(21),PU(4)}As a point on the Euler line, X(4232) has Shinagawa coefficients (3F, -2E - 2F).
X(4232) lies on these lines: {2, 3}, {107, 111}, {110, 193}, {253, 1301}, {391, 1474}, {1302, 3563}, {1304, 2770}, {1829, 3622}, {2356, 3240}, {3066, 3618}, {3199, 3291}
X(4232) = circumcircle-inverse of X(37962)
X(4232) = polar conjugate of X(5485)
As a point on the Euler line, X(4233) has Shinagawa coefficients (($aSA$ + 2abc)F, -($aSA$ + abc)(E + F)).
X(4233) lies on these lines: {2, 3}, {19, 1621}, {81, 2299}, {105, 107}, {110, 2203}, {162, 1396}, {915, 1302}, {1172, 2346}, {1304, 2752}, {1474, 2287}
X(4232) = isogonal conjugate of X(14417)
X(4233) = circumcircle-inverse of X(37963)
As a point on the Euler line, X(4234) has Shinagawa coefficients ((E + F)2 + $bc$(E + F) - 2S2, 3S2).
X(4234) lies on these lines: {2, 3}, {35, 1220}, {58, 519}, {81, 3241}, {86, 99}, {112, 2322}, {333, 3550}, {391, 1285}, {691, 2758}
As a point on the Euler line, X(4235) has Shinagawa coefficients (2(E+F)2F-6FS2, -(E-8F)S2).
X(4235) lies on these lines: {2, 3}, {74, 287}, {99, 112}, {107, 1296}, {525, 2420}, {691, 935}, {1289, 3565}, {1304, 2696}, {2394, 2966}
X(4235) = isotomic conjugate of X(14977)
X(4235) = polar conjugate of X(5466)
X(4235) = trilinear pole of line X(468)X(524)
X(4235) = pole wrt polar circle of trilinear polar of X(5466) (line X(115)X(523))
As a point on the Euler line, X(4236) has Shinagawa coefficients ($abSC2$(E+F) -$abSC$[(E+F)2-2S2] , -3$abSC$S2+$ab$(E+F)S2).
X(4236) lies on these lines: {2, 3}, {81, 3110}, {99, 100}, {108, 3565}, {110, 1292}, {476, 2691}, {662, 1633}, {691, 1290}
As a point on the Euler line, X(4237) has Shinagawa coefficients ($aSA2$(E+F) -$aSA$[(E+F)2-2S2] , -3$aSA$S2+$a$(E+F)S2).
X(4237) lies on these lines: {2, 3}, {99, 101}, {112, 653}, {163, 664}, {673, 759}, {691, 2690}, {1019, 1025}
As a point on the Euler line, X(4238) has Shinagawa coefficients (2$aSA$(E+F)F-2$a$FS2, $aSA3$ +$aSBSC$(E+F) -$aSA$[(E+F)2-S2]+2$a$FS2).
X(4238) lies on these lines: {2, 3}, {99, 108}, {100, 112}, {107, 1292}, {691, 2766}, {935, 1290}, {1304, 2691}
X(4238) = isogonal conjugate of X(10099)
As a point on the Euler line, X(4239) has Shinagawa coefficients ((E - 2F)S2 + (E + F)abc$a$, 2(E + F)S2).
X(4239) lies on these lines: {2, 3}, {37, 100}, {108, 1441}, {612, 3743}, {1290, 2770}, {1376, 3712}
As a point on the Euler line, X(4240) has Shinagawa coefficients ((E - 8F)F,6(E + F)F - 2S2).
X(4240) is the point of intersection of the Euler lines of twelve triangles, constructed as in the next few sentences.
Let E be the Euler line of a triangle ABC. Let A1 = E∩BC, and define B1 and C1 cyclically. Let AB be the reflection of A in B1, and define BC and CA cyclically. Let AC be the reflection of C in B1, and define BA and CB cyclically. The Euler lines of the four triangles ABC, AABAC, BBCBA, CCACB concur in X(4240). (Dao Thanh Oai, Problem 1 in attachment to ADGEOM #1709, September 15, 2014). See also Telv Cohl, 'Dao's Theorem on the Concurrence of Three Euler Lines,' International Journal of Geometry 3 (2014) 70-73: Dao's Theorem.
Continuing, let A*B*C* be the paralogic triangle of ABC whose perspectrix is E. Then X(4240) lies on the Euler line of A*B*C*. (Dao Thanh Oai, noted just after Figure 1 in attachment to ADGEOM #1709, September 15, 2014).
Continuing, redefine AB as the point on line AC and AC as the point on line AB such that B1, A1, AB, AC line on a circle and A1, AB, AC are collinear. Define BC and BA cyclically, and define CA and CB cyclically. Let A2 = BABC∩CACB and define B2 and C2 cyclically. The Euler lines of the five triangles ABC, A2B2C2, AABAC, BBCBA, CCACB concur in X(4240). (Dao Thanh Oai, Problem 2 in attachment to ADGEOM #1709, September 15, 2014).
Continuing, The Euler lines of the triangles A2BACA, B2CBAB, C2ACBC concur in X(4240). (Dao Thanh Oai, September 17, 2014).
X(4240) is the only point on the Euler line whose trilinear polar is parallel to the Euler line. (Randy Hutson, January 29, 2015)
X(4240) is one of three points used to define a 10-point circle, denoted by D at X(7740). (Dao Thanh Oai, June 24, 2015)
The Gossard triangle is homothetic to the anticomplementary triangle, and the homothetic center is X(4240). See Sava Grozdev and Deko Dekov, International Journal of Computer Discovered Mathematics, Computer Discovered Mathematics: A Note on the Gossard Triangle.
X(4240) lies on these lines: {2,3}, {107,110}, {112,1302}, {146,5667}, {476,1304}, {523,5502}, {685,5466}, {877,5468}, {925,1301}, {2407,3233}
X(4240) = midpoint of X(1561) and X(3081)
X(4240) = reflection of X(i) in X(j) for these (i,j): (2,1651), (1650,402)
X(4240) = isogonal conjugate of X(14380)
X(4240) = isotomic conjugate of X(34767)
X(4240) = anticomplement of X(1650)
X(4240) = X(687)-Ceva conjugate of X(112)
X(4240) = crosspoint of X(36306) and X(36309)
X(4240) = X(i)-cross conjugate of X(j) for these (i,j): (1511, 250), (2420, 2407)
X(4240) = trilinear pole of line X(30)X(1990)
X(4240) = homothetic center of Gossard and anticomplementary triangles
X(4240) = pole wrt polar circle of trilinear polar of X(2394) (line X(125)X(523))
X(4240) = polar conjugate of X(2394)
X(4240) = {X(i),X(j)}harmonic conjugate of X(k) for these (i,j,k): (402,1650,2), (1650,1651,402), (2409,4230,4226)
X(4240) = X(i)-isoconjugate of X(j) for these (i,j): (48,2394), (63,2433), (74,656), (525,2159), (647,2349), (656,74), (810,1494), (1304,2632), (1494,810), (2159,525), (2349,647), (2394,48), (2433,63), (2632,1304)
As a point on the Euler line, X(4241) has Shinagawa coefficients ($aSA2$F+2$aSBSC$F -$aSA$(E+F)F, -$aSA3$ -$aSBSC$(E+F) + $aSA$[(E+F)2-2S2]+$a$FS2).
X(4241) lies on these lines: {2, 3}, {101, 107}, {110, 3732}, {162, 658}, {1301, 1305}, {1304, 2690}
As a point on the Euler line, X(4242) has Shinagawa coefficients (2$aSA$F+3abcF, $aSBSC$ -$aSA$(E+F)-abc(E+F)).
X(4242) lies on these lines: {2, 3}, {100, 108}, {162, 662}, {214, 1845}, {1290, 2766}, {1309, 2222}
X(4242) = circumcircle-inverse of X(37964)
As a point on the Euler line, X(4243) has Shinagawa coefficients ($aSA2$E-2$aSBSC$F -$aSA$[(E+F)E-2S2], 2$aSBSC$(E+F)-2$aSA$S2).
X(4243) lies on these lines: {2, 3}, {100, 1624}, {101, 110}, {107, 1305}, {476, 2690}
X(4243) = Euler line intercept, other than X(376), of circle {X(376),PU(4)}
As a point on the Euler line, X(4244) has Shinagawa coefficients (2$abSC3$F+2$abSASB$(E+F)F -2$abSC$[(E+F)2-2S2]F -2$ab$F2S2, -2$abSC4$ -$abSASB$(E+F)2 +$abSC2$[(E+F)2-2S2] +$abSC$[(E+F)3-(3E+4F)S2] +3$ab$(E+F)FS2).
X(4244) lies on these lines: {2, 3}, {100, 1289}, {108, 112}, {935, 2766}, {1292, 1301}
X(4244) = circumcircle-inverse of X(37965)
As a point on the Euler line, X(4245) has Shinagawa coefficients ($a$(E + F) - $aSA$ - 2$a$F, $a$(E + F) + $aSA$).
X(4245) lies on these lines: {2, 3}, {6, 101}, {56, 1724}, {517, 1730}, {2453, 2690}
As a point on the Euler line, X(4246) has Shinagawa coefficients (2$abSASB$F+$abSC2$F -$abSC$(E+F)F, -$abSC3$ -$abSASB$(E+F) +$abSC$[(E+F)2-2S2]+$ab$FS2).
X(4246) lies on these lines: {2, 3}, {100, 107}, {108, 110}, {476, 2766}, {1290, 1304}
X(4246) = circumcircle-inverse of X(37966)As a point on the Euler line, X(4247) has Shinagawa coefficients (2[(E + F)2 + (E + F)$bc$ + S2]F, -2(E + F)3 - 2(E + F)2$bc$ + (E - 2F)S2).
X(4247) lies on these lines: {2, 3}, {106, 112}, {849, 1412}, {935, 2758}, {1289, 2370}
As a point on the Euler line, X(4248) has Shinagawa coefficients (3$a$F, -2$a$(E + F) + $aSA$ - abc).
X(4248) lies on these lines: {2, 3}, {106, 107}, {1222, 1474}, {1301, 2370}, {1304, 2758}
As a point on the Euler line, X(4249) has Shinagawa coefficients ($aSA3$F+$aSBSC$(E+F)F -$aSA$[(E+F)2+S2]F+$a$EFS2, -$aSA3$(E+F)-$aSBSC$(E+F)2 +$aSA$[(E+F)3-(2E-F)S2]).
X(4249) lies on these lines: {2, 3}, {101, 112}, {935, 2690}, {1289, 1305}
As a point on the Euler line, X(4250) has Shinagawa coefficients (2$aSA2$F+2$aSA$(E+F)F+4abc(E+F)F, 2$aSA3$-2$aSA2$(E+F) -2$aSA$[(E+F)2-S2]+$a$ES2 -2abc[(E+F)2-S2]).
X(4250) lies on these lines: {2, 3}, {101, 108}, {2690, 2766}
Points on the Brocard Axis: X(4251)-X(4291)
Many points are found as points of intersection of the Brocard Axis and other central lines.
X(4251) lies on these lines: {1, 41}, {3, 6}, {4, 2332}, {7, 1803}, {9, 943}, {10, 3684}, {35, 672}, {36, 1475}, {37, 1731}, {40, 2266}, {42, 251}, {48, 1449}, {55, 218}, {60, 163}, {65, 2301}, {81, 1730}, {172, 2251}, {213, 595}, {220, 3295}, {380, 1766}, {391, 2327}, {519, 2329}, {604, 3361}, {609, 1468}, {662, 1509}, {673, 2140}, {758, 3496}, {909, 2364}, {910, 942}, {999, 3207}, {1100, 2174}, {1206, 2205}, {1334, 3746}, {1582, 2663}, {1621, 3294}, {1630, 2262}, {1743, 2268}, {1763, 2172}, {2176, 2241}, {2183, 2302}, {2702, 2711}, {3217, 3731}
X(4251) = {X(371),X(372)}-harmonic conjugate of X(991)
X(4252) lies on these lines: {1, 3052}, {3, 6}, {20, 1834}, {21, 940}, {31, 56}, {44, 936}, {45, 975}, {55, 1468}, {57, 1104}, {171, 958}, {172, 220}, {184, 1408}, {213, 3207}, {218, 609}, {376, 387}, {394, 593}, {474, 1724}, {595, 999}, {601, 3428}, {902, 3303}, {960, 1707}, {965, 1778}, {988, 1386}, {1279, 3333}, {1394, 1427}, {1418, 3361}, {1451, 1466}, {1453, 3752}, {2163, 3445}, {2177, 2334}, {3017, 3534}, {3192, 3515}
X(4253) lies on these lines: {1, 672}, {2, 2350}, {3, 6}, {7, 2140}, {9, 1125}, {35, 2280}, {36, 41}, {46, 2082}, {56, 101}, {57, 169}, {71, 1449}, {145, 1018}, {165, 1202}, {213, 995}, {220, 999}, {378, 2332}, {514, 3212}, {519, 3501}, {758, 3061}, {942, 1212}, {978, 1400}, {1015, 2176}, {1481, 2316}, {1698, 3691}, {1723, 2285}, {1759, 3218}, {1766, 2257}, {1802, 2323}, {2311, 2503}, {3208, 3244}, {3294, 3616}, {3555, 3693}
X(4254) lies on these lines: {1, 198}, {3, 6}, {9, 55}, {21, 391}, {25, 941}, {35, 1743}, {37, 169}, {41, 219}, {48, 1604}, {56, 1449}, {71, 218}, {101, 2256}, {222, 2199}, {294, 2335}, {405, 966}, {517, 3553}, {940, 1730}, {958, 3686}, {999, 1100}, {1334, 3217}, {1385, 3554}, {1400, 1617}, {1444, 1992}, {1696, 3247}, {1697, 2324}, {1763, 3666}, {2268, 2347}, {2301, 3197}, {2329, 3169}, {3731, 3746}
X(4254) = {X(3),X(6)}-harmonic conjugate of X(5120)
X(4254) = {X(371),X(372)}-harmonic conjugate of X(36746)
X(4254) = Brocard-circle-inverse of X(5120)
X(4254) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(36746)
X(4255) lies on these lines: {1, 474}, {2, 1043}, {3, 6}, {35, 3052}, {37, 936}, {42, 56}, {43, 958}, {55, 1191}, {73, 1407}, {78, 3666}, {220, 2276}, {387, 631}, {404, 940}, {405, 3216}, {518, 988}, {956, 3293}, {978, 1001}, {995, 1616}, {1086, 3487}, {1104, 2999}, {1125, 3755}, {1201, 2177}, {1393, 2099}, {1470, 2594}, {1593, 3192}, {2256, 3682}, {2975, 3240}
X(4256) lies on these lines: {1, 88}, {3, 6}, {21, 3216}, {35, 595}, {36, 42}, {43, 993}, {55, 995}, {101, 753}, {140, 1834}, {378, 3192}, {387, 3523}, {549, 3017}, {551, 3750}, {936, 3731}, {975, 3247}, {1064, 2077}, {1125, 1738}, {1126, 1468}, {1201, 3746}, {1450, 2078}, {1457, 3256}, {2650, 3336}, {2703, 2712}, {2975, 3293}
X(4257) lies on these lines: {1, 89}, {3, 6}, {31, 36}, {35, 1468}, {56, 106}, {171, 993}, {172, 3730}, {186, 3192}, {376, 3017}, {387, 3522}, {404, 1724}, {519, 3550}, {550, 1834}, {603, 3418}, {609, 672}, {975, 3731}, {997, 1707}, {999, 3052}, {1125, 3662}, {1308, 2382}, {2263, 3361}, {2701, 2712}
X(4258) lies on these lines: {1, 910}, {3, 6}, {25, 2332}, {35, 218}, {37, 380}, {41, 55}, {56, 2280}, {101, 3295}, {213, 3052}, {958, 3684}, {1055, 3304}, {1100, 3333}, {1146, 3486}, {1191, 1914}, {1212, 3601}, {1449, 3361}, {1616, 2241}, {2082, 2646}, {2174, 2256}, {3172, 3192}
X(4259) lies on these lines: {1, 674}, {3, 6}, {7, 8}, {22, 2194}, {37, 3781}, {46, 3293}, {56, 1813}, {63, 209}, {81, 2979}, {141, 442}, {181, 1376}, {916, 990}, {1155, 3240}, {1193, 3764}, {1386, 2646}, {1400, 1818}, {2097, 2810}, {2274, 3778}
X(4260) lies on these lines: {1, 3688}, {3, 6}, {10, 141}, {28, 1843}, {43, 57}, {65, 1738}, {69, 274}, {209, 3666}, {517, 3755}, {674, 1386}, {990, 2808}, {1175, 1176}, {1185, 1194}, {1193, 3778}, {1682, 3056}, {1818, 2260}, {2854, 3031}
X(4261) lies on these lines: {2, 37}, {3, 6}, {9, 3216}, {42, 2260}, {56, 2197}, {71, 1193}, {141, 980}, {171, 2214}, {194, 3770}, {387, 1108}, {404, 2303}, {1100, 2275}, {1393, 2171}, {1470, 2286}, {1796, 2221}, {2174, 2273}, {2309, 3764}
X(4262) lies on these lines: {1, 1055}, {3, 6}, {19, 3247}, {24, 2332}, {35, 41}, {36, 2280}, {42, 609}, {55, 101}, {112, 3192}, {169, 3601}, {869, 902}, {993, 3684}, {995, 1914}, {2082, 3612}, {2251, 2276}, {3207, 3295}
X(4263) lies on these lines: {3, 6}, {9, 1500}, {37, 519}, {42, 51}, {193, 980}, {198, 2242}, {213, 2269}, {391, 941}, {872, 3688}, {1015, 1449}, {1172, 3199}, {1196, 2670}, {1405, 2197}, {1743, 2276}, {3664, 3752}
X(4263) = complement of X(34282)
X(4264) lies on these lines: {3, 6}, {9, 31}, {37, 595}, {48, 609}, {57, 2199}, {109, 2285}, {172, 2300}, {251, 1400}, {966, 1724}, {995, 2178}, {1169, 2150}, {1449, 1468}, {1453, 2270}, {1630, 2288}
X(4265) lies on these lines: {1, 2916}, {3, 6}, {21, 141}, {22, 940}, {35, 518}, {36, 1386}, {37, 3220}, {38, 55}, {56, 1631}, {404, 3589}, {405, 3763}, {993, 3416}, {1818, 2174}
X(4266) lies on these lines: {1, 957}, {3, 6}, {8, 9}, {35, 2267}, {41, 2323}, {44, 3730}, {45, 1573}, {55, 2316}, {71, 380}, {672, 3240}, {879, 2966}, {1400, 1420}, {1405, 2078}, {2341, 2503}
X(4267) lies on these lines: {1, 859}, {3, 6}, {8, 21}, {35, 3293}, {56, 81}, {100, 1220}, {198, 2303}, {283, 1036}, {1010, 1376}, {1408, 1470}, {1437, 3435}, {1466, 1817}, {1829, 3666}
X(4268) lies on these lines: {3, 6}, {37, 604}, {44, 48}, {71, 1404}, {198, 909}, {672, 2317}, {1100, 2268}, {1631, 3271}, {1743, 2174}, {2256, 3451}, {2359, 3478}, {2478, 3615}
X(4269) lies on these lines: {3, 6}, {19, 27}, {21, 71}, {81, 2260}, {219, 859}, {283, 1474}, {672, 1778}, {1400, 2303}, {1486, 2328}, {1713, 1764}, {2183, 2287}
X(4270) lies on these lines: {1, 966}, {3, 6}, {4, 2331}, {9, 42}, {142, 2999}, {995, 1100}, {1172, 3192}, {1193, 1449}, {1696, 2334}, {2003, 2199}, {2345, 3293}
X(4271) lies on these lines: {3, 6}, {9, 80}, {37, 1953}, {44, 71}, {51, 55}, {219, 3204}, {966, 2478}, {993, 3032}, {1100, 1319}, {2174, 2323}, {2209, 3764}
X(4272) lies on these lines: {1, 1213}, {3, 6}, {37, 42}, {594, 3293}, {1100, 1193}, {1187, 2174}, {1400, 2594}, {1750, 1901}, {1834, 2910}, {1865, 2331}, {2345, 3240}
X(4273) lies on these lines: {3, 6}, {21, 44}, {37, 2287}, {42, 692}, {48, 2658}, {81, 88}, {607, 1474}, {1100, 2303}, {1172, 1389}, {1404, 1408}, {2173, 2650}
X(4273) = isogonal conjugate of X(30588)
X(4274) lies on these lines: {3, 6}, {9, 2295}, {10, 44}, {31, 51}, {172, 2323}, {213, 2183}, {593, 1994}, {1201, 1400}, {1468, 1682}, {2056, 2248}
X(4275) lies on these lines: {3, 6}, {31, 37}, {213, 3204}, {609, 2174}, {966, 2363}, {1100, 1468}, {1213, 1724}, {1399, 2285}, {1451, 2199}, {2217, 2262}
X(4276) lies on these lines: {1, 994}, {3, 6}, {10, 21}, {36, 81}, {55, 859}, {215, 501}, {283, 3422}, {333, 993}, {1412, 1470}
X(4277) lies on these lines: {3, 6}, {8, 37}, {9, 3293}, {42, 2183}, {44, 751}, {45, 1500}, {524, 980}, {1100, 2277}
X(4278) lies on these lines: {3, 6}, {21, 36}, {35, 81}, {191, 3724}, {501, 2194}, {993, 1010}, {1780, 1790}, {1817, 3008}
X(4279) lies on these lines: {1, 1258}, {3, 6}, {10, 82}, {31, 43}, {35, 2309}, {86, 1078}, {98, 2051}, {101, 731}
X(4279) = {X(1687),X(1688)}-harmonic conjugate of X(572)
X(4280) lies on these lines: {3, 6}, {37, 1474}, {38, 48}, {55, 2203}, {72, 2174}, {209, 2259}, {2194, 2911}
X(4281) lies on these lines: {1, 333}, {3, 6}, {21, 42}, {43, 1010}, {81, 1193}, {86, 978}, {1220, 3293}
X(4282) lies on these lines: {3, 6}, {47, 48}, {112, 2716}, {1731, 1951}, {2250, 2341}, {2600, 2602}
X(4283) lies on these lines: {2, 38}, {3, 6}, {238, 3778}, {1757, 3216}, {2275, 3781}
X(4284) lies on these lines: {3, 6}, {9, 38}, {995, 2911}, {1400, 3108}, {2273, 2275}
X(4285) lies on these lines: {3, 6}, {37, 3691}, {42, 44}, {1100, 1201}, {1405, 2594}
X(4286) lies on these lines: {2, 45}, {3, 6}, {37, 3670}, {44, 3216}, {71, 1201}
X(4287) lies on these lines: {3, 6}, {45, 48}, {484, 1449}, {604, 2099}, {2174, 2267}
X(4288) lies on these lines: {3, 6}, {19, 21}, {48, 283}, {71, 2327}, {1630, 2328}
X(4289) lies on these lines: {3, 6}, {41, 45}, {1449, 3337}, {2302, 3207}
X(4290) lies on these lines: {3, 6}, {31, 44}, {45, 595}
X(4291) lies on these lines: {3, 6}, {41, 47}, {906, 3730}
Points on the Soddy Line: X(4292)-X(4356)
Many points are found as points of intersection of the Soddy Line and other central lines.
A construction of X(4292) is given by Antreas Hatipolakis and Angel Montesdeoca at 24046.
X(4292) lies on these lines: {1, 7}, {3, 226}, {4, 57}, {8, 2093}, {9, 443}, {10, 46}, {11, 1354}, {12, 1155}, {21, 36}, {27, 58}, {28, 3220}, {30, 553}, {31, 1777}, {40, 388}, {56, 946}, {65, 515}, {72, 527}, {109, 3072}, {142, 405}, {165, 3085}, {225, 603}, {255, 1074}, {278, 1394}, {320, 1043}, {329, 936}, {376, 3487}, {379, 1724}, {386, 1745}, {404, 908}, {412, 1785}, {474, 3452}, {495, 3579}, {497, 3333}, {535, 3754}, {580, 1935}, {601, 1072}, {938, 3146}, {944, 3340}, {1004, 1259}, {1056, 1697}, {1086, 1104}, {1088, 1097}, {1254, 1735}, {1330, 3687}, {1364, 2816}, {1393, 1877}, {1400, 1765}, {1445, 1728}, {1453, 1890}, {1466, 3149}, {1479, 3338}, {1593, 1892}, {1699, 3086}, {1706, 3421}, {1737, 3336}, {1818, 3191}, {1839, 2260}, {1844, 2822}, {1876, 3575}, {1902, 2823}, {2475, 3218}, {2478, 3306}, {2646, 3649}, {3337, 3583}, {3485, 3576}, {3488, 3529}
X(4292) = X(52)-of-intouch-triangle
X(4292) = X(329)-of-Conway-triangle
X(4292) = {X(7),X(20)}-harmonic conjugate of X(1)
X(4292) = orthologic center of these triangles: Conway to 5th Conway
X(4292) = excentral-to-Conway similarity image of X(10)
X(4292) = inner-Conway-to-Conway similarity image of X(12527)
X(4293) lies on these lines: {1, 7}, {2, 36}, {3, 388}, {4, 11}, {8, 46}, {12, 631}, {28, 3433}, {30, 497}, {35, 3522}, {55, 376}, {57, 515}, {65, 944}, {79, 2320}, {84, 3427}, {145, 2802}, {151, 1795}, {226, 3576}, {278, 1455}, {329, 997}, {354, 3488}, {355, 1788}, {377, 2975}, {382, 496}, {387, 1468}, {404, 3436}, {443, 958}, {452, 1125}, {474, 2551}, {498, 3523}, {499, 3091}, {513, 957}, {517, 3474}, {519, 2093}, {529, 1376}, {550, 3295}, {595, 1777}, {602, 1935}, {603, 3072}, {840, 929}, {938, 3338}, {942, 3486}, {946, 1420}, {950, 3333}, {956, 2550}, {1010, 1444}, {1012, 1617}, {1058, 3304}, {1106, 3075}, {1111, 3598}, {1193, 1745}, {1210, 3361}, {1319, 1836}, {1385, 3485}, {1398, 3575}, {1450, 2635}, {1479, 3146}, {1587, 2067}, {2217, 3418}, {2242, 2549}, {2646, 3487}, {3189, 3555}, {3543, 3583}
X(4292) = excentral-to-intouch similarity image of X(4)
X(4293) = homothetic center of anti-Euler triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(4294) lies on these lines: {1, 7}, {2, 35}, {3, 496}, {4, 12}, {8, 90}, {10, 452}, {11, 631}, {21, 3434}, {28, 1486}, {30, 388}, {31, 387}, {36, 3522}, {40, 950}, {46, 938}, {56, 376}, {65, 3488}, {72, 3189}, {80, 3617}, {100, 2478}, {145, 758}, {165, 1210}, {212, 3073}, {377, 1621}, {382, 495}, {405, 2550}, {443, 1001}, {464, 1612}, {498, 3091}, {499, 3523}, {515, 1697}, {517, 3486}, {528, 958}, {550, 999}, {601, 1936}, {942, 3474}, {944, 3057}, {946, 3601}, {976, 3465}, {1056, 3303}, {1253, 3074}, {1453, 3755}, {1478, 3146}, {1587, 2066}, {1788, 3579}, {1834, 3052}, {1836, 3487}, {2241, 2549}, {3543, 3585}, {3612, 3616}
X(4294) = extangents-to-intangents similarity image of X(4)X(4295) lies on these lines: {1, 7}, {2, 46}, {3, 3474}, {4, 65}, {5, 1788}, {8, 79}, {10, 329}, {28, 3556}, {30, 3486}, {36, 3616}, {40, 226}, {42, 1745}, {55, 3487}, {57, 946}, {58, 1777}, {72, 2550}, {221, 278}, {281, 1901}, {354, 1058}, {376, 2646}, {382, 1159}, {387, 1838}, {388, 517}, {443, 960}, {484, 498}, {497, 942}, {499, 3336}, {515, 3340}, {553, 3333}, {631, 1155}, {938, 1479}, {944, 2099}, {1056, 3057}, {1086, 1191}, {1210, 1699}, {1245, 1246}, {1451, 3073}, {1452, 3089}, {1482, 3476}, {1588, 2362}, {1737, 3091}, {1892, 1902}, {2551, 3753}, {3295, 3475}, {3522, 3612}
X(4295) = polar conjugate of isotomic conjugate of X(6349)
X(4295) = 2nd-extouch-to-intouch similarity image of X(40)
X(4296) lies on these lines: {1, 7}, {2, 34}, {3, 1398}, {4, 1060}, {8, 1943}, {12, 858}, {21, 1214}, {22, 56}, {33, 3146}, {35, 2071}, {63, 1394}, {65, 81}, {72, 651}, {73, 3152}, {78, 223}, {100, 227}, {109, 283}, {171, 1254}, {201, 1935}, {221, 394}, {225, 2475}, {241, 1104}, {270, 1325}, {278, 377}, {280, 1219}, {330, 3164}, {376, 1062}, {388, 1370}, {404, 1465}, {511, 1425}, {517, 3562}, {603, 3218}, {664, 1043}, {741, 1305}, {934, 1297}, {960, 1456}, {982, 1106}, {1010, 1441}, {1040, 3522}, {1445, 1453}, {1455, 2975}, {1813, 2360}, {3153, 3585}
X(4296) = homothetic center of dual of orthic triangle and anti-tangential midarc triangle
X(4297) lies on these lines: {1, 7}, {3, 10}, {4, 1125}, {8, 165}, {30, 551}, {36, 411}, {40, 376}, {56, 950}, {57, 3486}, {72, 2801}, {99, 103}, {104, 3651}, {214, 2829}, {226, 2646}, {388, 3601}, {452, 1750}, {497, 1420}, {517, 550}, {548, 952}, {631, 3634}, {758, 1071}, {938, 3361}, {953, 2692}, {960, 971}, {997, 1490}, {1468, 1754}, {1478, 3612}, {1482, 3534}, {1610, 3220}, {1697, 3476}, {1698, 3523}, {1699, 3146}, {2360, 2822}, {2701, 2723}, {3086, 3586}, {3091, 3624}, {3333, 3488}, {3340, 3474}, {3528, 3626}, {3529, 3636}
X(4297) = midpoint of X(1) and X(20)
X(4297) = reflection of X(10) in X(3)
X(4298) lies on these lines: {1, 7}, {2, 3361}, {4, 1435}, {8, 3339}, {10, 57}, {12, 3634}, {40, 1056}, {56, 226}, {65, 519}, {79, 1476}, {142, 958}, {354, 950}, {379, 1468}, {515, 942}, {527, 960}, {535, 3660}, {551, 1420}, {651, 1203}, {946, 999}, {948, 1453}, {1010, 1434}, {1210, 1478}, {1254, 3670}, {1319, 3636}, {1398, 1892}, {1471, 1724}, {1496, 1754}, {1497, 1777}, {1697, 3474}, {1737, 3337}, {1836, 3304}, {2099, 3635}, {3189, 3243}, {3244, 3340}, {3296, 3488}, {3306, 3436}, {3475, 3601}, {3487, 3576}
X(4298) = X(389)-of-intouch-triangle
X(4298) = excentral-to-intouch similarity image of X(10)
X(4299) lies on these lines: {1, 7}, {2, 3585}, {3, 12}, {4, 36}, {8, 484}, {11, 382}, {30, 56}, {35, 376}, {46, 515}, {55, 550}, {79, 3485}, {80, 1788}, {84, 920}, {172, 2549}, {226, 3612}, {355, 1155}, {377, 993}, {452, 3624}, {495, 548}, {497, 3529}, {535, 3436}, {938, 3337}, {944, 3474}, {950, 3338}, {999, 1657}, {1056, 3746}, {1385, 1836}, {2829, 3149}, {3085, 3522}, {3086, 3146}, {3295, 3534}, {3361, 3586}, {3526, 3614}, {3543, 3582}
X(4300) lies on these lines: {1, 7}, {3, 31}, {12, 2635}, {38, 1071}, {40, 42}, {48, 3556}, {55, 64}, {72, 2340}, {165, 386}, {171, 411}, {185, 1409}, {201, 1858}, {222, 1496}, {500, 517}, {580, 2308}, {608, 1593}, {612, 1490}, {750, 3149}, {774, 1214}, {946, 3720}, {960, 1818}, {978, 3523}, {1006, 3073}, {1066, 3295}, {1149, 1385}, {1201, 3576}, {1457, 2646}, {1468, 3428}, {1745, 3085}, {1902, 2356}, {3072, 3651}
X(4300) = isogonal conjugate of polar conjugate of X(30686)
Let A' be the intersection of these three lines:
1) the perpendicular from the midpoint of CA to line BX(1);
2) the perpendicular from the midpoint of AB to line CX(1);
3) the perpendicular from the midpoint of AX(1) to line BC.
Define B' and C' cyclically. Then X(3817) = X(20)-of-A'B'C'. Triangle A'B'C' is the complement of the excentral triangle, and the extraversion triangle of X(10). (Randy Hutson, January 15, 2019)
For a discussion of this point as the center of a six-point circle associated with the 2nd Conway triangle (alias the outer Nagel triangle), see The outer Nagel paoints and unknown central circle".
X(4301) lies on these lines: {1, 7}, {2, 7991}, {3, 551}, {4, 519}, {5, 10}, {8, 1699}, {9, 6766}, {11, 4848}, {30, 5882}, {37, 10443}, {40, 631}, {42, 22392}, {46, 6966}, {65, 10866}, {72, 22019}, {76, 4673}, {140, 19883}, {142, 31793}, {144, 24644}, {145, 5691}, {149, 13253}, {153, 12653}, {165, 3616}, {226, 3057}, {329, 4853}, {355, 3625}, {381, 4669}, {382, 515}, {388, 7962}, {392, 7957}, {404, 5537}, {411, 3746}, {497, 3340}, {518, 9856}, {527, 12513}, {528, 12437}, {529, 33895}, {546, 34641}, {548, 1385}, {550, 15178}, {553, 3304}, {573, 3986}, {632, 38068}, {674, 31817}, {758, 12672}, {764, 3667}, {908, 6736}, {938, 18421}, {942, 21625}, {944, 3635}, {950, 2099}, {952, 3853}, {958, 8158}, {971, 34791}, {993, 11496}, {997, 6769}, {1000, 5714}, {1012, 8666}, {1058, 6744}, {1071, 3881}, {1072, 36250}, {1086, 45219}, {1159, 17706}, {1201, 24177}, {1210, 5903}, {1320, 34789}, {1387, 37582}, {1393, 45269}, {1420, 3474}, {1478, 30323}, {1479, 25415}, {1480, 5707}, {1483, 28160}, {1519, 10915}, {1537, 2802}, {1571, 31450}, {1572, 5319}, {1656, 3654}, {1657, 3655}, {1695, 19853}, {1697, 3485}, {1698, 7486}, {1706, 20103}, {1754, 3915}, {1829, 1906}, {1836, 2098}, {1837, 9671}, {1902, 1907}, {1953, 8804}, {2093, 3086}, {2095, 6705}, {2136, 25568}, {2262, 21068}, {2550, 12447}, {2784, 7983}, {2796, 23235}, {2800, 4084}, {2801, 3555}, {2807, 31732}, {3036, 38161}, {3068, 31440}, {3085, 31436}, {3090, 3828}, {3091, 3679}, {3146, 3241}, {3149, 8715}, {3214, 5400}, {3245, 37735}, {3303, 7580}, {3339, 14986}, {3428, 5248}, {3434, 6737}, {3475, 37556}, {3476, 9579}, {3486, 9580}, {3487, 31393}, {3522, 30389}, {3523, 25055}, {3526, 5886}, {3528, 3576}, {3530, 3579}, {3545, 4745}, {3582, 6972}, {3583, 11280}, {3584, 6960}, {3617, 7989}, {3621, 37712}, {3622, 7987}, {3626, 3855}, {3627, 28204}, {3634, 5067}, {3649, 5919}, {3651, 34486}, {3698, 5316}, {3741, 12435}, {3742, 31787}, {3753, 17575}, {3754, 9843}, {3812, 31798}, {3813, 8727}, {3816, 10107}, {3822, 15908}, {3839, 4677}, {3840, 31785}, {3851, 34718}, {3856, 18357}, {3859, 38140}, {3861, 5844}, {3868, 9949}, {3869, 4847}, {3871, 44425}, {3872, 11415}, {3873, 15071}, {3874, 6001}, {3876, 15104}, {3877, 4197}, {3884, 12609}, {3889, 9961}, {3890, 5249}, {3892, 12675}, {3898, 31786}, {3902, 4101}, {3911, 11376}, {3913, 19541}, {3919, 31870}, {3947, 5697}, {3950, 10445}, {3962, 7965}, {3993, 29054}, {4067, 5887}, {4082, 25253}, {4134, 20117}, {4229, 16712}, {4525, 5694}, {4656, 10459}, {4658, 37422}, {4662, 10157}, {4691, 5818}, {4701, 18492}, {4857, 6840}, {4861, 5180}, {4915, 5815}, {5045, 9943}, {5048, 7354}, {5056, 19875}, {5059, 34628}, {5068, 30308}, {5070, 18493}, {5079, 38066}, {5082, 6743}, {5119, 6962}, {5128, 7288}, {5183, 5433}, {5225, 5727}, {5229, 37709}, {5250, 41338}, {5252, 9656}, {5258, 6912}, {5267, 11249}, {5270, 37437}, {5289, 5805}, {5435, 18220}, {5436, 5759}, {5441, 37005}, {5450, 10680}, {5531, 9802}, {5563, 6909}, {5692, 38210}, {5709, 6892}, {5715, 45039}, {5728, 9848}, {5758, 12572}, {5784, 12446}, {5795, 24703}, {5840, 25485}, {5847, 15069}, {5850, 6762}, {5853, 12635}, {5883, 13374}, {5905, 36846}, {6147, 31792}, {6244, 25524}, {6260, 13600}, {6261, 12521}, {6282, 12436}, {6284, 11011}, {6702, 38038}, {6796, 10679}, {6825, 10197}, {6831, 24387}, {6838, 10056}, {6845, 10916}, {6847, 45700}, {6848, 45701}, {6885, 30144}, {6886, 38101}, {6890, 10072}, {6891, 10199}, {6936, 30147}, {6985, 37622}, {6999, 29574}, {7377, 29594}, {7387, 34642}, {7406, 16834}, {7503, 37546}, {7548, 31159}, {7672, 30294}, {7673, 21617}, {7967, 28158}, {7971, 12559}, {7972, 10724}, {7988, 9780}, {7993, 9809}, {7994, 8583}, {8196, 12459}, {8203, 12458}, {8226, 31165}, {8582, 41012}, {8983, 31454}, {9612, 31410}, {9614, 18391}, {9616, 13902}, {9623, 18250}, {9668, 37739}, {9698, 31396}, {9797, 36991}, {9845, 36996}, {9950, 24349}, {9957, 21620}, {10039, 18393}, {10167, 17609}, {10179, 20330}, {10246, 15696}, {10247, 13607}, {10248, 20050}, {10283, 13624}, {10306, 22753}, {10431, 11520}, {10541, 38023}, {10572, 11009}, {10698, 14217}, {10738, 12762}, {10896, 41687}, {10914, 21075}, {11230, 16239}, {11246, 20323}, {11260, 17768}, {11263, 37401}, {11373, 36279}, {11374, 31480}, {11499, 44455}, {11500, 25439}, {11533, 37443}, {11623, 12258}, {11727, 14690}, {12119, 33812}, {12261, 20379}, {12433, 14563}, {12514, 31458}, {12607, 12640}, {12608, 23340}, {12610, 21255}, {12700, 17647}, {12709, 17642}, {12812, 38083}, {12943, 37738}, {12953, 37740}, {13274, 41558}, {13334, 22475}, {13541, 34548}, {13605, 16003}, {13912, 35610}, {13975, 35611}, {14869, 38022}, {14872, 31871}, {14981, 21636}, {15022, 30315}, {15064, 34790}, {15606, 31737}, {15704, 28202}, {15934, 40270}, {15950, 37568}, {16126, 37433}, {16483, 37537}, {16491, 25406}, {17050, 24220}, {17552, 31435}, {17583, 17614}, {17613, 32636}, {17625, 17634}, {17747, 41006}, {18240, 37566}, {18249, 19843}, {19066, 31414}, {19708, 41150}, {19860, 40998}, {19876, 46936}, {19878, 31423}, {20060, 41698}, {21214, 24175}, {21669, 41691}, {22836, 37569}, {24393, 38158}, {24440, 45204}, {25917, 34501}, {27541, 44897}, {28146, 33179}, {28168, 32900}, {28224, 33697}, {28619, 37402}, {28647, 41705}, {29311, 45305}, {30556, 31486}, {31246, 44848}, {31400, 31431}, {31401, 31444}, {31663, 38028}, {31666, 33923}, {34379, 39898}, {34625, 37434}, {35514, 38036}, {35732, 36440}, {36458, 42282}, {37532, 40256}, {37533, 40257}, {37561, 45977}, {37570, 40091}, {38053, 43151}, {38200, 45085}, {39580, 39605}
X(4301) = midpoint of X(i) and X(j) for these {i,j}: {1, 962}, {4, 7982}, {8, 11531}, {20, 9589}, {145, 5691}, {149, 13253}, {153, 12653}, {355, 8148}, {382, 37727}, {944, 41869}, {1320, 34789}, {1482, 12699}, {3555, 12688}, {4295, 12651}, {5531, 9802}, {7972, 10724}, {7993, 9809}, {9812, 11224}, {10698, 14217}, {11278, 22793}, {13541, 34548}, {16126, 37433}
X(4301) = reflection of X(i) in X(j) for these {i,j}: {3, 13464}, {8, 19925}, {10, 946}, {40, 1125}, {355, 18483}, {550, 15178}, {551, 3656}, {944, 3635}, {946, 22791}, {1071, 3881}, {2951, 43176}, {3244, 1482}, {3579, 5901}, {3625, 355}, {3626, 12571}, {3878, 45776}, {4067, 5887}, {4084, 24474}, {4297, 1}, {4669, 381}, {5493, 3}, {5690, 9955}, {5836, 5806}, {5882, 10222}, {6361, 12512}, {7991, 43174}, {9943, 5045}, {10164, 5603}, {11362, 5}, {12119, 33812}, {12245, 3626}, {12512, 3636}, {12520, 12563}, {12640, 12607}, {12702, 6684}, {14110, 3884}, {14690, 11727}, {14872, 31871}, {18480, 40273}, {18481, 13607}, {21627, 13463}, {21635, 1537}, {24391, 3813}, {24393, 42356}, {24728, 4353}, {31673, 22793}, {31730, 1385}, {31788, 13374}, {31798, 3812}, {31803, 9856}, {33337, 25485}, {34773, 33179}, {37562, 31870}, {38127, 38034}, {38155, 1699}, {43161, 43179}, {43182, 5542}, {46684, 1387}
X(4301) = complement of X(7991)
X(4301) = anticomplement of X(43174)
X(4301) = X(9729)-of-2nd-Conway-triangle
X(4301) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7, 12577}, {1, 1770, 4311}, {1, 3671, 5542}, {1, 4292, 4315}, {1, 4295, 4298}, {1, 4312, 3600}, {1, 4338, 4317}, {1, 9589, 20}, {1, 10446, 43164}, {1, 10624, 4314}, {1, 12575, 30331}, {1, 30305, 12575}, {1, 43170, 43162}, {2, 7991, 43174}, {3, 3656, 13464}, {3, 13464, 551}, {5, 11362, 10}, {8, 1699, 19925}, {8, 3832, 37714}, {8, 19925, 38155}, {10, 946, 3817}, {20, 962, 9589}, {20, 5734, 1}, {40, 1125, 10164}, {40, 5603, 1125}, {40, 9624, 631}, {65, 12053, 11019}, {145, 9812, 5691}, {382, 1482, 37727}, {390, 4323, 1}, {497, 3340, 6738}, {551, 5493, 3}, {631, 5603, 9624}, {631, 9624, 1125}, {908, 14923, 6736}, {944, 16200, 3635}, {946, 10175, 9955}, {946, 11362, 5}, {962, 5734, 20}, {1058, 11529, 6744}, {1697, 3485, 13405}, {1699, 11531, 8}, {1699, 37714, 3832}, {1836, 2098, 10106}, {2099, 9670, 37724}, {2099, 12701, 950}, {2550, 15829, 12447}, {2886, 5837, 10}, {2951, 11038, 43176}, {3434, 11682, 6737}, {3452, 5836, 10}, {3522, 38314, 30389}, {3576, 6361, 12512}, {3576, 10595, 3636}, {3579, 5901, 10165}, {3600, 4345, 1}, {3616, 20070, 165}, {3617, 9779, 7989}, {3622, 9778, 7987}, {3626, 12571, 5587}, {3636, 12512, 3576}, {3663, 10446, 43172}, {3671, 4342, 1}, {3832, 37714, 19925}, {3872, 11415, 12527}, {4295, 4298, 30424}, {4317, 4338, 4292}, {5587, 12245, 3626}, {5657, 8227, 3634}, {5690, 9955, 10175}, {5690, 10175, 10}, {5691, 11224, 145}, {5697, 12047, 31397}, {5763, 5806, 3452}, {5886, 6684, 19862}, {5886, 12702, 6684}, {5901, 10165, 15808}, {5903, 30384, 1210}, {6361, 10595, 3576}, {7682, 11813, 3817}, {7982, 31162, 4}, {7991, 11522, 2}, {8148, 18483, 3625}, {9670, 37724, 950}, {9956, 38127, 10}, {9957, 39542, 21620}, {10247, 18481, 13607}, {10306, 22753, 25440}, {11376, 37567, 3911}, {11496, 22770, 993}, {12047, 31397, 3947}, {12699, 37727, 382}, {12701, 37724, 9670}, {13374, 31788, 5883}, {16200, 41869, 944}, {24393, 42356, 38158}, {31870, 37562, 3919}
X(4302) lies on these lines: {1, 7}, {2, 3583}, {3, 11}, {4, 35}, {8, 191}, {12, 382}, {30, 55}, {36, 376}, {40, 920}, {46, 950}, {56, 550}, {79, 3487}, {165, 1737}, {388, 3529}, {452, 1698}, {496, 548}, {515, 1709}, {528, 956}, {938, 3336}, {944, 2800}, {946, 3612}, {993, 3434}, {999, 3058}, {1657, 3295}, {1837, 3579}, {1914, 2549}, {3085, 3146}, {3086, 3522}, {3474, 3488}, {3543, 3584}
X(4303) lies on these lines: {1, 7}, {2, 1745}, {3, 73}, {5, 2635}, {30, 2654}, {36, 58}, {42, 46}, {55, 1066}, {56, 1064}, {57, 581}, {63, 3682}, {72, 1818}, {201, 912}, {226, 1076}, {411, 3075}, {500, 942}, {527, 3191}, {580, 2003}, {651, 3074}, {1006, 1935}, {1071, 1214}, {1155, 2594}, {1245, 2274}, {1385, 1457}, {1394, 3576}, {1464, 2646}, {1497, 1617}, {1936, 3651}, {2360, 3220}
X(4304) lies on these lines: {1, 7}, {2, 3586}, {3, 950}, {4, 3601}, {10, 21}, {30, 226}, {40, 3486}, {55, 515}, {57, 376}, {63, 519}, {84, 944}, {228, 855}, {319, 1043}, {377, 1125}, {452, 936}, {497, 3576}, {550, 942}, {553, 3534}, {938, 3522}, {946, 2646}, {1058, 1420}, {1071, 1317}, {1319, 3058}, {1385, 1387}, {1708, 3587}, {1765, 2269}, {3158, 3421}, {3487, 3529}
X(4304) = X(63)-of-X(1)-Brocard-triangle
X(4305) lies on these lines: {1, 7}, {2, 3612}, {3, 1788}, {4, 2646}, {8, 35}, {30, 3485}, {36, 938}, {46, 3522}, {55, 944}, {56, 3488}, {65, 376}, {452, 997}, {497, 1385}, {515, 3085}, {550, 3474}, {631, 1837}, {950, 3086}, {956, 3189}, {1058, 1319}, {1125, 3586}, {1155, 3528}, {1158, 1697}, {1388, 3058}, {1479, 2475}, {1737, 3523}, {1836, 3529}, {3295, 3476}
X(4306) lies on these lines: {1, 7}, {3, 951}, {36, 255}, {40, 1066}, {42, 3339}, {56, 58}, {57, 73}, {64, 103}, {72, 241}, {106, 1413}, {109, 1406}, {221, 595}, {223, 1467}, {580, 3157}, {581, 942}, {603, 3418}, {604, 2172}, {651, 1724}, {1064, 3333}, {1193, 3361}, {1203, 1471}, {1210, 1745}, {1394, 1420}, {1398, 1461}, {1419, 1453}, {1451, 2003}, {1754, 3562}
X(4307) lies on these lines: {1, 7}, {2, 31}, {4, 608}, {6, 2550}, {8, 193}, {10, 391}, {40, 2269}, {65, 3056}, {81, 3434}, {144, 984}, {145, 740}, {221, 388}, {226, 3424}, {329, 612}, {497, 940}, {553, 3677}, {611, 651}, {948, 1456}, {1449, 3755}, {1836, 3745}, {2345, 3416}, {3474, 3666}, {3475, 3744}, {3616, 3662}
X(4308) lies on these lines: {1, 7}, {2, 1420}, {8, 56}, {57, 145}, {65, 3241}, {226, 3622}, {388, 1319}, {519, 3361}, {604, 2329}, {651, 1191}, {938, 944}, {959, 3227}, {961, 1037}, {1000, 3579}, {1014, 1043}, {1056, 1385}, {1388, 3485}, {1617, 2975}, {1697, 3522}, {2098, 3474}, {3244, 3339}, {3304, 3486}, {3340, 3623}
X(4308) = {X(8),X(56)}-harmonic conjugate of X(5435)
X(4308) = perspector of Hutson intouch triangle and cross-triangle of ABC and Hutson intouch triangle
X(4309) lies on these lines: {1, 7}, {3, 3058}, {4, 3746}, {5, 55}, {8, 3467}, {11, 3526}, {30, 3303}, {35, 497}, {36, 1058}, {56, 548}, {79, 3475}, {382, 1478}, {405, 528}, {452, 3679}, {484, 938}, {496, 3530}, {550, 3304}, {912, 3057}, {950, 1728}, {3085, 3583}, {3091, 3584}, {3523, 3582}
X(4310) lies on these lines: {1, 7}, {2, 38}, {8, 1738}, {144, 238}, {226, 3677}, {329, 614}, {346, 726}, {497, 3782}, {613, 651}, {894, 3616}, {1086, 2550}, {1125, 3731}, {1463, 3056}, {1736, 3086}, {2285, 3333}, {2298, 3296}, {3085, 3670}, {3243, 3755}, {3474, 3744}, {3475, 3666}, {3617, 3775}
X(4311) lies on these lines: {1, 7}, {4, 1420}, {10, 36}, {40, 3476}, {46, 519}, {56, 515}, {57, 944}, {145, 2093}, {226, 1385}, {376, 1697}, {388, 3576}, {553, 3655}, {946, 1319}, {950, 999}, {995, 1745}, {1056, 3601}, {1106, 1771}, {1125, 1478}, {1388, 1836}, {1828, 2840}, {3333, 3486}
Let A' be the reflection of X(1) in A, and define B' and C' cyclically. Let A" be the reflection of X(1) in BC, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(4312). (Randy Hutson, July 20, 2016)
X(4312) lies on these lines: {1, 7}, {4, 3062}, {9, 46}, {10, 144}, {11, 57}, {35, 954}, {36, 1001}, {40, 495}, {65, 971}, {142, 3624}, {165, 226}, {497, 553}, {518, 3632}, {527, 1478}, {528, 3243}, {920, 1445}, {946, 3361}, {1249, 1838}, {1738, 1743}, {2829, 3577}, {3601, 3649}
X(4312) = reflection of X(1) in X(7)
X(4312) = X(6) of reflection triangle of X(1)
X(4313) lies on these lines: {1, 7}, {2, 950}, {3, 938}, {8, 21}, {30, 3487}, {57, 3522}, {63, 145}, {78, 452}, {226, 3146}, {376, 942}, {377, 497}, {943, 3560}, {944, 1012}, {1058, 1385}, {1210, 3523}, {1265, 3161}, {1486, 1610}, {3057, 3241}, {3086, 3612}, {3091, 3586}, {3303, 3476}
X(4313) = {X(1),X(20)}-harmonic conjugate of X(7)
X(4314) lies on these lines: {1, 7}, {9, 3189}, {10, 55}, {35, 1210}, {40, 3488}, {165, 938}, {200, 452}, {376, 3333}, {443, 497}, {515, 3295}, {519, 1697}, {551, 2646}, {758, 3057}, {1058, 3576}, {1104, 3755}, {1253, 1724}, {1864, 3678}, {2551, 3158}, {3021, 3026}, {3085, 3586}, {3361, 3522}
X(4314) = extangents-to-intangents similarity image of X(10)
X(4315) lies on these lines: {1, 7}, {8, 3361}, {10, 56}, {40, 1000}, {57, 519}, {65, 1317}, {80, 1210}, {145, 3339}, {226, 535}, {388, 1125}, {515, 999}, {529, 3452}, {553, 2099}, {944, 3333}, {946, 1387}, {950, 3304}, {993, 1617}, {1056, 3576}, {1788, 3626}, {3340, 3635}, {3485, 3636}
X(4316) lies on these lines: {1, 7}, {3, 3585}, {11, 30}, {12, 548}, {35, 495}, {55, 3534}, {56, 1657}, {79, 2646}, {80, 1155}, {100, 535}, {376, 1478}, {484, 515}, {498, 3522}, {499, 3146}, {609, 2549}, {950, 3337}, {952, 3245}, {1479, 3529}, {1776, 3065}, {3530, 3614}
X(4317) lies on these lines: {1, 7}, {5, 56}, {8, 3336}, {12, 3526}, {30, 3304}, {35, 1056}, {36, 388}, {55, 548}, {376, 3746}, {382, 999}, {474, 529}, {495, 3530}, {515, 3338}, {535, 2478}, {550, 3303}, {1657, 3058}, {1737, 3361}, {3086, 3585}, {3091, 3582}, {3523, 3584}
X(4318) lies on these lines: {1, 7}, {8, 34}, {59, 517}, {65, 82}, {100, 1465}, {109, 3218}, {219, 608}, {241, 1279}, {278, 3434}, {518, 651}, {655, 1411}, {902, 1758}, {1038, 3616}, {1214, 1621}, {1404, 1449}, {1419, 3243}, {1427, 3744}, {1457, 1818}, {2835, 3220}, {3315, 3660}
X(4319) is the point of concurrence of three lines associated with Soddy hyperbolas; see Angel Montesdeoca's construction at Hyacinthos 21290
X(4319) lies on these lines: {1, 7}, {9, 294}, {12, 1906}, {19, 25}, {31, 2257}, {34, 1885}, {40, 774}, {78, 3685}, {84, 1496}, {200, 346}, {220, 3059}, {497, 614}, {692, 2261}, {950, 3755}, {1086, 2191}, {1697, 2292}, {1854, 3057}, {2324, 2340}, {2876, 3056}, {3022, 3688}
X(4319) = {X(1),X(20)}-harmonic conjugate of X(4320)
X(4319) = extangents-to-intangents similarity image of X(19)
X(4319) = trilinear product X(33)*X(1040)
X(4319) = trilinear product X(55)*X(497)
Let EA be the ellipse passing through A, with foci at B and C. Let PA be the perspector of EA. Let LA be the polar of PA with respect to EA. Define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The lines AA', BB', CC' concur in X(4320). (Hyacinthos #21297, Nov 13, 2012, Randy Hutson)
X(4320) lies on these lines: {1, 7}, {11, 1906}, {25, 34}, {31, 1394}, {33, 1885}, {40, 1496}, {57, 961}, {65, 1407}, {84, 774}, {223, 1193}, {227, 1466}, {241, 958}, {388, 612}, {1191, 1456}, {1425, 1469}, {1453, 1471}, {2285, 2286}
X(4320) = trilinear product X(34)*X(1038)
X(4320) = trilinear product X(56)*X(388)
X(4320) = {X(1),X(20)}-harmonic conjugate of X(4319)
X(4321) lies on these lines: {1, 7}, {9, 56}, {57, 200}, {65, 2136}, {142, 388}, {936, 1445}, {954, 3576}, {971, 999}, {1001, 1420}, {1386, 1419}, {1418, 3242}, {1427, 3677}, {1471, 1743}, {1476, 3062}, {1490, 3333}, {1497, 2956}
X(4322) lies on these lines: {1, 7}, {42, 56}, {55, 1106}, {73, 1104}, {227, 244}, {354, 1254}, {388, 3720}, {603, 902}, {1066, 1385}, {1193, 1420}, {1388, 1457}, {1398, 2356}, {1407, 3303}, {1450, 2594}, {1466, 2177}, {1468, 1617}
X(4323) lies on these lines: {1, 7}, {2, 3340}, {8, 12}, {57, 3622}, {65, 3616}, {145, 226}, {388, 3241}, {496, 938}, {551, 3339}, {1058, 3656}, {1482, 3487}, {2093, 3523}, {2098, 3475}, {2171, 3061}, {3361, 3636}, {3476, 3649}
X(4324) lies on these lines: {1, 7}, {3, 3583}, {11, 548}, {12, 30}, {36, 496}, {40, 1727}, {55, 1657}, {56, 3534}, {80, 3579}, {90, 3587}, {376, 1479}, {498, 3146}, {499, 3522}, {950, 3336}, {1478, 3529}
X(4325) lies on these lines: {1, 7}, {3, 3584}, {4, 3582}, {5, 36}, {12, 3530}, {35, 548}, {56, 382}, {79, 1385}, {388, 3528}, {404, 535}, {515, 3336}, {550, 3746}, {631, 1478}, {1657, 3304}, {3303, 3534}
X(4326) lies on these lines: {1, 7}, {9, 55}, {142, 497}, {165, 1445}, {518, 1697}, {610, 1486}, {950, 2550}, {954, 1490}, {971, 3295}, {1001, 3601}, {1253, 1743}, {1438, 2268}, {2310, 3731}, {2346, 3062}, {3057, 3243}
X(4326) = extangents-to-intangents similarity image of X(9)
X(4327) lies on these lines: {1, 7}, {9, 1471}, {33, 354}, {34, 1892}, {37, 56}, {38, 57}, {65, 3242}, {226, 614}, {968, 1617}, {975, 3361}, {984, 1445}, {1040, 3475}, {1041, 3296}, {1407, 3745}, {1468, 2257}
X(4328) lies on these lines: {1, 7}, {33, 1119}, {37, 57}, {75, 200}, {142, 2324}, {226, 2999}, {241, 3247}, {388, 3755}, {612, 3598}, {894, 2297}, {1086, 3553}, {1100, 1419}, {1122, 2099}, {1445, 3731}, {3242, 3340}
X(4328) = {X(1),X(7)}-harmonic conjugate of X(269)
X(4329) lies on these lines: {1, 7}, {2, 19}, {4, 1441}, {8, 2893}, {22, 1486}, {40, 307}, {75, 1370}, {102, 1305}, {159, 1633}, {192, 3151}, {253, 306}, {281, 857}, {322, 3436}, {1891, 3146}
X(4329) = isogonal conjugate of X(7169)
X(4329) = isotomic conjugate of X(7219)
X(4329) = anticomplement of X(19)
X(4329) = crosssum of X(667) and X(3270)
X(4329) = polar conjugate of isogonal conjugate of X(22119)
X(4330) lies on these lines: {1, 7}, {3, 3582}, {4, 3584}, {5, 35}, {11, 3530}, {30, 3746}, {36, 548}, {55, 382}, {484, 950}, {497, 3528}, {550, 3058}, {631, 1479}, {1657, 3303}, {3304, 3534}
X(4331) lies on these lines: {1, 7}, {2, 1758}, {4, 774}, {12, 45}, {19, 208}, {31, 278}, {56, 1086}, {226, 968}, {388, 2292}, {1068, 3072}, {1284, 1486}, {1427, 1836}, {1445, 1738}, {1936, 3474}
X(4332) lies on these lines: {1, 7}, {8, 2647}, {31, 65}, {34, 42}, {55, 1254}, {221, 2099}, {226, 976}, {227, 2177}, {244, 1466}, {354, 1106}, {601, 942}, {608, 2171}, {948, 3189}, {1038, 3720}
X(4333) lies on these lines: {1, 7}, {30, 46}, {65, 1657}, {79, 3601}, {165, 3585}, {382, 1155}, {484, 1158}, {550, 1836}, {1060, 1717}, {1698, 2475}, {1737, 3146}, {2646, 3534}, {3336, 3586}, {3474, 3529}
X(4334) lies on these lines: {1, 7}, {43, 57}, {56, 87}, {171, 1407}, {241, 984}, {518, 1418}, {651, 1471}, {978, 1400}, {982, 1427}, {1284, 1420}, {1434, 3736}, {1445, 1757}, {1476, 3551}, {1745, 3338}
X(4335) lies on these lines: {1, 7}, {9, 43}, {42, 144}, {46, 2938}, {55, 1423}, {165, 1400}, {941, 3062}, {984, 3059}, {1001, 1740}, {1284, 3601}, {1463, 3303}, {1469, 1697}, {2346, 3551}
X(4336) lies on these lines: {1, 7}, {6, 2310}, {31, 1108}, {33, 42}, {37, 1253}, {41, 1827}, {55, 199}, {1040, 3720}, {1486, 1953}, {1824, 2187}, {1854, 2650}, {2171, 3209}, {2658, 2667}
X(4337) lies on these lines: {1, 7}, {3, 47}, {35, 73}, {36, 1064}, {42, 484}, {46, 581}, {65, 500}, {191, 3682}, {498, 1745}, {1066, 3746}, {1214, 1725}, {2594, 3579}
X(4338) lies on these lines: {1, 7}, {5, 46}, {40, 79}, {65, 382}, {548, 3612}, {631, 3474}, {1155, 3526}, {1158, 1699}, {2093, 3585}, {2475, 3679}, {3339, 3583}, {3485, 3528}
X(4339) lies on these lines: {1, 7}, {6, 3189}, {8, 31}, {171, 938}, {221, 3476}, {281, 3172}, {329, 976}, {388, 3744}, {452, 612}, {988, 3522}, {1104, 2550}, {2650, 3241}
X(4340) lies on these lines: {1, 7}, {2, 58}, {4, 940}, {6, 443}, {69, 1010}, {81, 377}, {142, 1453}, {171, 255}, {222, 388}, {226, 1394}, {315, 1509}, {329, 975}
X(4341) lies on these lines: {1, 7}, {48, 57}, {56, 1439}, {219, 241}, {307, 997}, {651, 1723}, {1100, 1427}, {1214, 1407}, {1419, 2257}, {1444, 2328}, {1617, 3433}, {1708, 2911}
X(4342) lies on these lines: {1, 7}, {10, 11}, {55, 551}, {495, 946}, {497, 519}, {535, 3058}, {950, 2098}, {1125, 1697}, {1837, 3625}, {2551, 3680}, {3486, 3635}, {3601, 3636}
X(4343) lies on these lines: {1, 7}, {9, 42}, {37, 2340}, {55, 1400}, {142, 3720}, {256, 2346}, {518, 2292}, {612, 3174}, {1001, 1193}, {1149, 1964}, {1334, 3779}, {1469, 3303}
X(4344) lies on these lines: {1, 7}, {6, 8}, {145, 894}, {171, 1471}, {388, 1456}, {497, 3745}, {536, 3241}, {940, 1462}, {1279, 3616}, {1386, 2550}, {1697, 2285}, {3622, 3662}
X(4345) lies on these lines: {1, 7}, {8, 11}, {145, 908}, {153, 1479}, {495, 1532}, {497, 3241}, {938, 1482}, {944, 1537}, {950, 3623}, {1056, 3656}, {1697, 3306}, {3057, 3616}
X(4346) lies on these lines: {1, 7}, {2, 45}, {8, 1266}, {44, 144}, {69, 3621}, {75, 3617}, {145, 320}, {346, 3662}, {982, 2310}, {1155, 3598}, {1278, 3620}, {3091, 3670}
X(4347) lies on these lines: {1, 7}, {10, 34}, {40, 1870}, {56, 2922}, {65, 1397}, {72, 1456}, {221, 758}, {946, 1060}, {1038, 1125}, {1069, 2800}, {2835, 3556}
X(4348) lies on these lines: {1, 7}, {12, 34}, {38, 1394}, {55, 1398}, {223, 976}, {496, 1060}, {614, 1038}, {961, 3340}, {1106, 3677}, {1425, 3056}
X(4349) lies on these lines: {1, 7}, {6, 10}, {142, 1386}, {226, 1456}, {388, 1419}, {551, 752}, {740, 3244}, {1100, 3755}, {1449, 2550}, {3717, 3758}
In the plane of a triangle ABC, let
I = X(1) = incenter;
DEF = intouch triangle;
(O) = circumcircle;
A' = the point, other than A, where the circle {{A,I,D}} meets (O), and define B' and C' cyclically;
Oa = center of the circle {{B',E,F,C'}}, and define Ob and Oc cyclically;
Oaa = A-extraversion of Oa, and define Obb and Occ cyclically;
T = affine transfomation that maps ABC onto OaObOc;
TT = affine transformation that maps ABC onto OaaObbOcc.
Then X(4350) = finite fixed point of T, and X(5250) = finite fixed point of TT. (Angel Montesdeoca, December 29, 2023)
X(4350) lies on these lines: {1, 7}, {34, 1847}, {41, 57}, {56, 3423}, {63, 220}, {218, 1445}, {738, 934}, {1014, 2360}, {1467, 3598}, {2125, 2371}
X(4351) lies on these lines: {1, 7}, {26, 56}, {34, 499}, {36, 186}, {78, 1079}, {498, 1038}, {920, 1394}, {1060, 1478}, {1735, 1795}
X(4352) lies on these lines: {1, 7}, {2, 39}, {85, 3666}, {86, 1975}, {144, 213}, {192, 304}, {940, 1434}, {986, 3212}, {988, 1447}
X(4353) lies on these lines: {1, 7}, {37, 39}, {519, 599}, {527, 1386}, {553, 3745}, {984, 3008}, {1766, 3333}, {3616, 3729}, {3626, 3775}
X(4353) = complement, wrt incircle-circles triangle, of X(12722)
X(4354) lies on these lines: {1, 7}, {26, 55}, {33, 498}, {35, 186}, {226, 1717}, {499, 1040}, {1062, 1479}
X(4355) lies on these lines: {1, 7}, {12, 57}, {65, 3632}, {226, 3361}, {388, 553}, {496, 1699}, {1420, 3649}
X(4356) lies on these lines: {1, 7}, {10, 37}, {2550, 3247}, {3242, 3244}
More Central Triangles and Combos
This section consists of points of the form X(k)com[T(f(a,b,c),g(b,c,a)], for choices of the pairs (f(a,b,c), g(b,c,a)) as indicated in the following list:
T(a, c), T(-a, c), T(a, b - c), T(a, c - b), T(a2, bc), T(-a2, bc), T(bc, bc + c2), T(-bc, bc + c2), T(bc, b2 + bc), T(-bc, b2 + bc) )
For notation and precedents, see the preambles just before X(3748) and X(3862).
X(4357) lies on these lines: {1, 69}, {2, 7}, {6,4416}, {8, 3672}, {10, 75}, {21, 3220}, {36, 1444}, {37, 141}, {38, 3778}, {43, 4104}, {44, 3589}, {45, 3763}, {58, 86}, {72, 4260}, {81, 4001}, {85, 3668}, {192, 2321}, {193, 1449}, {239, 1654}, {256, 314}, {264, 1785}, {269, 348}, {286, 1838}, {319, 519}, {325, 3847}, {326, 997}, {334, 1221}, {344, 3619}, {518, 4026}, {524, 1100}, {536, 594}, {732, 1107}, {740, 3775}, {756, 3263}, {903, 1268}, {942, 4205}, {950, 2893}, {960, 3674}, {980, 2277}, {988, 3926}, {1010, 4292}, {1045, 3783}, {1086, 1213}, {1122, 3665}, {1211, 2092}, {1743, 3618}, {1763, 2339}, {1848, 2354}, {2269, 3882}, {2345, 3729}, {2897, 3100}, {3122, 4022}, {3123, 3728}, {3247, 3620}, {3616, 3945}, {3631, 3723}, {3644, 4058}, {3836, 3842}, {3844, 3932}
X(4357) = isotomic conjugate of X(1220)
X(4357) = complement of X(894)
X(4357) = anticomplement of X(5750)
X(4357) = polar conjugate of isogonal conjugate of X(22097)
X(4357) = {X(2),X(7)}-harmonic conjugate of X(10436)
X(4358) lies on these lines: {1, 996}, {2, 37}, {8, 392}, {9, 1150}, {10, 3702}, {11, 3006}, {31, 4011}, {38, 3840}, {43, 3896}, {100, 2726}, {101, 2863}, {142, 4054}, {145, 341}, {190, 3218}, {239, 1016}, {244, 726}, {306, 3452}, {354, 3967}, {514, 661}, {518, 3952}, {519, 3992}, {551, 4125}, {614, 3891}, {672, 3985}, {740, 899}, {750, 3923}, {756, 3741}, {964, 975}, {1089, 1125}, {1210, 3710}, {1215, 3720}, {1921, 1978}, {1999, 2300}, {2235, 3231}, {2292, 3831}, {2325, 3911}, {2899, 3436}, {2901, 3216}, {3035, 3712}, {3120, 3836}, {3159, 3670}, {3264, 3943}, {3306, 3729}, {3679, 3902}, {3687, 3969}, {3695, 4187}, {3699, 3935}, {3703, 3816}, {3706, 3740}, {3834, 4080}
X(4358) = isogonal conjugate of X(9456)
X(4358) = isotomic conjugate of X(88)
X(4358) = X(6)-isoconjugate of X(106)
X(4358) = polar conjugate of X(36125)
X(4358) = pole wrt polar circle of trilinear polar of X(36125) (line X(19)X(4394))
X(4358) = trilinear pole of line X(900)X(1145) (the tangent at X(4738) to the inellipse centered at X(10))
X(4358) = perspector of Gemini triangle 27 and cross-triangle of ABC and Gemini triangle 27
X(4359) lies on these lines: {1, 3896}, {2, 37}, {8, 443}, {10, 38}, {31, 3980}, {57, 1150}, {63, 169}, {81, 239}, {100, 3757}, {142, 306}, {242, 4211}, {244, 3741}, {310, 1921}, {320, 2895}, {333, 2160}, {354, 3696}, {518, 4113}, {553, 3578}, {594, 3726}, {612, 3891}, {726, 756}, {740, 3720}, {748, 3923}, {899, 1215}, {940, 3187}, {1086, 1211}, {1089, 3634}, {1125, 1962}, {1213, 1230}, {1698, 3701}, {2355, 3916}, {3006, 3925}, {3120, 3847}, {3305, 3729}, {3452, 4054}, {3687, 3936}, {3703, 3826}, {3706, 3742}, {3740, 3952}, {3828, 3992}, {3842, 3989}, {3912, 3969}, {3957, 3996}
X(4359) = isotomic conjugate of X(1255)
X(4359) = complement of X(3995)
X(4359) = polar conjugate of isogonal conjugate of X(3916)
X(4359) = trilinear pole of line X(4977)X(4983) (the perspectrix of ABC and Gemini triangle 22)
X(4359) = perspector of Gemini triangle 14 and cross-triangle of ABC and Gemini triangle 14
X(4359) = perspector of Gemini triangle 21 and cross-triangle of ABC and Gemini triangle 21
X(4360) lies on these lines: {1, 75}, {2, 594}, {6, 190}, {7, 528}, {8, 4026}, {9, 3759}, {37, 239}, {38, 3571}, {42, 308}, {69, 145}, {99, 757}, {144, 1992}, {238, 3993}, {264, 1897}, {319, 519}, {320, 3244}, {322, 3870}, {333, 3187}, {346, 3618}, {386, 3596}, {536, 894}, {595, 4065}, {596, 1509}, {648, 2326}, {846, 3791}, {940, 3210}, {1193, 3264}, {1266, 3635}, {1269, 1909}, {1386, 3685}, {1434, 3889}, {1449, 3644}, {1621, 4068}, {1655, 3780}, {1999, 3666}, {3240, 3699}, {3589, 3943}, {3623, 3945}, {3723, 3739}, {3883, 4356}, {3896, 3920}, {3912, 3946}, {3948, 4272}
X(4360) = {X(17233),X(17380)}-harmonic conjugate of X(2)
X(4361) lies on these lines: {1, 3696}, {2, 594}, {6, 75}, {7, 524}, {8, 141}, {9, 536}, {10, 3946}, {37, 3875}, {38, 4042}, {44, 3729}, {45, 192}, {69, 1086}, {142, 519}, {144, 545}, {190, 1278}, {319, 599}, {333, 3210}, {344, 3943}, {590, 1267}, {613, 1733}, {614, 3706}, {740, 1001}, {940, 3187}, {966, 3672}, {1269, 3765}, {1279, 3886}, {1407, 1943}, {1722, 3714}, {1738, 3416}, {2321, 3008}, {2345, 3589}, {3175, 3305}, {3632, 3834}, {3661, 3763}, {3663, 3686}, {3687, 3772}, {3791, 3980}, {3914, 3966}
X(4361) = isogonal conjugate of X(30651)
X(4361) = complement of X(17314)
X(4361) = anticomplement of X(17243)
X(4362) lies on these lines: {1, 2}, {6, 1215}, {9, 3971}, {19, 2319}, {31, 321}, {38, 1150}, {44, 3967}, {55, 740}, {63, 726}, {75, 171}, {76, 1965}, {192, 846}, {238, 312}, {314, 983}, {333, 984}, {752, 1836}, {968, 3993}, {1089, 1724}, {1104, 3714}, {1707, 3729}, {2177, 3896}, {2887, 3416}, {3175, 3683}, {3686, 4104}, {3702, 3915}, {3706, 3744}, {3715, 4096}, {3749, 3886}, {3767, 4109}, {3847, 3966}
X(4362) = complement of X(33088)
X(4362) = anticomplement of X(29671)
X(4363) lies on these lines: {1, 536}, {2, 45}, {6, 75}, {7, 141}, {8, 524}, {9, 3739}, {10, 527}, {37, 980}, {69, 594}, {86, 192}, {144, 966}, {220, 1944}, {274, 2176}, {320, 599}, {321, 940}, {514, 996}, {611, 1733}, {615, 1267}, {869, 2234}, {1001, 3923}, {1100, 3875}, {1215, 1376}, {1958, 2174}, {2321, 3664}, {3052, 3757}, {3264, 3765}, {3596, 3770}, {3662, 3763}, {3696, 3751}
X(4363) = isogonal conjugate of X(30650)
X(4363) = complement of X(4419)
X(4363) = anticomplement of X(4364)
X(4364) lies on these lines: {1, 524}, {2, 45}, {9, 3589}, {10, 536}, {37, 141}, {38, 3122}, {44, 597}, {75, 1213}, {86, 1931}, {142, 3986}, {192, 594}, {344, 3763}, {527, 1125}, {551, 3246}, {756, 3123}, {966, 3672}, {984, 4026}, {1100, 3629}, {2486, 2886}, {3247, 3631}, {3630, 3723}, {3661, 3943}, {3663, 3739}, {3686, 4021}, {3703, 3989}, {3775, 3993}, {3821, 3826}, {3844, 4078}
X(4364) = complement of X(4363)
X(4364) = anticomplement of X(4472)
X(4365) lies on these lines: {2, 3993}, {10, 3995}, {38, 536}, {42, 321}, {75, 3720}, {192, 3989}, {210, 3994}, {226, 4062}, {306, 3120}, {312, 899}, {756, 3175}, {1089, 3214}, {1201, 3702}, {2308, 3187}, {2321, 3914}, {2887, 3969}, {3293, 4066}, {3632, 4067}, {3886, 3938}, {3925, 3943}, {3952, 4135}, {4028, 4054}
X(4365) = anticomplement of X(4970)
X(4366) lies on these lines: {1, 335}, {2, 11}, {6, 190}, {56, 3552}, {76, 2241}, {81, 1977}, {83, 1500}, {86, 1086}, {99, 1015}, {238, 239}, {257, 3512}, {287, 3270}, {330, 1975}, {350, 385}, {401, 3100}, {940, 1979}, {999, 1003}, {2242, 3972}, {2276, 3329}, {3027, 4027}
X(4366) = barycentric square of X(239)X(4367) lies on these lines: {1, 512}, {513, 663}, {514, 659}, {523, 1325}, {649, 4083}, {693, 814}, {804, 1966}, {830, 2530}, {885, 1476}, {891, 4063}, {905, 1491}, {1125, 4129}, {1577, 2787}, {1938, 4091}, {1960, 4040}, {2533, 3907}, {3287, 3805}
X(4367) = bicentric difference of PU(90)
X(4367) = PU(90)-harmonic conjugate of X(2238)
X(4367) = isogonal conjugate of X(3903)
X(4368) lies on these lines: {1, 1655}, {2, 846}, {10, 1018}, {42, 3952}, {190, 291}, {238, 350}, {659, 812}, {730, 3230}, {740, 2238}, {3027, 4154}, {3029, 3124}, {3685, 3783}, {3747, 3948}, {4082, 4133}
X(4369) lies on these lines: {2, 661}, {10, 4160}, {57, 4077}, {241, 514}, {513, 3716}, {522, 3798}, {523, 2487}, {649, 693}, {824, 4025}, {1019, 1577}, {1215, 3805}, {2533, 3907}, {2786, 3700}
X(4369) = isotomic conjugate of X(27805)X(4370) lies on the Steiner inellipse and the inellipse centered at X(4422) (the barycentric square of the Nagel line), and also lies on these lines: {2, 45}, {6, 644}, {9, 80}, {37, 537}, {44, 519}, {100, 3196}, {115, 121}, {678, 4152}, {752, 3932}, {900, 1635}, {918, 1642}, {2482, 2786}
X(4370) = midpoint of X(2) and X(190)
X(4370) = reflection of X(2) in X(4422)
X(4370) = isogonal conjugate of X(2226)
X(4370) = isotomic conjugate of isogonal conjugate of X(1017)
X(4370) = complement of X(903)
X(4370) = polar conjugate of isogonal conjugate of X(22371)
X(4370) = complementary conjugate of X(21241)
X(4370) = crosspoint of X(2) and X(519)
X(4370) = crosssum of X(6) and X(106)
X(4370) = antipode of X(2) in inellipse that is the barycentric square of the Nagel line
X(4370) = projection from Steiner circumellipse to Steiner inellipse of X(190)
X(4370) = Steiner inellipse antipode of X(1086)
X(4370) = center of circumconic that is locus of trilinear poles of lines parallel to Nagel line (i.e. lines that pass through X(519))
X(4370) = perspector of circumconic centered at X(519)
X(4370) = X(2)-Ceva conjugate of X(519)
X(4370) = barycentric square of X(519)
X(4370) = trilinear pole of line X(3251)X(4543)
X(4370) = crossdifference of every pair of points on line X(106)X(1960)
X(4371) lies on these lines: {2, 3723}, {8, 141}, {75, 193}, {142, 3632}, {145, 3739}, {239, 2345}, {391, 536}, {940, 3646}, {966, 3875}, {3008, 4007}, {3663, 4034}, {3679, 3946}
X(4372) lies on these lines: {32, 1930}, {41, 742}, {75, 172}, {141, 976}, {239, 2275}, {304, 1914}, {524, 3665}, {712, 1759}, {754, 4056}, {1089, 3734}, {3721, 3905}
X(4373) lies on these lines: {2, 2415}, {7, 145}, {69, 903}, {75, 3617}, {86, 3445}, {144, 673}, {335, 1278}, {346, 1086}, {675, 1293}, {3623, 3664}
X(4373) = isogonal conjugate of X(3052)
X(4373) = isotomic conjugate of X(145)
X(4373) = anticomplement of X(3161)
X(4373) = X(8)-cross conjugate of X(2)
X(4373) = antitomic conjugate of X(36807)
X(4373) = X(19)-isoconjugate of X(20818)
X(4373) = trilinear pole of line X(514)X(4521) (the complement of the Gergonne line, and the radical axis of incircle and de Longchamps circle)
X(4374) lies on these lines: {2, 3709}, {75, 523}, {86, 2605}, {513, 3261}, {522, 693}, {649, 802}, {804, 1966}, {894, 3287}, {1577, 2786}, {2533, 3805}
X(4374) = isotomic conjugate of X(3903)
X(4374) = anticomplement of X(3709)
X(4375) lies on these lines: {1, 514}, {2, 649}, {513, 875}, {659, 812}, {874, 3570}, {1019, 3249}, {1281, 2786}, {3294, 4063}, {3766, 4107}, {3995, 4024}
X(4375) = anticomplement of X(25381)
X(4376) lies on these lines: {1, 712}, {31, 742}, {32, 1930}, {75, 1914}, {172, 304}, {524, 3703}, {536, 3744}, {626, 4056}, {894, 2276}, {1111, 3734}
X(4377) lies on these lines: {10, 714}, {37, 313}, {44, 3765}, {76, 536}, {321, 4033}, {1230, 3175}, {3596, 3739}, {3943, 4044}
X(4378) lies on these lines: {1, 513}, {514, 659}, {649, 891}, {693, 2787}, {830, 3777}, {1019, 4083}, {1491, 3960}, {2526, 2530}
X(4379) lies on these lines: {2, 514}, {42, 2533}, {523, 1638}, {649, 693}, {663, 3720}, {4024, 4025}
X(4380) lies on these lines: {2, 4106}, {649, 693}, {1635, 3835}, {2402, 3474}
X(4380) = anticomplement of X(4106)
X(4381) lies on these lines: {1, 696}, {32, 744}, {75, 2210}, {894, 3764}
X(4381) = anticomplement of X(25346)
X(4382) lies on these lines: {514, 4024}, {649, 693}, {661, 4106}
X(4383) lies on these lines: {1, 210}, {2, 6}, {3, 1724}, {5, 1714}, {8, 1191}, {9, 2999}, {10, 3966}, {21, 4255}, {31, 899}, {37, 3305}, {42, 748}, {43, 55}, {44, 63}, {51, 4259}, {56, 978}, {57, 1122}, {58, 474}, {65, 1722}, {78, 1104}, {100, 3052}, {145, 1616}, {182, 2194}, {190, 3210}, {200, 3744}, {218, 226}, {219, 3452}, {221, 1788}, {222, 3911}, {223, 241}, {239, 312}, {278, 1783}, {329, 3782}, {354, 3751}, {386, 405}, {404, 4252}, {518, 614}, {580, 3149}, {582, 1780}, {607, 1848}, {612, 1386}, {651, 1407}, {740, 4011}, {750, 2308}, {908, 2911}, {936, 1453}, {956, 995}, {958, 1193}, {982, 1757}, {984, 3715}, {1040, 1864}, {1125, 4104}, {1155, 1707}, {1203, 1698}, {1214, 1723}, {1279, 3870}, {1332, 1997}, {1427, 1445}, {1465, 1708}, {1466, 1935}, {1480, 3654}, {1621, 3240}, {1738, 1836}, {1746, 2050}, {1751, 2051}, {1778, 1817}, {1790, 4268}, {1834, 2478}, {1861, 3195}, {1999, 3759}, {3066, 4228}, {3175, 3875}, {3214, 3913}, {3242, 3681}, {3286, 4191}, {3293, 3295}, {3670, 3927}, {3687, 3713}, {3689, 3749}, {3711, 3961}, {3720, 3789}, {3741, 4042}, {3891, 3952}, {3929, 3973}
X(4383) = {X(2),X(6)}-harmonic conjugate of X(940)
Let E be the cicumellipse centered at X(9). E also circumscribes the Gemini triangle 2. X(4384) is the perspector of E wrt the Gemini triangle 2. (Randy Hutson, November 30, 2018)
X(4384) lies on these lines: {1, 2}, {6, 3739}, {7, 391}, {9, 75}, {37, 3875}, {45, 536}, {57, 85}, {63, 169}, {69, 142}, {76, 3975}, {86, 1449}, {87, 2665}, {192, 3731}, {193, 3664}, {273, 2322}, {312, 728}, {319, 4034}, {321, 3294}, {344, 2321}, {354, 4042}, {527, 3707}, {572, 1958}, {599, 3834}, {748, 3747}, {870, 2279}, {894, 1743}, {980, 1107}, {1001, 3696}, {1010, 1453}, {1150, 3306}, {1229, 3692}, {1376, 2223}, {1386, 3846}, {1441, 1445}, {1654, 3662}, {1707, 3980}, {1751, 2339}, {2550, 3883}, {3416, 3826}, {3760, 3948}, {3761, 3765}, {3925, 3966}, {3986, 4021} X(4384) = isogonal conjugate of X(2279)X(4385) lies on these lines: {1, 312}, {2, 3701}, {4, 8}, {5, 3705}, {10, 75}, {12, 3703}, {40, 3729}, {42, 1008}, {58, 3769}, {78, 1065}, {83, 3759}, {85, 1930}, {145, 3702}, {192, 3931}, {273, 1235}, {304, 1909}, {314, 989}, {315, 319}, {345, 3085}, {346, 3991}, {388, 3974}, {405, 3757}, {495, 3695}, {518, 3714}, {519, 4066}, {612, 1010}, {938, 1229}, {960, 3967}, {964, 3920}, {982, 3831}, {998, 1222}, {1043, 3811}, {1125, 4125}, {1330, 3416}, {1698, 3992}, {2476, 3006}, {2901, 4043}, {3295, 3685}, {3501, 4095}, {3621, 3902}, {3840, 3976}, {3876, 3952}, {3948, 4205}
X(4385) = isogonal conjugate of X(1472)
X(4385) = anticomplement of X(37592)
X(4386) lies on these lines: {1, 1929}, {2, 1914}, {3, 1107}, {6, 43}, {8, 172}, {9, 3550}, {10, 32}, {19, 25}, {31, 2238}, {41, 2295}, {75, 385}, {100, 743}, {187, 993}, {230, 2886}, {333, 1333}, {404, 2275}, {519, 2242}, {609, 3679}, {750, 2280}, {940, 1100}, {958, 3053}, {976, 3721}, {985, 3783}, {997, 1572}, {1125, 2241}, {1468, 3780}, {1759, 3954}, {2214, 4272}, {3035, 3815}, {3290, 3744}, {3509, 3961}, {3726, 3938}
X(4387) lies on these lines: {1, 3175}, {2, 3712}, {6, 4037}, {8, 3058}, {9, 3706}, {11, 345}, {55, 312}, {200, 4009}, {210, 3886}, {321, 1001}, {344, 3925}, {346, 497}, {354, 3729}, {390, 3974}, {536, 614}, {740, 4011}, {940, 3923}, {958, 3702}, {1089, 3295}, {1479, 3695}, {1836, 3912}, {2321, 3966}, {2478, 3704}, {3305, 3696}, {3434, 3932}, {3701, 3913}, {3711, 3996}, {3870, 3967}, {3938, 3994}
X(4388) lies on these lines: {1, 1330}, {2, 31}, {4, 8}, {9, 4071}, {38, 256}, {63, 147}, {69, 350}, {75, 1836}, {100, 4192}, {149, 2895}, {190, 3703}, {226, 3757}, {239, 3914}, {306, 3685}, {312, 3416}, {319, 3706}, {320, 354}, {333, 2651}, {516, 3687}, {528, 3996}, {595, 3454}, {614, 1716}, {1193, 4201}, {1621, 3936}, {3006, 3219}, {3496, 4109}, {3888, 3917}
X(4388) = isogonal conjugate of X(34250)
X(4388) = isotomic conjugate of X(7224)
X(4388) = complement of X(20101)
X(4388) = anticomplement of X(171)
X(4388) = crossdifference of every pair of points on line X(3250)X(22383)
X(4388) = polar conjugate of isogonal conjugate of X(23150)
X(4389) lies on these lines: {1, 320}, {2, 45}, {7, 21}, {10, 75}, {37, 3662}, {38, 1227}, {63, 1732}, {69, 145}, {141, 192}, {144, 3618}, {319, 3632}, {346, 3619}, {527, 3758}, {536, 3661}, {594, 1278}, {982, 3122}, {1211, 3210}, {1621, 1623}, {2321, 3644}, {3007, 3890}, {3635, 3879}, {3636, 3664}, {3759, 3946}, {3790, 3844}, {3883, 4353}, {3912, 4029}
X(4389) = isotomic conjugate of X(996)
X(4389) = anticomplement of X(17369)
X(4390) lies on these lines: {1, 1390}, {8, 41}, {9, 644}, {21, 3208}, {48, 594}, {101, 3679}, {220, 2654}, {284, 4007}, {519, 2280}, {604, 2345}, {672, 956}, {748, 3230}, {750, 2242}, {958, 1334}, {993, 1018}, {1055, 1376}, {1468, 2295}, {2268, 2321}, {2975, 3501}, {3217, 3686}, {3632, 4251}, {3871, 4050}
X(4391) is the barycentric multiplier for the Feuerbach hyperbola. (The barycentric product of X(4391) and the circumcircle is the Feuerbach hyperbola.) (Randy Hutson, August 19, 2019)
X(4391) lies on these lines: {2, 905}, {8, 885}, {10, 1734}, {21, 1946}, {92, 2399}, {100, 929}, {190, 655}, {297, 525}, {513, 2517}, {514, 661}, {521, 1948}, {522, 3717}, {644, 666}, {650, 3975}, {659, 814}, {663, 3716}, {667, 2787}, {918, 3261}, {3700, 3910}, {3701, 3810}, {3777, 3837}, {4010, 4083}
X(4391) = isogonal conjugate of X(1415)
X(4391) = isotomic conjugate of X(651)
X(4391) = anticomplement of X(905)
X(4391) = trilinear pole of line X(11)X(123) (polar of X(108) wrt polar cirle, and the perspectrix of the outer and inner Garcia triangles)
X(4391) = pole wrt polar circle of trilinear polar of X(108) (line X(6)X(19))
X(4391) = polar conjugate of X(108)
X(4391) = X(i)-isoconjugate of X(j) for these {i,j}: {1,1415}, {6,109}, {31,651}, {48,108}
X(4391) = barycentric product of Feuerbach hyperbola intercepts of de Longchamps line
X(4391) = crossdifference of every pair of points on line X(31)X(184) (the isogonal conjugate of the isotomic conjugate of line X(1)X(3))
X(4392) lies on these lines: {1, 89}, {2, 38}, {8, 3670}, {37, 3999}, {57, 3920}, {63, 3677}, {100, 3242}, {145, 986}, {192, 4022}, {518, 3240}, {614, 3219}, {976, 4188}, {1001, 3315}, {1401, 2979}, {2087, 3735}, {2094, 4344}, {2292, 3622}, {3006, 3662}, {3616, 3953}, {3666, 3873}, {3681, 3752}, {3889, 3931}
X(4392) = anticomplement of X(32931)
X(4393) lies on these lines: {1, 2}, {6, 190}, {37, 3759}, {75, 1100}, {81, 330}, {89, 3227}, {193, 3672}, {194, 712}, {213, 3995}, {350, 3765}, {458, 1897}, {536, 3758}, {597, 3943}, {598, 4080}, {894, 1278}, {1386, 3797}, {1429, 3212}, {1468, 3552}, {2280, 2344}, {3662, 3879}, {3889, 4209}
X(4393) = isotomic conjugate of X(27494)
X(4393) = complement of X(20055)
X(4393) = anticomplement of X(3661)
X(4393) = polar conjugate of isogonal conjugate of X(23095)
X(4394) lies on these lines: {2, 4106}, {44, 513}, {101, 2743}, {523, 2527}, {647, 4132}, {665, 4083}, {667, 3900}, {900, 2490}, {905, 4063}, {918, 3798}, {1021, 3733}, {1734, 3803}, {1743, 2441}, {2487, 3676}, {2976, 3667}, {3669,4498}
X(4394) = isogonal conjugate of X(27834)
X(4394) = complement of X(4106)
X(4394) = anticomplement of X(4816)
X(4394) = homothetic center of inner-Conway triangle and incircle-circles triangle
X(4395) lies on these lines: {2, 3943}, {7, 3629}, {8, 141}, {44, 545}, {75, 3589}, {142, 3244}, {239, 320}, {519, 3834}, {523, 2487}, {536, 2325}, {740, 1125}, {1698, 3932}, {3631, 3662}, {3663, 3707}
X(4396) lies on these lines: {11, 524}, {32, 3760}, {36, 538}, {76, 172}, {100, 536}, {183, 2276}, {239, 3570}, {350, 385}, {543, 4316}, {609, 3734}, {732, 1428}, {754, 3583}, {1911, 2230}, {2242, 3761}
X(4397) lies on these lines: {8, 521}, {75, 2400}, {100, 1309}, {280, 2417}, {321, 4064}, {325, 523}, {346, 4130}, {522, 3717}, {646, 4076}, {650, 4140}, {657, 4148}, {2345, 2509}, {3239, 4171}, {3667, 3762}
X(4397) = isotomic conjugate of X(934)
X(4397) = anticomplement of X(6129)
X(4398) lies on these lines: {7, 528}, {10, 75}, {69, 3621}, {86, 3445}, {141, 1278}, {192, 1086}, {320, 3633}, {527, 3759}, {536, 3662}, {3210, 3782}, {3644, 3912}, {3758, 3946}
X(4398) = anticomplement of X(17340)
X(4399) lies on these lines: {7, 3630}, {8, 141}, {75, 524}, {142, 3625}, {239, 594}, {319, 1086}, {519, 3739}, {536, 3686}, {597, 2345}, {3008, 4060}, {3626, 3946}
X(4399) = complement of X(17388)
X(4400) lies on these lines: {12, 524}, {32, 3761}, {35, 538}, {76, 1914}, {172, 385}, {183, 2275}, {536, 3871}, {543, 4324}, {732, 2330}, {754, 3585}, {2241, 3760}
X(4401) lies on these lines: {100, 2748}, {514, 659}, {522, 1324}, {649, 2664}, {650, 830}, {663, 4063}, {669, 4151}, {1635, 1734}, {1960, 4083}, {2832, 3669}
X(4401) = pole wrt circumcircle of line X(10)X(1001)
X(4402) lies on these lines: {2, 2321}, {7, 193}, {8, 141}, {75, 3618}, {142, 145}, {144, 1266}, {346, 3008}, {391, 3663}, {536, 3161}, {3616, 3739}
X(4403) lies on these lines: {39, 85}, {115, 1565}, {279, 3767}, {512, 4014}, {514, 1086}, {538, 3797}, {1015, 1111}
X(4404) lies on these lines: {10, 4017}, {72, 4132}, {100, 2692}, {522, 3762}, {523, 1577}, {956, 3733}, {4139, 4170}
X(4404) = isotomic conjugate of isogonal conjugate of X(4729)
X(4405) lies on these lines: {8, 141}, {75, 3629}, {239, 597}, {320, 3630}, {536, 3707}, {3244, 3739}, {3625, 3834}
X(4406) lies on these lines: {75, 514}, {86, 663}, {313, 2533}, {513, 3261}, {693, 900}, {889, 1978}
X(4407) lies on these lines: {10, 537}, {551, 3707}, {597, 1125}, {984, 3661}
X(4408) lies on these lines: {513, 3261}, {693, 4036}, {786, 3835}, {850, 4106}
X(4409) lies on these lines: {2, 45}, {528, 3633}, {537, 3625}, {2796, 3635}
X(4410) lies on these lines: {536, 1909}, {1100, 1269}, {3739, 3770}
X(4411) lies on these lines: {75, 693}, {513, 3261}, {650, 3739}
X(4412) lies on these lines: {696, 1760}, {744, 2175}
X(4413) lies on these lines: {1, 3689}, {2, 11}, {3, 1698}, {6, 750}, {8, 3304}, {9, 1155}, {10, 56}, {12, 443}, {43, 940}, {57, 210}, {63, 3715}, {65, 936}, {69, 4023}, {78, 3812}, {142, 480}, {165, 3683}, {200, 354}, {244, 3242}, {344, 3712}, {377, 1329}, {404, 958}, {405, 3634}, {518, 3306}, {612, 3752}, {650, 1027}, {748, 3052}, {851, 1213}, {984, 1054}, {993, 3828}, {997, 2099}, {999, 3679}, {1125, 3303}, {1259, 4197}, {1388, 4002}, {1478, 3820}, {1696, 2345}, {1706, 3057}, {1836, 3452}, {2334, 3293}, {2999, 3745}, {3158, 3748}, {3295, 3624}, {3336, 3927}, {3339, 3962}, {3340, 3922}, {3616, 3913}, {3729, 4009}, {3742, 3870}
X(4414) lies on these lines: {1, 89}, {2, 846}, {3, 2292}, {6, 896}, {7, 1758}, {9, 899}, {10, 3977}, {21, 986}, {31, 1386}, {35, 976}, {37, 750}, {38, 55}, {42, 63}, {43, 3219}, {57, 968}, {69, 4062}, {100, 753}, {141, 3712}, {165, 612}, {191, 386}, {240, 1013}, {244, 1001}, {518, 2177}, {740, 1150}, {748, 3683}, {756, 1376}, {940, 1962}, {980, 3747}, {982, 1621}, {988, 1201}, {991, 1768}, {1011, 1403}, {1158, 4300}, {1279, 4003}, {1468, 3916}, {1707, 2308}, {1724, 3647}, {1754, 4137}, {1757, 3240}, {1936, 4336}, {2239, 2276}, {3011, 3663}, {3550, 3920}, {3679, 4141}, {3750, 3873}, {4001, 4028}
X(4414) = anticomplement of X(25385)
X(4415) lies on these lines: {1, 529}, {2, 45}, {6, 329}, {9, 3772}, {10, 3967}, {11, 38}, {12, 2292}, {37, 226}, {53, 92}, {72, 1834}, {141, 312}, {210, 3914}, {225, 1868}, {278, 2256}, {306, 3175}, {313, 321}, {497, 3242}, {524, 1999}, {528, 3961}, {536, 3687}, {612, 1836}, {726, 3847}, {756, 3120}, {908, 3666}, {982, 3756}, {984, 2886}, {986, 1329}, {1146, 3735}, {1213, 4054}, {1215, 4026}, {1738, 3740}, {1762, 2161}, {1828, 3057}, {2887, 3932}, {3011, 3683}, {3058, 3938}, {3159, 3454}, {3452, 3663}, {3670, 4187}, {3696, 4104}, {3773, 4135}, {3936, 3995}, {3950, 4035}, {4078, 4138}, {4085, 4090}
X(4415) = complement of X(32939)
X(4416) lies on these lines: {1, 193}, {2, 1743}, {6,4357}, {7, 391}, {8, 144}, {9, 69}, {10, 894}, {21, 4101}, {37, 524}, {44, 141}, {63, 573}, {71, 3882}, {72, 511}, {75, 527}, {78, 991}, {142, 320}, {171, 4104}, {190, 319}, {192, 519}, {200, 1742}, {226, 333}, {239, 3663}, {306, 2895}, {307, 651}, {314, 4044}, {321, 3578}, {345, 3929}, {348, 1419}, {518, 3883}, {846, 4028}, {908, 1150}, {1100, 3629}, {1155, 4023}, {1449, 1992}, {1812, 2003}, {1836, 4042}, {2792, 3732}, {3008, 3662}, {3416, 3717}, {3620, 3973}, {3666, 4263}, {3678, 4019}, {3759, 3946}
X(4416) = complement of X(17364)
X(4416) = anticomplement of X(3664)
X(4416) = {X(17331),X(17364)}-harmonic conjugate of X(2)
X(4417) lies on these lines: {1, 3847}, {2, 6}, {3, 1330}, {4, 1043}, {8, 12}, {27, 317}, {34, 78}, {43, 2887}, {57, 320}, {75, 226}, {76, 2051}, {92, 264}, {190, 329}, {223, 326}, {238, 3771}, {239, 3772}, {311, 1230}, {321, 3262}, {386, 3454}, {411, 1792}, {518, 3705}, {540, 4257}, {637, 2048}, {740, 3944}, {752, 3550}, {1699, 3886}, {1738, 4138}, {1760, 1763}, {1764, 3882}, {2979, 3909}, {3006, 3681}, {3061, 3452}, {3210, 3782}, {3434, 3996}, {3662, 3752}, {3696, 3838}, {3757, 3966}, {3790, 3967}, {3925, 4023}, {4201, 4255}
X(4417) = complement of X(37683)
X(4417) = anticomplement of X(37646)
X(4418) lies on these lines: {1, 596}, {2, 846}, {8, 1046}, {10, 191}, {31, 75}, {42, 894}, {79, 3454}, {81, 740}, {86, 1962}, {100, 1215}, {171, 321}, {190, 756}, {239, 2308}, {274, 3747}, {310, 1966}, {312, 750}, {333, 896}, {516, 2941}, {524, 4046}, {536, 3745}, {612, 3729}, {726, 3920}, {867, 2886}, {873, 4094}, {893, 2229}, {902, 3757}, {964, 986}, {1010, 2292}, {1043, 2650}, {1580, 2206}, {1761, 2345}, {1762, 2550}, {1961, 3995}, {2234, 3725}, {3218, 3741}, {3683, 3739}, {3685, 3720}, {3924, 4195}
X(4418) = complement of X(33100)
X(4418) = anticomplement of X(4425)
X(4419) lies on these lines: {1, 527}, {2, 45}, {6, 144}, {7, 37}, {8, 536}, {9, 3008}, {38, 497}, {55, 1633}, {69, 192}, {71, 1423}, {75, 966}, {141, 346}, {142, 3731}, {145, 524}, {319, 3644}, {329, 3666}, {344, 3662}, {347, 2256}, {387, 3927}, {388, 2292}, {390, 3242}, {514, 1000}, {599, 3943}, {612, 3474}, {968, 3475}, {984, 2550}, {986, 2551}, {1001, 4310}, {1265, 4201}, {1278, 1654}, {1429, 2267}, {1449, 4021}, {1743, 3946}, {2176, 4352}, {2345, 3729}, {3177, 3727}, {3247, 3664}
X(4419) = complement of X(4454)
X(4419) = anticomplement of X(4363)
X(4419) = {X(17325),X(17369)}-harmonic conjugate of X(2)
X(4420) lies on these lines: {1, 2}, {3, 3681}, {21, 210}, {35, 3219}, {40, 3984}, {44, 2220}, {45, 3713}, {55, 3876}, {60, 2330}, {69, 1443}, {72, 74}, {101, 4006}, {165, 3951}, {191, 4134}, {283, 3939}, {319, 1273}, {404, 518}, {452, 2900}, {474, 3873}, {480, 1259}, {484, 4067}, {500, 3578}, {958, 3711}, {960, 3689}, {1043, 3699}, {1320, 3893}, {1376, 3868}, {2173, 3949}, {2287, 3694}, {2478, 3189}, {3100, 3790}, {3702, 3996}, {3869, 3940}, {3877, 3913}, {3974, 4123}
X(4421) lies on these lines: {1, 4004}, {2, 11}, {3, 519}, {6, 3550}, {35, 958}, {43, 3052}, {56, 3241}, {63, 3689}, {165, 518}, {200, 3929}, {228, 3175}, {376, 529}, {404, 3303}, {474, 3746}, {535, 3534}, {551, 3295}, {631, 3813}, {678, 3938}, {940, 2177}, {1155, 3870}, {1319, 3895}, {1329, 4294}, {1388, 3885}, {3207, 3208}, {3219, 3711}, {3304, 4188}, {3306, 3748}, {3501, 4258}, {3576, 3880}, {3579, 3811}, {3749, 3752}, {4187, 4309}, {4234, 4267}
X(4421) = X(2)-of-anti-Mandart-incircle-triangle
X(4422) is the center of the inellipse that is the barycentric square of the Nagel line. (Randy Hutson, October 15, 2018)
X(4422) lies on these lines: {1, 597}, {2, 45}, {6, 344}, {9, 141}, {10, 528}, {37, 3589}, {44, 524}, {238, 3932}, {239, 3943}, {516, 3823}, {519, 3246}, {527, 3834}, {536, 2325}, {537, 1125}, {620, 2786}, {673, 2345}, {692, 1083}, {748, 3703}, {899, 3712}, {900, 3035}, {918, 3960}, {1211, 2503}, {1279, 3717}, {1386, 4078}, {1743, 3629}, {2161, 2339}, {2796, 3634}, {2886, 4011}, {3011, 4009}, {3630, 3973}, {3826, 3923}, {3938, 4126}
X(4422) = midpoint of X(2) and X(4370)
X(4422) = complement of X(1086)
X(4423) lies on these lines: {1, 210}, {2, 11}, {3, 1699}, {6, 748}, {9, 354}, {10, 3303}, {37, 614}, {38, 45}, {56, 226}, {57, 3683}, {63, 3742}, {142, 1836}, {200, 3748}, {238, 940}, {344, 3703}, {392, 2099}, {518, 3305}, {551, 956}, {612, 1279}, {750, 3052}, {756, 3242}, {958, 3304}, {968, 3752}, {1159, 3899}, {1697, 3698}, {1698, 3295}, {1750, 3576}, {3306, 3848}, {3677, 3731}, {3711, 3740}, {3912, 3966}
X(4424) lies on these lines: {1, 3}, {2, 1739}, {10, 321}, {37, 1018}, {38, 519}, {39, 3727}, {42, 758}, {72, 3293}, {80, 256}, {244, 551}, {386, 3869}, {392, 3752}, {495, 3782}, {536, 984}, {960, 3216}, {995, 3877}, {1064, 2800}, {1074, 1111}, {1149, 3898}, {1193, 3878}, {1201, 3884}, {1254, 3671}, {1500, 3721}, {1743, 4047}, {1962, 3919}, {2276, 3735}, {2650, 4084}, {3120, 3822}, {3214, 3678}, {3743, 3754}, {3954, 4006}
X(4425) lies on these lines: {1, 1330}, {2, 846}, {10, 321}, {21, 36}, {37, 744}, {210, 4085}, {226, 1284}, {256, 314}, {306, 3993}, {320, 4038}, {516, 4220}, {740, 1211}, {752, 3745}, {968, 3771}, {982, 3122}, {1001, 1626}, {1213, 3985}, {1215, 4026}, {1283, 1621}, {1962, 3936}, {2486, 2886}, {2922, 3145}, {3006, 3989}, {3175, 3773}, {3178, 3454}, {3666, 3847}, {3706, 3775}, {3735, 3981}, {3755, 4104}, {3842, 3925}, {4028, 4356}
X(4425) = complement of X(4418)
X(4426) lies on these lines: {1, 6}, {2, 172}, {3, 1575}, {8, 1914}, {10, 32}, {21, 2276}, {25, 2053}, {31, 2295}, {39, 993}, {41, 2238}, {48, 992}, {75, 384}, {187, 1574}, {205, 2112}, {230, 1329}, {333, 3661}, {519, 2241}, {609, 1698}, {1010, 1333}, {1125, 2242}, {1376, 3053}, {1759, 3125}, {1861, 1968}, {2275, 2975}, {2280, 3780}, {2345, 4195}, {3496, 3959}, {3721, 3924}
X(4427) lies on these lines: {2, 846}, {8, 191}, {63, 3886}, {99, 110}, {100, 190}, {109, 835}, {145, 1046}, {171, 3995}, {346, 1761}, {513, 3909}, {516, 3006}, {537, 3722}, {726, 902}, {740, 896}, {799, 874}, {1045, 3240}, {1293, 2415}, {1330, 3648}, {1331, 2398}, {1707, 3187}, {1781, 3161}, {3052, 3891}, {3218, 3685}, {3578, 4046}, {3579, 3701}, {3702, 3916}, {3712, 3936}
X(4427) = isotomic conjugate of X(4608)
X(4427) = trilinear pole of line X(1100)X(1125)
X(4427) = complement of X(3120)
X(4428) lies on these lines: {1, 3052}, {2, 11}, {3, 551}, {6, 3750}, {21, 3241}, {37, 3749}, {45, 3961}, {56, 4323}, {63, 3748}, {165, 3742}, {405, 3679}, {442, 4309}, {518, 3929}, {519, 958}, {534, 1486}, {553, 1617}, {846, 3242}, {902, 940}, {968, 3744}, {2800, 3898}, {2999, 3246}, {3158, 3740}, {3304, 4189}, {3305, 3689}, {3683, 3870}, {3715, 3935}
X(4429) lies on these lines: {1, 3836}, {2, 11}, {8, 141}, {10, 75}, {37, 3823}, {43, 2887}, {45, 1213}, {46, 1760}, {192, 3932}, {239, 3416}, {312, 3914}, {320, 3751}, {377, 1220}, {404, 2933}, {518, 3662}, {536, 3790}, {958, 4201}, {1737, 4008}, {3008, 3883}, {3210, 3703}, {3240, 3936}, {3618, 4307}, {3661, 3696}, {3679, 3775}, {3705, 3752}, {3755, 3912}
X(4430) lies on these lines: {1, 2308}, {2, 210}, {7, 2890}, {8, 2891}, {20, 145}, {57, 3935}, {63, 3243}, {65, 3621}, {72, 3622}, {81, 3242}, {89, 171}, {149, 152}, {165, 3218}, {519, 3894}, {758, 3241}, {942, 3617}, {982, 3240}, {2810, 3060}, {3244, 3901}, {3616, 3881}, {3623, 3869}, {3633, 4084}, {3648, 4309}, {3811, 4188}, {3890, 3962}
X(4430) = anticomplement of X(3681)X(4431) lies on these lines: {1, 4133}, {2, 3950}, {8, 144}, {10, 192}, {69, 4007}, {75, 142}, {76, 4110}, {141, 1266}, {190, 3686}, {193, 3632}, {319, 527}, {321, 908}, {519, 894}, {536, 594}, {553, 4102}, {1145, 2783}, {1269, 4033}, {1278, 3661}, {1654, 3626}, {1738, 3773}, {2345, 3875}, {3264, 4043}, {3596, 4044}, {3696, 3717}, {3739, 3943}
X(4432) is the intersection of the orthic axes of the 1st & 2nd Montesdeoca bisector triangles. (Randy Hutson, December 2, 2017)
X(4432) lies on these lines: {1, 190}, {10, 528}, {44, 519}, {55, 4011}, {214, 900}, {238, 239}, {516, 3836}, {536, 3246}, {545, 551}, {659, 812}, {726, 1279}, {752, 3912}, {1001, 3923}, {1086, 1125}, {1215, 1621}, {1386, 3993}, {1914, 3985}, {2235, 3230}, {3683, 3741}, {3722, 3952}, {3744, 3971}, {3773, 3883}, {3961, 4096}
X(4432) = complement of X(24715)
X(4432) = anticomplement of X(25351)
X(4433) lies on these lines: {8, 21}, {9, 4111}, {31, 3780}, {42, 2229}, {200, 3208}, {210, 1334}, {519, 2223}, {612, 3291}, {650, 663}, {740, 1284}, {851, 4062}, {862, 4037}, {874, 3978}, {1213, 4068}, {2238, 3747}, {2321, 4097}, {3056, 3169}, {3158, 4050}, {3683, 3691}, {3685, 3975}
X(4434) lies on these lines: {37, 1908}, {43, 1964}, {100, 740}, {171, 385}, {172, 4095}, {214, 519}, {230, 4071}, {312, 3550}, {537, 3218}, {726, 1155}, {752, 908}, {896, 3952}, {1757, 3699}, {2533, 3907}, {3011, 3836}, {3052, 4011}, {3219, 4096}, {3647, 4075}, {3744, 3840}
X(4435) lies on these lines: {1, 665}, {6, 900}, {239, 3766}, {294, 885}, {522, 3063}, {644, 1639}, {649, 4083}, {650, 663}, {654, 2170}, {812, 4107}, {918, 1814}, {926, 1024}, {1635, 3722}, {1643, 3887}, {2483, 4132}, {2511, 4272}, {3716, 4148}, {3904, 3910}
X(4436) lies on these lines: {2, 2486}, {3, 2783}, {6, 1045}, {45, 846}, {86, 4068}, {99, 670}, {100, 190}, {513, 3882}, {536, 2223}, {662, 3573}, {740, 3286}, {851, 3712}, {1018, 2284}, {1046, 3913}, {1580, 3285}, {1761, 2938}, {2234, 3747}, {3875, 3941}
X(4436) = anticomplement of X(2486)X(4437) lies on these lines: {2, 1280}, {6, 344}, {8, 673}, {69, 144}, {75, 141}, {116, 4103}, {313, 1229}, {321, 1233}, {338, 1234}, {518, 3717}, {528, 3416}, {537, 3773}, {545, 599}, {644, 1814}, {668, 1146}, {742, 3943}, {1211, 3124}, {3030, 3687}
X(4437) = isotomic conjugate of X(6185)
X(4437) = barycentric square of X(3912)
X(4438) lies on these lines: {2, 38}, {3, 10}, {9, 3847}, {11, 4011}, {31, 3006}, {57, 3836}, {63, 2887}, {190, 3944}, {238, 3705}, {345, 740}, {518, 3771}, {527, 4138}, {726, 3772}, {752, 1707}, {2886, 3923}, {3914, 3977}, {3925, 3980}
X(4439) lies on these lines: {10, 536}, {44, 519}, {190, 752}, {192, 4085}, {537, 3912}, {726, 1086}, {740, 3717}, {824, 1491}, {984, 3661}, {2886, 4135}, {3006, 3994}, {3264, 3992}, {3687, 4096}, {3703, 3847}, {3792, 3799}, {4037, 4119}
X(4440) lies on Steiner circumellipse of anticomplementary triangle (aka the permutation ellipse E(X(4440)) and on these lines: {1, 2796}, {2, 45}, {3, 24817}, {4, 24814}, {5, 24844}, {6, 4398}, {7, 192}, {8, 537}, {9, 29628}, {10, 17254}, {11, 36237}, {20, 24813}, {22, 24822}, {37, 7321}, {38, 24463}, {44, 4912}, {69, 1278}, {75, 1654}, {86, 17246}, {100, 21320}, {142, 17261}, {144, 673}, {145, 528}, {148, 150}, {149, 900}, {193, 4452}, {194, 17753}, {238, 17767}, {239, 527}, {244, 17777}, {291, 3123}, {319, 4686}, {320, 536}, {321, 21600}, {329, 17490}, {344, 25269}, {346, 17232}, {388, 24816}, {497, 24837}, {513, 25048}, {514, 6630}, {524, 17160}, {594, 17273}, {645, 24617}, {726, 4645}, {812, 9263}, {894, 3663}, {918, 37781}, {962, 17480}, {966, 4772}, {984, 24693}, {1018, 27295}, {1054, 21093}, {1100, 4796}, {1111, 20881}, {1155, 37764}, {1227, 21427}, {1270, 24832}, {1271, 24831}, {1320, 20098}, {1655, 24404}, {1836, 29840}, {2321, 17288}, {2325, 17266}, {2345, 17236}, {2481, 30228}, {2550, 31302}, {2896, 24825}, {2975, 24826}, {3008, 4480}, {3085, 24845}, {3086, 24846}, {3091, 24828}, {3122, 24338}, {3146, 11851}, {3151, 16099}, {3210, 5905}, {3218, 16560}, {3248, 24722}, {3271, 4499}, {3434, 24834}, {3436, 24835}, {3616, 4432}, {3620, 4437}, {3621, 9041}, {3644, 4851}, {3661, 4659}, {3662, 3729}, {3664, 17319}, {3672, 4366}, {3685, 24231}, {3731, 27147}, {3739, 17258}, {3758, 17301}, {3821, 36478}, {3834, 17264}, {3875, 17364}, {3888, 4014}, {3891, 20101}, {3912, 4887}, {3923, 29660}, {3943, 7238}, {3946, 17120}, {3950, 17312}, {3952, 26073}, {3995, 26842}, {4000, 17350}, {4025, 17036}, {4029, 29575}, {4054, 24627}, {4240, 24830}, {4310, 24280}, {4357, 17116}, {4360, 17365}, {4361, 17347}, {4384, 17333}, {4388, 17155}, {4393, 4644}, {4402, 32096}, {4416, 17117}, {4427, 24416}, {4431, 17287}, {4436, 24405}, {4439, 31151}, {4459, 5992}, {4479, 24691}, {4488, 26685}, {4579, 5091}, {4641, 19796}, {4648, 4704}, {4664, 4675}, {4665, 17271}, {4667, 29584}, {4670, 17320}, {4674, 21290}, {4681, 17317}, {4684, 28557}, {4688, 17256}, {4693, 28542}, {4699, 17257}, {4716, 17771}, {4718, 17315}, {4726, 5564}, {4756, 24988}, {4764, 17299}, {4788, 17314}, {4821, 17343}, {4859, 17338}, {4873, 29577}, {4888, 17391}, {4896, 29574}, {4902, 17242}, {4919, 17089}, {4966, 28556}, {4967, 17252}, {4969, 28333}, {5057, 5211}, {5224, 17118}, {5484, 17164}, {5601, 24823}, {5602, 24824}, {5749, 17383}, {5750, 17324}, {6173, 17244}, {6462, 24838}, {6463, 24839}, {6547, 32028}, {6549, 6633}, {6625, 11611}, {6651, 10436}, {6999, 29069}, {7222, 17321}, {7227, 17307}, {7232, 17233}, {7263, 17277}, {7585, 24819}, {7586, 24818}, {7613, 27549}, {7779, 24724}, {7787, 24815}, {8287, 31057}, {8972, 24842}, {9295, 17217}, {9359, 27846}, {9780, 25351}, {9791, 24325}, {9965, 30699}, {10528, 24847}, {10529, 24848}, {11246, 32926}, {12701, 34860}, {13941, 24843}, {14594, 24465}, {15590, 30332}, {16561, 27003}, {16676, 29581}, {16706, 17351}, {16710, 17183}, {17067, 29607}, {17086, 28968}, {17119, 17346}, {17138, 17157}, {17140, 33100}, {17147, 17483}, {17148, 17220}, {17151, 17363}, {17165, 33102}, {17227, 17281}, {17234, 17262}, {17235, 17289}, {17237, 29591}, {17238, 26582}, {17248, 25590}, {17249, 17303}, {17260, 24199}, {17268, 21255}, {17278, 17336}, {17282, 17339}, {17283, 17340}, {17291, 17355}, {17304, 17368}, {17310, 28301}, {17313, 29589}, {17318, 17378}, {17323, 17381}, {17373, 21296}, {17484, 17495}, {17488, 36588}, {17491, 32842}, {17759, 20347}, {17761, 27348}, {17768, 32922}, {17794, 35025}, {18150, 24004}, {18159, 21888}, {18830, 32035}, {19789, 20078}, {20068, 33110}, {20081, 21281}, {20171, 39126}, {20244, 21226}, {20927, 26567}, {21220, 21224}, {24165, 33099}, {24248, 24349}, {24620, 31018}, {25521, 25601}, {25536, 31059}, {26109, 28606}, {26626, 35578}, {28516, 32846}, {28582, 32850}, {29837, 33154}, {29839, 32934}, {30823, 32851}, {31647, 32094}, {32845, 32856}, {32935, 33149}, {32940, 33145}, {36606, 36807}
X(4440) = isotomic conjugate of X(6630)
X(4440) = anticomplement of X(190)
X(4440) = polar conjugate of isogonal conjugate of X(22148)
X(4440) = trilinear pole wrt anticomplementary triangle of Nagel line
X(4440) = {X(17305),X(17369)}-harmonic conjugate of X(2)
X(4441) lies on these lines: {2, 37}, {7, 310}, {8, 76}, {10, 3760}, {42, 3875}, {69, 674}, {100, 183}, {145, 1909}, {274, 3616}, {304, 3702}, {519, 3761}, {561, 4087}, {672, 3729}, {1975, 2975}, {3663, 3741}, {3696, 3789}
X(4441) = isotomic conjugate of X(1002)
X(4441) = polar conjugate of isogonal conjugate of X(23151)
X(4441) = anticomplement of X(2276)
X(4442) lies on these lines: {2, 3712}, {10, 321}, {75, 3266}, {100, 1284}, {145, 388}, {226, 3896}, {522, 693}, {536, 3006}, {740, 3120}, {896, 2796}, {1010, 3616}, {1824, 4018}, {1836, 3187}, {2887, 3969}, {3755, 4054}, {3925, 3995}
X(4443) lies on these lines: {1, 674}, {2, 3122}, {6, 256}, {38, 1227}, {75, 700}, {76, 714}, {239, 3764}, {982, 1086}, {984, 4026}, {986, 1834}, {1045, 4261}, {1582, 1631}, {2228, 3661}, {2486, 3944}, {3662, 4022}, {3923, 4283}
X(4443) = complement of X(24351)
X(4443) = anticomplement of X(24327)
Let P be a point in the plane of triangle ABC. Let A' be the intersection, other than A, of the Kiepert hyperbola and line AP. Define B' and C' cyclically. Triangle A'B'C' is here named the Kiepert-cevian triangle of P. X(4444) is the orthocenter of the Kiepert-cevian triangle of X(1). (Randy Hutson, October 15, 2018)
X(4444) lies on the Kiepert hyperbola and these lines: {2, 661}, {10, 514}, {76, 1577}, {83, 1019}, {98, 741}, {226, 3676}, {291, 812}, {292, 3960}, {321, 693}, {334, 3762}, {335, 4080}, {513, 875}, {813, 927}, {918, 3837}, {2786, 4010}
X(4444) = trilinear pole of line X(523)X(1086)
X(4444) = isotomic conjugate of X(3570)
X(4444) = anticomplement of X(27929)
X(4445) lies on these lines: {6, 319}, {7, 3631}, {8, 141}, {45, 1654}, {69, 594}, {75, 599}, {142, 3626}, {239, 3763}, {524, 2345}, {527, 4058}, {536, 4007}, {1086, 3620}, {3625, 3946}, {3663, 4060}, {3679, 3739}
X(4446) lies on these lines: {1, 4261}, {6, 291}, {75, 700}, {141, 982}, {192, 3122}, {674, 1740}, {714, 3596}, {894, 3764}, {984, 1213}, {986, 4026}, {1580, 1631}, {2228, 3662}, {2663, 4277}, {3661, 4022}
X(4447) lies on these lines: {8, 56}, {43, 2275}, {165, 3208}, {171, 172}, {241, 518}, {291, 3507}, {292, 2238}, {1001, 4344}, {1961, 4199}, {2223, 3912}, {2239, 3009}, {2533, 3907}, {3286, 3932}, {3691, 3740}
X(4448) lies on these lines: {2, 513}, {514, 551}, {519, 3251}, {522, 3971}, {523, 1962}, {650, 2276}, {659, 812}, {667, 993}, {764, 1125}, {900, 1635}, {1960, 3762}, {2533, 4040}, {3570, 3573}, {3877, 4083}
X(4449) lies on these lines: {1, 514}, {2, 4147}, {42, 2533}, {513, 4162}, {521, 4017}, {522, 4318}, {523, 1459}, {649, 4083}, {667, 891}, {693, 3907}, {905, 4041}, {1734, 3960}, {2254, 3669}, {3676, 4105}
X(4449) = anticomplement of X(4147)
X(4450) lies on these lines: {2, 3052}, {8, 30}, {42, 752}, {55, 3936}, {100, 4192}, {320, 3957}, {321, 516}, {595, 4202}, {902, 2887}, {1150, 3434}, {1330, 3871}, {2308, 4085}, {2895, 3996}, {3416, 3969}
X(4451) lies on these lines: {8, 192}, {37, 893}, {75, 325}, {78, 904}, {318, 429}, {335, 1581}, {341, 3704}, {346, 3985}, {518, 1222}, {726, 3865}, {960, 1043}, {1219, 1432}, {3687, 3797}
X(4452) lies on these lines: {2, 37}, {7, 145}, {8, 3663}, {69, 3621}, {144, 239}, {740, 4310}, {1119, 1897}, {1122, 3880}, {3008, 3161}, {3241, 3664}, {3616, 4021}, {3623, 3945}, {3870, 4328}
X(4452) = anticomplement of X(346)
X(4452) = polar conjugate of isogonal conjugate of X(23089)
X(4453) lies on these lines: {2, 918}, {88, 2401}, {100, 658}, {244, 1111}, {514, 1635}, {522, 693}, {649, 3776}, {654, 3218}, {659, 3004}, {900, 903}, {926, 3873}, {2610, 3936}, {3904, 3960}
X(4453) = centroid of (degenerate) side-triangle of Gemini triangles 1 and 13X(4454) lies on these lines: {2, 45}, {7, 346}, {8, 527}, {75, 144}, {142, 3161}, {145, 536}, {192, 3945}, {193, 742}, {524, 3621}, {726, 4307}, {894, 3672}, {962, 1219}, {3923, 4310}
X(4454) = anticomplement of X(4419)
X(4455) lies on these lines: {36, 238}, {512, 798}, {659, 812}, {661, 669}, {663, 788}, {810, 3221}, {875, 2107}, {890, 1635}, {1015, 1960}, {1018, 4069}, {2054, 3572}, {4094, 4155}
X(4455) = isogonal conjugate of X(4589)
X(4455) = crossdifference of every pair of points on line X(37)X(86)
X(4456) lies on these lines: {4, 9}, {22, 2172}, {42, 251}, {101, 1297}, {209, 218}, {306, 1763}, {315, 1760}, {672, 1724}, {1018, 3710}, {1453, 4253}, {2200, 3425}
X(4457) lies on these lines: {8, 982}, {10, 4046}, {210, 4135}, {354, 3625}, {726, 4113}, {740, 756}, {1215, 3696}, {2887, 4061}, {3290, 4060}, {3626, 3666}, {3842, 3896}
X(4458) lies on these lines: {1, 2785}, {2, 4088}, {244, 1109}, {513, 3776}, {514, 659}, {522, 693}, {523, 2487}, {676, 918}, {2774, 3874}, {2786, 4010}, {3669, 3810}
X(4458) = complement of X(4088)
X(4459) lies on these lines: {1, 2783}, {11, 244}, {75, 3056}, {511, 1733}, {804, 3023}, {894, 2330}, {1111, 4014}, {1469, 4008}, {1837, 2475}, {3022, 3026}, {3271, 4124}
X(4460) lies on these lines: {2, 4007}, {7, 145}, {8, 4026}, {75, 3241}, {519, 3672}, {740, 4344}, {1992, 3644}, {3161, 3759}, {3244, 3945}, {3632, 4021}, {3633, 3663}
X(4460) = anticomplement of X(4007)
X(4461) lies on these lines: {2, 37}, {7, 2321}, {8, 144}, {141, 4346}, {145, 894}, {190, 391}, {193, 3621}, {527, 4007}, {3305, 3646}, {3617, 3717}, {3695, 4208}
X(4461) = anticomplement of X(3672)
X(4462) lies on these lines: {2, 3669}, {8, 3309}, {63, 4063}, {100, 2737}, {145, 4162}, {514, 661}, {649, 4148}, {667, 2975}, {2254, 4147}, {3810, 4088}, {3869, 4083}
X(4462) = isogonal conjugate of X(34080)
X(4462) = isotomic conjugate of X(27834)
X(4462) = anticomplement of X(3669)
X(4463) lies on these lines: {4, 8}, {22, 1760}, {37, 82}, {42, 4137}, {63, 990}, {100, 1297}, {228, 3425}, {387, 3868}, {390, 3995}, {518, 3187}, {758, 3914}
X(4464) lies on these lines: {2, 4060}, {7, 145}, {69, 3633}, {75, 3244}, {86, 3635}, {319, 519}, {1125, 1268}, {1999, 3911}, {3210, 4031}, {3759, 3950}
X(4464) = anticomplement of X(4060)
X(4465) lies on these lines: {2, 45}, {43, 4069}, {192, 3807}, {239, 350}, {536, 899}, {659, 812}, {2230, 3009}, {3248, 3720}, {3294, 3934}, {3912, 4144}
X(4466) lies on these lines: {2, 1762}, {4, 2822}, {11, 244}, {71, 307}, {116, 2973}, {122, 125}, {226, 1020}, {1375, 2173}, {1565, 3942}, {1826, 3668}
X(4466) = crossdifference of every pair of points on line X(101)X(112)
X(4467) lies on these lines: {2, 3700}, {63, 1021}, {75, 850}, {448, 525}, {522, 693}, {649, 824}, {661, 2786}, {900, 3004}, {2605, 3268}, {3900, 4131}
X(4467) = isotomic conjugate of X(6742)
X(4468) lies on these lines: {2, 3676}, {63, 649}, {92, 3064}, {100, 2736}, {449, 525}, {514, 661}, {522, 3935}, {650, 918}, {1635, 3798}, {2402, 3870}
X(4468) = isotomic conjugate of X(37206)
X(4468) = anticomplement of X(3676)
X(4469) lies on these lines: {75, 4016}, {81, 518}, {274, 335}, {698, 1211}, {869, 984}, {1333, 1931}, {1959, 3728}, {2205, 3219}, {3807, 3862}
X(4470) lies on these lines: {2, 45}, {144, 1213}, {524, 3617}, {527, 1698}, {536, 3616}, {594, 3945}, {742, 3618}, {894, 966}, {2345, 3912}
X(4471) lies on these lines: {6, 560}, {41, 674}, {45, 55}, {692, 4266}, {1486, 4254}, {2174, 3056}, {2175, 4271}, {2278, 3271}, {3204, 3688}
X(4472) lies on these lines: {2, 45}, {10, 524}, {86, 594}, {527, 3634}, {536, 1125}, {742, 3008}, {894, 1213}, {1268, 1654}, {3631, 3664}
X(4472) = complement of X(4364)
X(4472) = anticomplement of X(25358)
X(4473) lies on these lines: {2, 45}, {9, 1654}, {192, 3161}, {239, 2325}, {528, 3617}, {537, 3616}, {1698, 2796}, {3271, 3799}
X(4473) = anticomplement of X(27191)
X(4474) lies on these lines: {8, 522}, {514, 4088}, {649, 2787}, {657, 3691}, {663, 3716}, {1334, 4140}, {1459, 4036}, {1909, 3261}
X(4475) lies on these lines: {42, 2809}, {141, 4118}, {244, 665}, {523, 1086}, {760, 2239}, {984, 3799}, {1111, 3120}, {3661, 3864}
X(4476) lies on these lines: {58, 672}, {86, 291}, {310, 726}, {1326, 2206}, {2227, 2292}, {2276, 3736}, {3783, 3786}, {3799, 3864}
X(4477) lies on these lines: {55, 3700}, {200, 1021}, {521, 1491}, {522, 659}, {612, 647}, {650, 663}, {884, 1261}, {2533, 3907}
X(4478) lies on these lines: {8, 141}, {75, 3631}, {319, 524}, {536, 4060}, {1654, 3943}, {2345, 3629}, {3589, 3661}, {3626, 3739}
X(4479) lies on these lines: {2, 37}, {76, 519}, {314, 3551}, {320, 1836}, {325, 3829}, {1266, 3840}, {1909, 3241}, {3679, 3760}
X(4480) lies on these lines: {8, 144}, {44, 545}, {75, 3707}, {190, 320}, {192, 3244}, {193, 3633}, {894, 1125}, {1026, 3000}
X(4481) lies on these lines: {2, 661}, {36, 238}, {514, 1921}, {788, 1491}, {824, 3250}, {984, 3805}, {2786, 4079}
X(4482) lies on these lines: {2, 106}, {69, 544}, {101, 668}, {190, 514}, {830, 3888}, {3799, 4160}, {3907, 3939}
X(4483) lies on these lines: {21, 3886}, {58, 740}, {81, 3875}, {86, 3946}, {284, 314}, {333, 2321}, {1010, 3755}
X(4484) lies on these lines: {6, 560}, {45, 3122}, {55, 4286}, {599, 2228}, {674, 2275}, {1001, 4283}, {3763, 4022}
X(4485) lies on these lines: {8, 1237}, {75, 982}, {92, 264}, {350, 561}, {700, 3116}, {718, 4116}, {1502, 1926}
X(4486) lies on these lines: {514, 661}, {659, 812}, {786, 4036}, {816, 2605}, {824, 1491}, {918, 3837}, {2254, 2786}
X(4486) = isotomic conjugate of X(37207)
X(4487) lies on these lines: {4, 8}, {341, 3621}, {519, 3992}, {3617, 3999}, {3625, 3702}, {3632, 3701}, {3880, 3952}
X(4488) lies on these lines: {7, 190}, {8, 144}, {192, 537}, {346, 527}, {894, 3616}, {1992, 3644}
X(4488) = complement of X(4452)
X(4488) = anticomplement of X(4862)
X(4489) lies on these lines: {1, 2}, {171, 3780}, {740, 3975}, {846, 3691}, {1045, 3686}, {1580, 3684}
X(4490) lies on these lines: {10, 514}, {513, 4041}, {659, 3803}, {661, 4083}, {667, 4160}, {784, 3762}
X(4491) lies on these lines: {3, 2827}, {6, 3768}, {36, 238}, {100, 190}, {106, 1960}, {1635, 3196}
X(4491) = isogonal conjugate of isotomic conjugate of X(21297)
X(4491) = polar conjugate of isotomic conjugate of X(23141)
X(4491) = crosspoint of X(100) and X(106) (circumcircle-X(1)-antipodes)
X(4492) lies on these lines: {45, 899}, {239, 751}, {536, 984}, {674, 1469}, {749, 894}, {995, 1001}
X(4492) = isogonal conjugate of X(17126)
X(4493) lies on these lines: {1, 766}, {38, 1227}, {55, 3571}, {76, 2085}, {350, 3116}, {561, 700}
X(4494) lies on these lines: {9, 3596}, {312, 646}, {313, 3729}, {314, 4007}, {536, 3760}
X(4495) lies on these lines: {171, 561}, {238, 1921}, {292, 716}, {350, 874}, {984, 3403}
X(4496) lies on these lines: {2, 893}, {257, 3175}, {524, 1431}, {538, 3865}, {599, 3863}
X(4497) lies on these lines: {6, 560}, {56, 976}, {579, 692}, {583, 2175}, {604, 674}
X(4498) lies on these lines:
{239, 514}, {513, 4041}, {659, 663}, {667, 891}, {905, 1635}, {3669, 4394}
X(4498) = pole, with respect to Bevan circle, of line X(8)X(9)
X(4499) lies on these lines: {2,4014}, {37,9339}, {63,9355}, {100,1293}, {190,513}, {651,3573}, {1633,4579}, {2801,3869}, {3123,9359}, {3271,4440}, {3766,4572}, {3939,6163}
X(4499) = reflection of X(i) in Xj) for these (i,j): (3888,190), (4440,3271)
X(4499) = anticomplement of X(4014)
X(4499) = {X(190),X(3888)}-harmonic conjugate of X(3799)
X(4500) lies on these lines: {321, 693}, {514, 3700}, {522, 3798}, {523, 3835}
X(4501) lies on these lines: {522, 3063}, {649, 4139}, {667, 4155}, {1643, 2509}
X(4502) lies on these lines: {37, 513}, {522, 661}, {649, 3709}, {834, 3768}
X(4503) lies on these lines: {527, 2295}, {3271, 3720}, {3666, 3882}, {3888, 3920}
X(4504) lies on these lines: {1, 4170}, {514, 3803}, {2533, 3907}, {3667, 4162}
X(4505) lies on these lines: {190, 646}, {693, 1978}, {3773, 3864}
X(4506) lies on these lines: {2, 37}, {44, 3264}, {513, 3762}
X(4507) lies on these lines: {42, 649}, {512, 650}, {1635, 2978}
X(4508) lies on these lines: {239, 812}, {804, 3747}, {3766, 4107}
X(4509) lies on these lines: {522, 693}, {812, 1019}, {824, 1577}
X(4509) = isotomic conjugate of X(36147)
X(4510) lies on these lines: {106, 789}, {320, 519}
X(4511) lies on these lines: {1, 2}, {3, 3417}, {7, 326}, {9, 2320}, {20, 224}, {21, 60}, {35, 3878}, {36, 214}, {41, 3061}, {46, 4188}, {54, 72}, {55, 3877}, {56, 1259}, {59, 518}, {63, 3576}, {65, 404}, {69, 1442}, {80, 3814}, {100, 517}, {104, 912}, {153, 515}, {320, 1443}, {377, 3485}, {392, 1621}, {497, 4123}, {522, 663}, {644, 3693}, {651, 1455}, {662, 1325}, {860, 1870}, {934, 2750}, {944, 3436}, {956, 3681}, {958, 3715}, {993, 3219}, {999, 1260}, {1043, 3615}, {1055, 3509}, {1064, 4388}, {1100, 3965}, {1178, 4451}, {1257, 2982}, {1318, 1320}, {1376, 2099}, {1392, 3680}, {1420, 1708}, {1457, 1818}, {1727, 3612}, {1792, 3193}, {1807, 4358}, {1829, 4231}, {1837, 4193}, {2077, 2800}, {2098, 3885}, {2170, 3684}, {2324, 3161}, {2478, 3486}, {3057, 3871}, {3158, 3895}, {3185, 4216}, {3295, 3890}, {3304, 3889}, {3336, 4084}, {3746, 3884}, {3902, 3996}, {4190, 4295}, {4256, 4424}
X(4511) = isogonal conjugate of X(1411)
X(4511) = anticomplement of X(1737)
X(4511) = inverse-in-circumconic-centered-at-X(1) of X(8)
X(4511) = X(13851)-of-excentral-triangle
X(4511) = endo-homothetic center of Ehrmann side-triangle and X(3)-Ehrmann triangle; the homothetic center is X(10540)
X(4512) lies on these lines: {1, 21}, {2, 165}, {8, 4314}, {9, 55}, {10, 452}, {19, 4183}, {35, 936}, {37, 3052}, {40, 405}, {42, 1743}, {57, 1001}, {100, 3305}, {142, 3474}, {154, 392}, {228, 3294}, {238, 2999}, {333, 3886}, {345, 3883}, {354, 3928}, {391, 4061}, {518, 3929}, {527, 3475}, {551, 2094}, {572, 2187}, {610, 3185}, {612, 902}, {614, 4414}, {631, 1519}, {958, 1697}, {960, 3601}, {984, 3749}, {1005, 1750}, {1103, 3074}, {1125, 4295}, {1155, 4423}, {1279, 3677}, {1449, 4047}, {1453, 3931}, {1617, 4321}, {1698, 2478}, {2093, 3919}, {2177, 3973}, {2195, 2339}, {2293, 3190}, {2325, 3974}, {2956, 4303}, {3161, 4082}, {3219, 3870}, {3243, 3748}, {3247, 3745}, {3333, 3916}, {3361, 3616}, {3712, 3966}, {3729, 3757}, {3740, 4421}, {3750, 3751}, {3811, 4134}, {3841, 4193}, {4034, 4046}
X(4513) lies on these lines: {1, 728}, {3, 1018}, {6, 145}, {8, 220}, {9, 3057}, {37, 3692}, {41, 3913}, {45, 3727}, {55, 2053}, {56, 3501}, {76, 1016}, {100, 3207}, {190, 3177}, {218, 519}, {219, 2321}, {239, 312}, {345, 940}, {594, 965}, {884, 3900}, {956, 3730}, {958, 1334}, {1212, 3872}, {1229, 4361}, {2082, 3880}, {2098, 3061}, {2256, 2345}, {2324, 3965}, {2348, 3893}, {3684, 4050}, {3871, 4258}, {3940, 4006}
X(4514) lies on these lines: {1, 977}, {2, 1279}, {8, 210}, {9, 4119}, {11, 4030}, {33, 3872}, {55, 3705}, {75, 1370}, {149, 321}, {319, 350}, {320, 3873}, {333, 643}, {345, 390}, {518, 4388}, {614, 4429}, {1219, 3146}, {1233, 2481}, {1330, 3555}, {1479, 4385}, {1621, 3006}, {2886, 3757}, {3058, 3685}, {3218, 4450}, {3452, 3699}, {3687, 3996}, {3790, 4387}, {3847, 3961}, {3870, 4417}, {3936, 3957}
X(4515) lies on these lines: {8, 1212}, {9, 3913}, {10, 37}, {41, 3689}, {65, 3930}, {72, 1018}, {200, 220}, {210, 1334}, {306, 1427}, {341, 346}, {518, 3501}, {536, 3673}, {644, 4420}, {960, 3208}, {2170, 3893}, {2646, 4390}, {3061, 3880}, {3158, 4258}, {3294, 3697}, {3692, 3713}, {3753, 3970}, {3969, 3998}
X(4515) = complement of X(17158)
X(4516) lies on these lines: {1, 2648}, {4, 2791}, {6, 2870}, {11, 123}, {37, 4068}, {65, 1020}, {115, 2971}, {210, 4069}, {314, 4451}, {522, 4459}, {523, 2486}, {692, 2161}, {926, 2170}, {1086, 3675}, {1365, 2611}, {1402, 1824}, {2087, 3248}, {2175, 4336}, {2642, 2643}, {2805, 4436}, {3123, 4475}, {3942, 4014}
X(4516) = trilinear product of extraversions of X(65)
X(4516) = crossdifference of every pair of points on line X(651)X(662)
X(4517) lies on these lines: {8, 210}, {9, 3056}, {37, 3779}, {41, 55}, {42, 2176}, {45, 674}, {63, 4447}, {181, 3340}, {200, 3208}, {219, 2330}, {612, 2295}, {869, 2276}, {984, 1469}, {2223, 3730}, {2275, 3009}, {3661, 3789}, {3691, 3715}, {3765, 3952}, {3786, 3790}, {4007, 4111}
X(4518) lies on these lines: {1, 3329}, {2, 38}, {8, 2170}, {10, 257}, {11, 312}, {12, 85}, {92, 427}, {190, 1281}, {210, 333}, {292, 1107}, {295, 660}, {325, 3932}, {385, 1757}, {612, 1911}, {813, 1311}, {1921, 3263}, {3685, 3693}, {3706, 4102}
X(4518) = isotomic conjugate of X(1447)
X(4519) lies on these lines: {8, 210}, {11, 2321}, {321, 354}, {536, 4003}, {1125, 2901}, {1215, 3244}, {1698, 3931}, {3175, 3741}, {3452, 4046}, {3555, 4066}, {3683, 4387}, {3689, 3886}, {3696, 4358}, {3707, 3985}, {3752, 4365}
X(4520) lies on these lines: {9, 3057}, {37, 3869}, {210, 3208}, {392, 3730}, {517, 3294}, {960, 1334}, {1212, 3877}, {2176, 3666}, {2329, 3683}, {3247, 4047}, {3691, 3880}, {4009, 4095}
X(4521) lies on these lines: {2, 3676}, {9, 649}, {101, 2731}, {124, 1566}, {281, 3064}, {513, 2490}, {514, 4885}, {522, 650}, {663, 4163}, {900, 2516}, {2976, 3667}, {3452, 3835}
X(4521) = isogonal conjugate of X(38828)
X(4521) = complement of X(3676)
X(4521) = complementary conjugate of X(17059)
X(4521) = trilinear pole of line X(4534)X(4953)
X(4521) = crosspoint of X(514) and X(3667)
X(4521) = crosssum of X(i) and X(j) for these {i,j}: {101, 1293}, {649, 32577}
X(4521) = crossdifference of every pair of points on line X(56)X(1149)
X(4521) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 38828}, {7, 34080}, {56, 27834}, {57, 1293}, {101, 19604}, {190, 16945}, {664, 38266}
X(4521) = trilinear product X(i)*X(j) for these {i,j}: {2, 4162}, {8, 4394}, {9, 3667}, {21, 14321}, {55, 4462}, {57, 4546}, {145, 650}, {284, 4404}, {312, 8643}, {333, 4729}, {513, 3161}, {514, 3158}, {522, 1743}, {663, 18743}, {3052, 4391}
X(4521) = trilinear quotient X(i)/X(j) for these (i,j): (1, 38828), (8, 27834), (9, 1293), (55, 34080), (145, 651), (514, 19604), (522, 8056), (649, 16945), (650, 3445), (663, 38266), (1743, 109), (3052, 1415), (3158, 101), (3161, 100), (3667, 57), (4162, 6), (4391, 4373), (4394, 56), (4404, 226), (4462, 7), (4546, 9), (4729, 1400), (8643, 604), (14321, 65), (18743, 664)
X(4521) = barycentric product X(i)*X(j) for these {i,j}: {7, 4546}, {8, 3667}, {9, 4462}, {21, 4404}, {75, 4162}, {145, 522}, {312, 4394}, {314, 4729}, {333, 14321}, {514, 3161}, {650, 18743}, {693, 3158}, {1743, 4391}, {3052, 35519}, {3596, 8643}
X(4521) = barycentric quotient X(i)/X(j) for these (i,j): (6, 38828), (9, 27834), (41, 34080), (55, 1293), (145, 664), (522, 4373), (650, 8056), (663, 3445), (667, 16945), (1743, 651), (3052, 109), (3063, 38266), (3158, 100), (3161, 190), (3667, 7), (4162, 1), (4394, 57), (4404, 1441), (4462, 85), (4546, 8), (4729, 65), (8643, 56), (14321, 226), (18743, 4554)
X(4522) lies on these lines: {522, 650}, {523, 3835}, {693, 4088}, {824, 1491}, {960, 3900}, {2517, 4064}, {3701, 3810}, {3776, 3837}, {3904, 4474}, {3910, 4147}
X(4523) lies on these lines: {42, 4137}, {65, 4085}, {72, 740}, {210, 3773}, {758, 3755}, {760, 4260}, {1215, 1824}, {2321, 3678}, {3874, 3946}, {4133, 4134}
X(4524) lies on these lines: {42, 647}, {200, 1021}, {210, 3700}, {512, 661}, {650, 926}, {657, 4105}, {2495, 2509}, {3239, 3900}, {3681, 4467}
X(4524) = isogonal conjugate of X(4616)
X(4524) = crossdifference of every pair of points on line X(81)X(279)
X(4525) lies on these lines: {10, 12}, {518, 3898}, {519, 3899}, {551, 3873}, {960, 3892}, {1125, 3894}, {3244, 3877}, {3625, 3869}, {3634, 3901}
X(4526) lies on these lines: {9, 4435}, {37, 665}, {192, 3766}, {522, 650}, {657, 4501}, {891, 3768}, {926, 2170}, {2786, 3960}, {3310, 4120}
X(4527) lies on these lines: {8, 4439}, {10, 37}, {192, 3775}, {321, 4062}, {1757, 3632}, {2887, 3969}, {3624, 3875}, {3971, 4046}, {4061, 4096}
X(4528) lies on these lines: {8, 3904}, {10, 676}, {764, 2505}, {900, 1145}, {1146, 2310}, {1639, 4152}, {2804, 3036}, {3239, 3900}
X(4529) lies on these lines: {9, 4086}, {346, 4171}, {522, 650}, {656, 2345}, {657, 4148}, {812, 4391}, {3287, 3907}, {4369, 4374}
X(4530) lies on these lines: {8, 9}, {11, 1146}, {514, 4089}, {519, 1023}, {673, 1121}, {952, 2246}, {1566, 3328}, {1647, 2087}
X(4531) lies on these lines: {8, 4087}, {39, 766}, {72, 740}, {210, 4095}, {213, 872}, {960, 3688}, {1212, 3271}, {3056, 3061}
X(4532) lies on these lines: {10, 12}, {2802, 3681}, {3241, 3898}, {3621, 3878}, {3622, 3892}, {3633, 3877}, {3876, 3894}
X(4533) lies on these lines: {10, 12}, {145, 392}, {517, 3832}, {518, 3624}, {960, 3632}, {3555, 3616}, {3715, 3811}
X(4534) lies on these lines: {11, 1146}, {101, 1317}, {514, 1358}, {519, 2348}, {644, 3039}, {650, 1015}, {3271, 4081}
X(4535) lies on these lines: {8, 4432}, {10, 37}, {1089, 4033}, {1215, 3969}, {1268, 3875}, {3836, 4431}, {4066, 4377}
X(4536) lies on these lines: {10, 12}, {3241, 3884}, {3612, 3951}, {3622, 3881}, {3633, 3878}, {3833, 3876}
X(4537) lies on these lines: {10, 12}, {551, 3889}, {3244, 3890}, {3612, 3984}, {3625, 3681}, {3828, 3901}
X(4538) lies on these lines: {210, 2321}, {740, 4015}, {3678, 3773}, {3697, 3755}, {3740, 3946}, {3956, 4085}
X(4539) lies on these lines: {10, 12}, {392, 3241}, {517, 3839}, {960, 3633}, {3555, 3622}, {3621, 3877}
X(4540) lies on these lines: {10, 12}, {1698, 3881}, {3626, 3740}, {3634, 3848}, {3679, 3884}, {3833, 4430}
X(4541) lies on these lines: {8, 41}, {519, 3290}, {2321, 2348}, {3686, 3693}, {3689, 4061}
X(4542) lies on these lines: {11, 522}, {655, 3322}, {2310, 4041}, {2325, 3689}, {3271, 3900}
X(4542) = trilinear pole wrt extouch triangle of line X(8)X(11)
X(4543) lies on these lines: {8, 522}, {200, 663}, {210, 3900}, {1639, 4152}, {4041, 4046}
X(4544) lies on these lines: {8, 41}, {10, 230}, {37, 519}, {2784, 3696}
X(4545) lies on these lines: {8, 9}, {519, 1213}, {3008, 4478}, {3625, 3986}
X(4546) lies on these lines: {8, 514}, {200, 663}, {3239, 3900}, {3667, 4404}
X(4547) lies on these lines: {10, 12}, {517, 3856}, {3624, 3681}, {3632, 3876}
X(4548) lies on these lines: {32, 2352}, {55, 607}, {197, 3162}, {198, 2220}
X(4549) = (3J2 - 7)*X(4) - 4(J2 - 4)*X(1209) (Peter Moses, January 16 2012)
Let A'B'C' be the circumcevian triangle of X(3), and let A''B''C'' be the 2nd Euler triangle of ABC. The lines A'A'', B'B'', C'C'' concur in X(4549). (Antreas Hatzipolakis, January 14 2012; see Segovia Point.)
X(4549) lies on these lines: {4, 1209}, {20, 1216}, {30, 599}, {110, 376}, {550, 1498}, {1657, 3426}, {2777,3098}}
Let A'B'C' be the circumcevian triangle of X(3), and let A''B''C'' be the 2nd Euler triangle of ABC. Let A''' be the point, other than A', where the line A'A'' meets the circumcircle of ABC, and let NA be the nine-point center of A'''BC, and define NB and NC cyclically. Then the points X(5), NA, NB, NC lie on a circle having center X(4550). (Antreas Hatzipolakis, January 14 2012; see Picasso Point.)
X(4550) lies on the Jerabek circumhyperbola of the medial triangle, the cuibics K1265 and K1290, and these lines: {2, 74}, {3, 1495}, {4, 1209}, {5, 7689}, {6, 9818}, {15, 40581}, {16, 40580}, {20, 41469}, {22, 16194}, {23, 16261}, {26, 44870}, {30, 141}, {55, 1480}, {56, 6580}, {64, 7393}, {110, 3431}, {140, 2883}, {182, 5663}, {184, 399}, {186, 10546}, {206, 5092}, {323, 7527}, {376, 41462}, {378, 5891}, {381, 1531}, {511, 8542}, {512, 46608}, {517, 41454}, {546, 16254}, {549, 46817}, {550, 16621}, {569, 12111}, {574, 3016}, {578, 1493}, {631, 15062}, {842, 6787}, {942, 37729}, {960, 3579}, {1092, 14130}, {1147, 5907}, {1154, 37517}, {1204, 1656}, {1216, 1593}, {1352, 5181}, {1511, 9306}, {1514, 15760}, {1596, 44201}, {1597, 1843}, {1598, 46852}, {1658, 45958}, {1995, 32110}, {2777, 15116}, {2781, 25489}, {2937, 33539}, {2979, 13596}, {3090, 11440}, {3520, 15056}, {3524, 5888}, {3545, 10545}, {3619, 40196}, {3627, 46728}, {3628, 32138}, {3839, 48912}, {3854, 38848}, {5067, 43601}, {5071, 15053}, {5158, 51544}, {5167, 19602}, {5221, 7986}, {5447, 12085}, {5462, 11479}, {5890, 15018}, {5892, 10605}, {5899, 18551}, {5972, 18580}, {6102, 46084}, {6200, 10960}, {6241, 13336}, {6396, 10962}, {6636, 11455}, {6642, 33537}, {6759, 32391}, {6800, 41450}, {6815, 43577}, {7387, 46849}, {7395, 40647}, {7464, 7998}, {7485, 14855}, {7486, 43597}, {7495, 32111}, {7503, 11456}, {7509, 10575}, {7512, 11439}, {7516, 46850}, {7525, 32137}, {7530, 46847}, {7550, 15072}, {7687, 19479}, {7712, 14157}, {7999, 12086}, {9968, 44491}, {10282, 34472}, {10293, 38728}, {10296, 41466}, {10297, 45303}, {10539, 11464}, {10606, 12379}, {10620, 22112}, {10625, 35502}, {10639, 10645}, {10640, 10646}, {10984, 18439}, {11002, 14483}, {11003, 14094}, {11004, 15033}, {11064, 44218}, {11074, 50464}, {11130, 35469}, {11131, 35470}, {11204, 11598}, {11250, 14128}, {11403, 37486}, {11424, 12316}, {11425, 41597}, {11444, 14865}, {11591, 13346}, {11793, 12084}, {12038, 17814}, {12041, 16187}, {12083, 32062}, {12112, 15080}, {12140, 35481}, {12281, 51882}, {12290, 37126}, {12367, 47353}, {13154, 17704}, {13491, 37515}, {14002, 41398}, {14070, 41424}, {14643, 19402}, {14810, 33532}, {14831, 44107}, {14926, 15041}, {15035, 39083}, {15037, 34783}, {15045, 46202}, {15051, 35473}, {15055, 51942}, {15067, 37480}, {15081, 23293}, {15311, 34573}, {15361, 39487}, {15694, 44750}, {16003, 18911}, {16419, 35450}, {18431, 25563}, {18445, 44109}, {18451, 18475}, {18488, 37444}, {18537, 37643}, {18917, 43573}, {19576, 26316}, {20773, 39084}, {20987, 33534}, {21844, 43614}, {21905, 30230}, {23039, 37496}, {30209, 41167}, {32223, 44275}, {33884, 43576}, {35485, 38726}, {36749, 45187}, {37890, 45975}, {38789, 45619}, {39170, 40355}, {40685, 50140}, {41468, 44833}, {43596, 43651}, {44262, 51548}, {44287, 51391}, {44413, 44456}
X(4550) = midpoint of X(i) and X(j) for these {i,j}: {3, 11472}, {4, 4549}, {74, 45019}, {110, 34802}, {378, 44754}, {1352, 49669}, {3426, 35237}
X(4550) = reflection of X(i) in X(j) for these {i,j}: {182, 49671}, {3098, 33533}, {7706, 5}, {8717, 3}, {33532, 14810}, {50008, 24206}, {51993, 381}
X(4550) = complement of X(4846)
X(4550) = complement of the isogonal conjugate of X(378)
X(4550) = complement of the isotomic conjugate of X(44134)
X(4550) = medial isogonal conjugate of X(15760)
X(4550) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 15760}, {19, 37648}, {31, 5158}, {162, 8675}, {378, 10}, {1973, 5309}, {2190, 5892}, {5063, 1214}, {8675, 34846}, {15066, 18589}, {32676, 9209}, {36131, 46229}, {42660, 16573}, {44080, 37}, {44134, 2887}, {51833, 34825}
X(4550) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 5158}, {74, 3581}, {110, 8675}, {16077, 46229}, {40423, 46808}
X(4550) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 5158}, {381, 39263}, {4550, 4846}
X(4550) = crosspoint of X(2) and X(44134)
X(4550) = crosssum of X(4846) and X(51471)
X(4550) = barycentric product X(i)*X(j) for these {i,j}: {378, 37638}, {381, 15066}, {4993, 5891}, {5063, 44135}, {5158, 44134}, {10564, 46808}, {32833, 34417}
X(4550) = barycentric quotient X(i)/X(j) for these {i,j}: {378, 43530}, {381, 34289}, {5063, 3431}, {5158, 4846}, {10564, 46809}, {34417, 34288}
X(4550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 74, 37470}, {3, 3426, 35237}, {3, 15030, 46261}, {378, 15066, 10564}, {381, 3581, 34417}, {381, 44751, 15362}, {399, 14805, 184}, {578, 5876, 15083}, {5891, 10564, 15066}, {5907, 7526, 1147}, {5907, 11430, 15068}, {7503, 11456, 37513}, {7526, 15068, 11430}, {7527, 11459, 13352}, {11430, 15068, 1147}, {11464, 15052, 10539}, {11464, 15058, 15052}, {11472, 32620, 3}, {11472, 35237, 3426}, {11479, 12163, 5462}, {12111, 35500, 569}, {12112, 35921, 15080}, {12162, 37513, 11456}, {14118, 15052, 11464}, {14118, 15058, 10539}, {14805, 18435, 399}, {15060, 18570, 9306}, {15080, 15305, 12112}, {18439, 34864, 10984}
X(4551) lies on these lines: {1, 5}, {7, 3240}, {9, 1937}, {10, 73}, {33, 118}, {34, 3811}, {35, 1935}, {40, 1745}, {42, 226}, {43, 57}, {56, 3216}, {59, 110}, {65, 3293}, {72, 227}, {100, 109}, {101, 108}, {171, 2003}, {200, 223}, {201, 3678}, {210, 1214}, {222, 1376}, {225, 3191}, {238, 2078}, {278, 3190}, {386, 388}, {516, 2635}, {518, 1465}, {519, 1457}, {581, 3085}, {664, 668}, {872, 4032}, {899, 1458}, {912, 1735}, {978, 1420}, {995, 3476}, {1042, 3214}, {1066, 1210}, {1103, 1490}, {1393, 3874}, {1450, 4315}, {1456, 3689}, {1617, 4383}, {1754, 3173}, {1757, 1758}, {1771, 3157}, {1788, 4306}, {3935, 4318}, {3952, 4069}, {4296, 4420}
X(4551) = isogonal conjugate of X(3737)
X(4551) = isotomic conjugate of X(18155)
X(4551) = anticomplement of X(34589)
X(4551) = vertex conjugate of X(162) and X(163)
X(4551) = cevapoint of Kiepert hyperbola intercepts of antiorthic axis
X(4551) = SS(A → A') of X(2617), where A'B'C' is the excentral triangle
X(4552) lies on these lines: {2, 2006}, {7, 192}, {8, 201}, {20, 2823}, {37, 1441}, {77, 3729}, {100, 108}, {101, 1305}, {109, 835}, {190, 644}, {226, 3995}, {227, 3701}, {241, 536}, {279, 4099}, {307, 2321}, {321, 1214}, {346, 347}, {646, 1978}, {655, 662}, {726, 1458}, {894, 1442}, {1018, 1020}, {1427, 3175}, {1445, 3875}, {1446, 3991}, {1465, 4358}, {1617, 3891}, {1708, 3187}, {1813, 2406}, {1943, 3219}, {2197, 3963}, {2283, 4436}, {2398, 3939}, {3668, 3950}, {3685, 4318}, {3952, 4069}
X(4552) = isogonal conjugate of X(7252)
X(4552) = isotomic conjugate of X(4560)
X(4552) = anticomplement of X(4858)
X(4552) = anticomplementary conjugate of anticomplement of X(2149)
X(4552) = polar conjugate of isogonal conjugate of X(23067)
X(4553) lies on these lines: {2, 25048}, {8, 40624}, {10, 17197}, {37, 256}, {38, 46150}, {44, 9025}, {67, 72}, {69, 25050}, {75, 25279}, {80, 22306}, {86, 22279}, {99, 29071}, {100, 110}, {101, 831}, {141, 3688}, {190, 513}, {193, 25316}, {210, 4690}, {274, 22328}, {291, 16726}, {312, 25308}, {314, 22289}, {319, 20722}, {344, 25304}, {511, 3932}, {518, 3792}, {521, 3939}, {524, 20683}, {545, 4014}, {644, 1633}, {668, 670}, {674, 3912}, {894, 21865}, {1018, 2284}, {1026, 3882}, {1083, 16686}, {1086, 14839}, {1401, 4884}, {1463, 28582}, {1654, 40607}, {1930, 46158}, {2228, 3009}, {2245, 4447}, {2325, 29353}, {2530, 46162}, {2876, 4437}, {2979, 32862}, {3056, 17279}, {3254, 5559}, {3271, 4422}, {3309, 4752}, {3416, 3781}, {3681, 17360}, {3703, 3917}, {3717, 8679}, {3733, 40519}, {3779, 4851}, {3834, 20358}, {3873, 17387}, {3879, 22277}, {3909, 3952}, {3942, 4712}, {3943, 6007}, {3954, 46154}, {4033, 23354}, {4069, 21362}, {4111, 4478}, {4132, 21295}, {4411, 4572}, {4473, 24482}, {4482, 8678}, {4517, 4643}, {4567, 5389}, {4579, 4585}, {4589, 4623}, {4594, 4601}, {4645, 20718}, {4684, 9049}, {4735, 37596}, {4899, 9026}, {4966, 9052}, {5205, 38472}, {5360, 42713}, {6004, 32094}, {6012, 29099}, {6542, 44671}, {7064, 17332}, {7077, 24358}, {7155, 29542}, {7186, 33164}, {7270, 22299}, {7998, 33089}, {8287, 20531}, {9016, 17755}, {9355, 16561}, {13476, 17300}, {15523, 46160}, {15624, 22370}, {16696, 21035}, {17049, 17245}, {17142, 18143}, {17165, 25289}, {17243, 21746}, {17311, 35892}, {17384, 25144}, {17777, 38390}, {17934, 22311}, {18137, 21278}, {18165, 29653}, {18191, 33115}, {18743, 25306}, {20344, 21293}, {20723, 20947}, {21814, 46156}, {22031, 23821}, {22275, 33078}, {22292, 33297}, {22325, 33079}, {24410, 43135}, {30172, 37536}, {34790, 41822}, {35309, 46148}, {35333, 46163}, {35335, 46153}, {40966, 44419}
X(4553) = midpoint of X(i) and X(j) for these {i,j}: {69, 25050}, {190, 3888}, {3792, 32847}
X(4553) = reflection of X(i) in X(j) for these {i,j}: {190, 40521}, {3271, 4422}, {20358, 3834}, {46149, 141}
X(4553) = isogonal conjugate of X(18108)
X(4553) = complement of X(25048)
X(4553) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1016, 40007}, {1110, 40637}, {1252, 16552}, {4570, 16684}, {43076, 17154}
X(4553) = X(i)-Ceva conjugate of X(j) for these (i,j): {4576, 4568}, {4594, 190}, {4601, 37}
X(4553) = X(i)-cross conjugate of X(j) for these (i,j): {2530, 38}, {21123, 16696}, {35309, 4568}
X(4553) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18108}, {6, 10566}, {37, 39179}, {82, 513}, {83, 649}, {86, 18105}, {109, 18101}, {251, 514}, {308, 1919}, {667, 3112}, {733, 4107}, {827, 3120}, {893, 18111}, {1019, 18098}, {1086, 4628}, {1176, 7649}, {1293, 18113}, {1333, 18070}, {1459, 32085}, {1474, 4580}, {1980, 18833}, {2162, 18107}, {3121, 4593}, {3122, 4577}, {3125, 4599}, {3733, 18082}, {4164, 43763}, {4556, 34294}, {4630, 21207}, {6591, 34055}, {7252, 18097}, {9315, 18110}, {10547, 46107}, {16732, 34072}, {21832, 39276}, {27846, 36081}
X(4553) = cevapoint of X(i) and X(j) for these (i,j): {38, 2530}, {513, 22279}, {21035, 21123}
X(4553) = crosspoint of X(100) and X(668)
X(4553) = crosssum of X(i) and X(j) for these (i,j): {513, 667}, {10566, 18107}
X(4553) = trilinear pole of line {38, 39}
X(4553) = crossdifference of every pair of points on line {3125, 4164}
X(4553) = barycentric product X(i)*X(j) for these {i,j}: {1, 4568}, {37, 4576}, {38, 190}, {39, 668}, {72, 41676}, {75, 46148}, {86, 35309}, {99, 3954}, {100, 141}, {101, 1930}, {312, 46153}, {321, 1634}, {345, 46152}, {427, 1332}, {644, 3665}, {646, 1401}, {651, 3703}, {662, 15523}, {664, 33299}, {670, 21814}, {692, 8024}, {765, 16892}, {799, 21035}, {826, 4567}, {906, 1235}, {1016, 2530}, {1018, 16887}, {1331, 20883}, {1783, 3933}, {1964, 1978}, {2525, 5379}, {3005, 4601}, {3051, 6386}, {3263, 46163}, {3688, 4554}, {3903, 16720}, {3908, 23297}, {3912, 35333}, {3917, 6335}, {3952, 16696}, {3998, 46151}, {4033, 17187}, {4093, 4639}, {4357, 35334}, {4358, 46162}, {4557, 16703}, {4561, 17442}, {4572, 40972}, {4574, 16747}, {4592, 21016}, {4594, 16587}, {4600, 8061}, {4602, 41267}, {4884, 27834}, {5380, 7813}, {7035, 21123}, {7260, 40936}, {20021, 42717}, {20336, 35325}, {21817, 35137}, {24004, 46150}, {32008, 35335}, {36827, 42713}, {42701, 46155}, {42716, 46147}, {42720, 46149}, {42721, 46154}, {42723, 46158}
X(4553) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 10566}, {6, 18108}, {10, 18070}, {38, 514}, {39, 513}, {43, 18107}, {58, 39179}, {72, 4580}, {100, 83}, {101, 82}, {141, 693}, {171, 18111}, {190, 3112}, {213, 18105}, {427, 17924}, {650, 18101}, {668, 308}, {688, 3121}, {692, 251}, {732, 14296}, {826, 16732}, {906, 1176}, {1018, 18082}, {1110, 4628}, {1331, 34055}, {1332, 1799}, {1376, 18110}, {1401, 3669}, {1634, 81}, {1783, 32085}, {1843, 6591}, {1923, 1919}, {1930, 3261}, {1964, 649}, {1978, 18833}, {2084, 3122}, {2236, 4107}, {2530, 1086}, {3005, 3125}, {3051, 667}, {3313, 16757}, {3665, 24002}, {3688, 650}, {3703, 4391}, {3908, 10130}, {3917, 905}, {3933, 15413}, {3954, 523}, {4020, 1459}, {4093, 21832}, {4394, 18113}, {4436, 18089}, {4551, 18097}, {4557, 18098}, {4567, 4577}, {4568, 75}, {4570, 4599}, {4576, 274}, {4600, 4593}, {4601, 689}, {4705, 34294}, {4884, 4462}, {5379, 42396}, {6335, 46104}, {6386, 40016}, {7239, 16889}, {8024, 40495}, {8041, 2530}, {8061, 3120}, {8623, 4164}, {15523, 1577}, {16587, 2533}, {16696, 7192}, {16720, 4374}, {16887, 7199}, {16892, 1111}, {17187, 1019}, {17442, 7649}, {18715, 21205}, {20775, 22383}, {20883, 46107}, {21016, 24006}, {21035, 661}, {21123, 244}, {21249, 27712}, {21752, 7234}, {21814, 512}, {21817, 7927}, {21839, 22105}, {33299, 522}, {35309, 10}, {35319, 18180}, {35325, 28}, {35333, 673}, {35334, 1220}, {35335, 142}, {35338, 18087}, {40972, 663}, {41267, 798}, {41331, 1980}, {41676, 286}, {42717, 20022}, {46148, 1}, {46150, 1022}, {46152, 278}, {46153, 57}, {46162, 88}, {46163, 105}
X(4553) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 1332, 692}, {190, 3799, 40521}, {692, 3908, 1332}, {1018, 35338, 4436}, {1026, 3882, 4557}, {3271, 4422, 16482}, {3799, 3888, 190}, {4069, 21362, 23343}, {16726, 22323, 291}
As the trilinear product of Steiner circumellipse antipodes, X(4554) lies on conic {{A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75). (Randy Hutson, July 11, 2019)
X(4554) lies on these lines: {7, 1357}, {11, 2481}, {76, 348}, {99, 108}, {100, 693}, {109, 789}, {190, 658}, {274, 349}, {307, 314}, {344, 1996}, {347, 3596}, {645, 651}, {646, 1978}, {664, 668}, {833, 1305}, {883, 3952}, {1214, 1920}, {1465, 1921}, {1758, 1966}
X(4554) = isogonal conjugate of X(3063)X(4555) lies on these lines: {88, 239}, {99, 901}, {100, 4378}, {106, 3226}, {190, 514}, {320, 519}, {659, 898}, {660, 891}, {664, 3676}, {668, 693}, {671, 4080}, {874, 889}, {1022, 4444}, {1121, 3912}, {1320, 2481}
X(4555) = reflection of X(35168) in X(2)
X(4555) = isogonal conjugate of X(1960)
X(4555) = isotomic conjugate of X(900)
X(4555) = complement of X(39349)
X(4555) = anticomplement of X(35092)
X(4555) = X(19)-isoconjugate of X(22086)
X(4555) = Steiner circumellipse antipode of X(35168)
X(4556) lies on these lines: {58, 3122}, {86, 4466}, {110, 351}, {163, 662}, {757, 2150}
X(4556) = isogonal conjugate of X(4024)
X(4556) = crossdifference of every pair of points on line X(115)X(2643)
X(4557) lies on these lines: {3, 2801}, {6, 292}, {37, 4068}, {42, 3122}, {44, 2223}, {45, 55}, {100, 190}, {101, 692}, {197, 1260}, {198, 480}, {200, 3185}, {210, 228}, {523, 4552}, {524, 4447}, {651, 2283}, {660, 662}, {674, 2183}, {1001, 3842}, {1011, 3715}, {1018, 4069}, {1020, 4551}, {1026, 3882}, {1215, 1376}, {1259, 2933}, {1732, 2352}, {1743, 3941}, {1757, 3286}, {2174, 2330}, {2175, 3204}, {2178, 4497}, {3688, 4271}, {3943, 4433}, {3950, 4097}, {4096, 4421}
X(4557) = isogonal conjugate of X(7192)
X(4557) = X(7668)-of-excentral-triangle
X(4557) = crossdifference of every pair of points on line X(812)X(1015)
X(4557) = complement of anticomplementary conjugate of X(31290)
X(4558) lies on the MacBeath circumconic and on the inscribed parabola with focus X(107), and on these lines: {2, 2986}, {3, 895}, {6, 2987}, {32, 1992}, {50, 524}, {69, 248}, {81, 2990}, {86, 2989}, {97, 343}, {99, 112}, {110, 351}, {193, 571}, {333, 2988}, {401, 3260}, {417, 1942}, {651, 662}, {670, 2966}, {691, 1296}, {771, 823}, {827, 907}, {906, 1332}, {1333, 2991}, {1444, 1814}, {1632, 4226}, {1790, 1797}, {1815, 2327}, {1983, 3882}, {2965, 3629}
X(4558) = reflection of X(2987) in X(6)
X(4558) = isogonal conjugate of X(2501)
X(4558) = isotomic conjugate of X(14618)
X(4558) = MacBeath circumconic antipode of X(2987)
X(4558) = crossdifference of every pair of points on line X(115)X(2971)
X(4558) = X(3)-cross conjugate of X(647)
X(4558) = X(92)-isoconjugate of X(512)
X(4558) = barycentric product X(3)*X(99)
X(4558) = barycentric product of circumcircle intercepts of line X(3)X(69)
X(4559) lies on these lines: {6, 1411}, {12, 2295}, {37, 1409}, {55, 1945}, {56, 292}, {65, 213}, {73, 1334}, {101, 109}, {163, 1625}, {172, 1399}, {190, 644}, {201, 220}, {219, 478}, {226, 3997}, {607, 1825}, {608, 2911}, {672, 1457}, {692, 2498}, {1018, 4551}, {1319, 3230}, {1500, 2594}, {1935, 2329}, {1950, 2174}
X(4559) = isogonal conjugate of X(4560)
X(4559) = polar conjugate of isotomic conjugate of X(23067)
X(4559) = X(92)-isoconjugate of X(23189)
The line X(11)X(124) is the radical axis of incircle and Mandart circle.
X(4560) lies on these lines: {1, 4151}, {2, 1577}, {8, 3907}, {21, 884}, {81, 2401}, {99, 666}, {110, 929}, {239, 514}, {448, 525}, {522, 663}, {523, 1325}, {650, 3975}, {655, 662}, {667, 784}, {693, 905}, {812, 4481}, {814, 1491}, {3125, 3271}, {3904, 3910}, {4147, 4474}
X(4560) = isogonal conjugate of X(4559)
X(4560) = isotomic conjugate of X(4552)
X(4560) = anticomplement of X(1577)
X(4560) = polar conjugate of anticomplement of X(34588)
X(4560) = polar conjugate of isogonal conjugate of X(23189)
X(4560) = anticomplementary conjugate of X(21294)
X(4560) = X(19)-isoconjugate of X(23067)
X(4560) = intersection of perspectrices of [ABC and Gemini triangle 1] and [ABC and Gemini triangle 2]
X(4561) lies on these lines: {69, 1565}, {72, 295}, {75, 997}, {78, 304}, {99, 101}, {100, 1310}, {326, 3718}, {332, 1807}, {336, 3682}, {664, 668}, {811, 1897}, {883, 934}, {978, 3905}, {995, 4360}, {1265, 3926}, {1332, 4574}, {3263, 4511}, {3570, 3732}, {4103, 4482}
X(4561) = isotomic conjugate of X(7649)
X(4561) = X(92)-isoconjugate of X(1919)
X(4562) lies on the Steiner circumellipse and these lines: {75, 3252}, {99, 813}, {190, 513}, {239, 292}, {291, 519}, {334, 350}, {335, 536}, {514, 668}, {664, 3669}, {666, 1026}, {670, 4033}, {730, 3864}, {742, 3862}, {889, 3572}, {1022, 4444}, {1121, 4518}, {3225, 3507}
X(4562) = isogonal conjugate of X(8632)
X(4562) = isotomic conjugate of X(812)
X(4562) = complement of X(39362)
X(4562) = anticomplement of X(35119)
X(4562) = X(19)-isoconjugate of X(22384)
X(4563) lies on the MacBeath circumconic and these lines: {2, 2987}, {69, 125}, {76, 2986}, {81, 2991}, {99, 110}, {107, 877}, {126, 193}, {274, 2990}, {287, 305}, {310, 2989}, {323, 3266}, {645, 651}, {648, 670}, {805, 3222}, {1799, 3917}, {1812, 1814}
X(4563) = isogonal conjugate of X(2489)
X(4563) = isotomic conjugate of X(2501)
X(4563) = anticomplement of X(6388)
X(4563) = X(92)-isoconjugate of X(669)
X(4563) = X(525)-cross conjugate of X(69)
X(4563) = X(3049)-cross conjugate of X(3)
X(4563) = orthocorrespondent of X(99)
The line X(100)X(109) is the tangent at X(100) to the circumconic centered at X(1) (conic {{A,B,C,X(100),X(664),X(1120),X(1320)}}); it is also the locus of trilinear poles of tangents at P to hyperbola {{A,B,C,X(9),P}} as P moves on line X(1)X(6). (Randy Hutson, March 21, 2019)
X(4564) lies on these lines: {1, 1053}, {59, 518}, {100, 3900}, {101, 514}, {109, 898}, {190, 1813}, {241, 1252}, {249, 2185}, {527, 1275}, {651, 3257}, {655, 662}, {1016, 1429}, {1023, 1025}, {2283, 3573}, {2397, 2406}
X(4564) = isogonal conjugate of X(2170)
X(4564) = isotomic conjugate of X(4858)
X(4564) = anticomplement of complementary conjugate of X(21232)
X(4564) = X(6)-isoconjugate of X(11)
X(4565) lies on these lines: {57, 593}, {60, 603}, {99, 4552}, {109, 110}, {112, 934}, {163, 1461}, {651, 662}, {1014, 1333}, {1019, 1020}, {1325, 1455}, {1326, 1458}, {1442, 1950}, {1576, 3733}, {1634, 2283}
X(4565) = isogonal conjugate of X(3700)
X(4565) = anticomplement of complementary conjugate of X(17069)
X(4566) lies on these lines: {2, 1952}, {7, 80}, {46, 3188}, {65, 1446}, {100, 658}, {109, 1305}, {151, 4329}, {279, 291}, {651, 653}, {668, 883}, {1018, 1020}, {1435, 3187}, {1439, 1441}, {1847, 3868}
X(4566) = isogonal conjugate of X(21789)
X(4566) = isotomic conjugate of X(7253)
X(4566) = anticomplement of X(34591)
X(4566) = polar conjugate of X(17926)
X(4566) = X(19)-isoconjugate of X(23090)
X(4567) lies on these lines: {99, 666}, {100, 691}, {110, 898}, {172, 763}, {249, 1931}, {250, 2074}, {662, 1019}, {668, 906}, {765, 1110}, {1016, 1252}, {1262, 1275}, {1983, 3570}, {2397, 2407}, {2421, 2427}
X(4567) = isogonal conjugate of X(3125)
X(4567) = isotomic conjugate of X(16732)
X(4567) = X(19)-isoconjugate of X(18210)
X(4567) = trilinear product of extraversions of X(81)
X(4568) lies on these lines: {10, 257}, {99, 101}, {100, 831}, {192, 995}, {514, 668}, {664, 4482}, {712, 1575}, {997, 3729}, {1026, 3952}, {1565, 4437}, {1655, 4075}, {3699, 3732}, {4134, 4416}
X(4569) lies on the Steiner circumellipse and these lines: {7, 2481}, {85, 1121}, {99, 934}, {190, 658}, {269, 3226}, {279, 3227}, {290, 1439}, {651, 666}, {668, 883}, {671, 1446}, {903, 1088}, {927, 1633}, {1427, 3228}
X(4569) = isogonal conjugate of X(8641)
X(4569) = isotomic conjugate of X(3900)
X(4569) = anticomplement of X(35508)
X(4570) lies on these lines: {59, 3286}, {101, 691}, {110, 901}, {190, 2966}, {249, 1101}, {250, 2073}, {662, 3737}, {687, 1897}, {692, 4556}, {765, 1110}, {1252, 3285}, {2398, 2407}, {2421, 2426}
X(4570) = isogonal conjugate of X(3120)
X(4570) = isotomic conjugate of X(21207)
X(4571) lies on these lines: {3, 1811}, {55, 4126}, {63, 1810}, {100, 190}, {109, 1026}, {345, 1260}, {643, 644}, {1259, 1265}, {1331, 1332}, {1792, 1793}, {1808, 1812}, {1897, 2397}
X(4572) lies on these lines: {85, 3020}, {99, 1305}, {109, 689}, {190, 3261}, {646, 1978}, {653, 799}, {664, 670}, {668, 883}, {1231, 1952}, {3888, 4374}, {4411, 4553}
X(4572) = isotomic conjugate of X(663)
X(4573) lies on these lines: {7, 1365}, {57, 552}, {77, 261}, {81, 1462}, {85, 1509}, {99, 109}, {110, 927}, {645, 651}, {658, 662}, {931, 934}, {1434, 4031}
X(4573) = isogonal conjugate of X(3709)
X(4573) = isotomic conjugate of X(3700)
X(4573) = anticomplement of complementary conjugate of X(17066)
X(4574) lies on these lines: {3, 295}, {101, 692}, {213, 3991}, {219, 1807}, {220, 3678}, {643, 1625}, {644, 1783}, {906, 1331}, {1018, 4551}, {1089, 4513}, {1332,4561}, {3694, 3990}
X(4574) = X(92)-isoconjugate of X(3733)
X(4574) = crossdifference of every pair of points on line X(1086)X(2969)
X(4575) lies on these lines: {58, 1800}, {63, 293}, {109, 110}, {162, 662}, {163, 1983}, {283, 1795}, {501, 4337}, {849, 4257}, {1789, 2169}, {1822, 2584}, {1823, 2585}
X(4575) = isogonal conjugate of X(24006)
X(4575) = trilinear product X(48)*X(662)
X(4575) = trilinear product of MacBeath circumconic intercepts of Brocard axis
X(4575) = X(92)-isoconjugate of X(661)
X(4576) lies on these lines: {2, 694}, {6, 6665}, {38, 46159}, {39, 46156}, {67, 69}, {76, 7998}, {81, 32029}, {86, 3315}, {99, 110}, {126, 6388}, {141, 31078}, {183, 21766}, {194, 9463}, {249, 17708}, {305, 2979}, {310, 32035}, {314, 38478}, {323, 12215}, {339, 13416}, {376, 38641}, {511, 3266}, {518, 16741}, {523, 14607}, {524, 25325}, {543, 40915}, {660, 799}, {668, 8050}, {670, 888}, {689, 805}, {698, 3231}, {826, 35359}, {850, 14221}, {858, 6393}, {873, 17140}, {877, 6331}, {883, 4573}, {892, 42367}, {925, 35575}, {930, 10425}, {1078, 41462}, {1112, 34336}, {1370, 4176}, {1509, 17141}, {1613, 8267}, {1634, 14424}, {1915, 35929}, {1930, 46160}, {1975, 15066}, {2086, 25327}, {2421, 11794}, {2669, 17495}, {3005, 46161}, {3051, 14778}, {3098, 26233}, {3222, 25424}, {3265, 23181}, {3313, 46165}, {3589, 25322}, {3618, 25315}, {3763, 25334}, {3787, 19568}, {3819, 39998}, {3917, 8024}, {3933, 46147}, {3978, 11673}, {3995, 39915}, {4481, 7239}, {4558, 34211}, {4568, 46148}, {4577, 35137}, {4625, 35312}, {5026, 20976}, {5108, 20998}, {5189, 5207}, {5640, 11059}, {5650, 26235}, {5972, 7664}, {6337, 34834}, {6515, 19583}, {6636, 37894}, {7257, 21272}, {7304, 17150}, {7782, 15080}, {8033, 17165}, {8591, 14916}, {8617, 35288}, {9019, 36824}, {9066, 39639}, {9465, 35275}, {10754, 39024}, {14588, 14999}, {14683, 38940}, {14928, 24981}, {14957, 39266}, {15059, 30786}, {15589, 40911}, {15958, 17932}, {16703, 46149}, {16887, 46150}, {16951, 34945}, {17135, 20351}, {17154, 18827}, {17204, 21241}, {18061, 27189}, {18155, 21580}, {19599, 35296}, {20023, 34095}, {20045, 25302}, {23285, 46155}, {30966, 31117}, {32032, 33297}, {33014, 43714}, {33260, 37889}, {33756, 41309}, {35325, 41676}
X(4576) = midpoint of X(69) and X(25052)
X(4576) = reflection of X(i) in X(j) for these {i,j}: {2, 45672}, {69, 36792}, {25047, 3124}, {25322, 3589}, {46154, 141}
X(4576) = isogonal conjugate of X(18105)
X(4576) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {99, 21221}, {110, 21220}, {163, 25054}, {249, 192}, {250, 21216}, {662, 148}, {670, 21294}, {799, 3448}, {1101, 194}, {2185, 17036}, {4556, 9263}, {4567, 1654}, {4570, 1655}, {4590, 8}, {4592, 39352}, {4593, 25051}, {4599, 25047}, {4600, 2895}, {4601, 1330}, {4610, 149}, {4612, 39351}, {4620, 2475}, {4623, 150}, {4631, 33650}, {6064, 329}, {7340, 7}, {18020, 5905}, {23357, 17486}, {23889, 39356}, {23995, 8264}, {23999, 6515}, {24000, 6392}, {24037, 69}, {24041, 2}, {31614, 7192}, {34537, 6327}, {36085, 45291}, {44168, 21275}
X(4576) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 1634}, {249, 69}, {4590, 7794}, {17708, 5468}, {34537, 2}, {35137, 99}
X(4576) = X(i)-cross conjugate of X(j) for these (i,j): {688, 39}, {826, 141}, {1634, 41676}, {2525, 8024}, {3005, 2}, {4074, 34537}, {7794, 4590}, {8711, 6}, {14406, 46156}, {14424, 31125}
X(4576) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18105}, {32, 18070}, {42, 18108}, {82, 512}, {83, 798}, {115, 34072}, {163, 34294}, {213, 10566}, {251, 661}, {308, 1924}, {649, 18098}, {667, 18082}, {669, 3112}, {689, 4117}, {810, 32085}, {827, 2643}, {923, 22105}, {1084, 4593}, {1109, 4630}, {1500, 39179}, {1973, 4580}, {2179, 39182}, {2422, 3405}, {2489, 34055}, {3063, 18097}, {3124, 4599}, {3125, 4628}, {5027, 43763}, {9426, 18833}, {9427, 37204}, {10547, 24006}, {18107, 21759}, {18111, 40729}
X(4576) = cevapoint of X(i) and X(j) for these (i,j): {39, 688}, {141, 826}, {512, 3589}, {523, 3934}, {525, 11574}, {2525, 3917}, {3005, 8041}, {4553, 4568}, {6665, 8711}, {7192, 17140}
X(4576) = crosspoint of X(i) and X(j) for these (i,j): {99, 670}, {4577, 6573}
X(4576) = crosssum of X(i) and X(j) for these (i,j): {512, 669}, {887, 38366}, {3005, 8711}
X(4576) = trilinear pole of line {39, 141}
X(4576) = crossdifference of every pair of points on line {3124, 5027}
XX(4576) = intersection of tangents to Steiner circumellipse at X(99) and X(670)
X(4576) = barycentric product X(i)*X(j) for these {i,j}: {38, 799}, {39, 670}, {69, 41676}, {76, 1634}, {86, 4568}, {99, 141}, {100, 16703}, {110, 8024}, {190, 16887}, {249, 23285}, {274, 4553}, {305, 35325}, {310, 46148}, {427, 4563}, {645, 3665}, {648, 3933}, {662, 1930}, {668, 16696}, {688, 44168}, {689, 8041}, {732, 18829}, {805, 35540}, {826, 4590}, {873, 35309}, {892, 7813}, {1235, 4558}, {1332, 16747}, {1964, 4602}, {1978, 17187}, {2396, 20021}, {2525, 18020}, {2530, 4601}, {3005, 34537}, {3051, 4609}, {3266, 36827}, {3703, 4573}, {3917, 6331}, {3926, 46151}, {3954, 4623}, {3978, 46161}, {4175, 42396}, {4561, 17171}, {4577, 7794}, {4592, 20883}, {4594, 16720}, {4600, 16892}, {4610, 15523}, {4625, 33299}, {5468, 31125}, {6292, 35137}, {6573, 6665}, {7799, 46155}, {7953, 42554}, {8061, 24037}, {9146, 23297}, {16739, 35334}, {18157, 35333}, {23642, 35567}, {27853, 46159}, {28660, 46153}, {31614, 39691}, {34384, 35319}
X(4576) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 18105}, {38, 661}, {39, 512}, {69, 4580}, {75, 18070}, {81, 18108}, {86, 10566}, {95, 39182}, {99, 83}, {100, 18098}, {110, 251}, {141, 523}, {190, 18082}, {249, 827}, {427, 2501}, {523, 34294}, {524, 22105}, {648, 32085}, {662, 82}, {664, 18097}, {670, 308}, {688, 1084}, {732, 804}, {757, 39179}, {799, 3112}, {805, 733}, {826, 115}, {1101, 34072}, {1235, 14618}, {1401, 7180}, {1634, 6}, {1843, 2489}, {1923, 1924}, {1930, 1577}, {1964, 798}, {2076, 17997}, {2396, 20022}, {2525, 125}, {2528, 39691}, {2530, 3125}, {3005, 3124}, {3051, 669}, {3313, 2485}, {3665, 7178}, {3688, 3709}, {3703, 3700}, {3787, 8651}, {3917, 647}, {3933, 525}, {3954, 4705}, {4020, 810}, {4175, 2525}, {4553, 37}, {4558, 1176}, {4560, 18101}, {4563, 1799}, {4568, 10}, {4570, 4628}, {4590, 4577}, {4592, 34055}, {4602, 18833}, {4609, 40016}, {4884, 14321}, {6292, 7927}, {6331, 46104}, {7779, 18010}, {7794, 826}, {7813, 690}, {8024, 850}, {8041, 3005}, {8061, 2643}, {8362, 3800}, {8623, 5027}, {9019, 2492}, {9146, 10130}, {9494, 9427}, {11205, 8664}, {11794, 30505}, {14096, 3288}, {14406, 1645}, {14424, 1648}, {14570, 17500}, {14994, 23878}, {15523, 4024}, {16030, 2623}, {16696, 513}, {16703, 693}, {16720, 2533}, {16747, 17924}, {16887, 514}, {16892, 3120}, {17103, 18111}, {17171, 7649}, {17187, 649}, {17708, 9076}, {18020, 42396}, {18047, 18099}, {18829, 14970}, {19174, 15422}, {20021, 2395}, {20775, 3049}, {20883, 24006}, {21035, 4079}, {21123, 3122}, {23285, 338}, {23297, 8599}, {23357, 4630}, {23642, 2514}, {24037, 4593}, {24041, 4599}, {26714, 42288}, {30489, 46001}, {31125, 5466}, {32449, 25423}, {32661, 10547}, {33296, 18107}, {33299, 4041}, {33946, 16889}, {34211, 21458}, {34537, 689}, {35137, 40425}, {35309, 756}, {35319, 51}, {35325, 25}, {35333, 18785}, {35335, 21808}, {35540, 14295}, {36066, 39276}, {36827, 111}, {37134, 43763}, {39292, 41209}, {39691, 8029}, {41328, 3050}, {41331, 9426}, {41622, 32472}, {41657, 28470}, {41676, 4}, {41677, 10550}, {44168, 42371}, {44766, 16277}, {45215, 10551}, {46147, 2433}, {46148, 42}, {46151, 393}, {46152, 1880}, {46153, 1400}, {46154, 9178}, {46155, 1989}, {46157, 14998}, {46159, 3572}, {46161, 694}, {46164, 34212}, {46166, 8106}, {46167, 8105}
X(4576) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 25047, 3124}, {99, 110, 10330}, {99, 4563, 110}, {99, 5468, 35356}, {99, 9146, 5468}, {110, 4563, 5468}, {110, 9146, 4563}, {110, 10330, 35356}, {877, 6331, 35360}, {3952, 7192, 799}, {4074, 8041, 2}, {5468, 10330, 110}, {9464, 33884, 69}, {30508, 30509, 2396}, {36793, 41673, 69}
X(4577) is the center of the conic that passes through the following six points: the vertices of the antipedal triangle of X(3) and the vertices of the antipedal triangle of X(4). Other points on this conic are X(i) for these i: 2, 6, 22, 5596, 8264, 8266, 8267, 8272. See the preamble to X(8268). Note added: this conic is the bianticevian conic of X(2) and X(6). (Randy Hutson, July 11, 2019)
X(4577) lies on the Steiner circumellipse and these lines: {82,757}, {83,597}, {99,827}, {110,670}, {190,4599}, {206,1502}, {251,3228}, {290,308}, {662,4562}, {668,692}, {733,1084}, {782,880}, {1494,1799}, {1503,8928}, {1634,4590}, {2966,4580}, {6573,7953}
X(4577) = isogonal conjugate of X(3005)
X(4577) = isotomic conjugate of X(826)
X(4577) = complement of X(39346)
X(4577) = trilinear pole of the line X(2)X(32)
X(4577) = cevapoint of X(i) and X(j) for these {i,j}: {2,826}, {99,110}, {512,1194}, {523,3589}, {525,6676}, {688,8265}, {690,7664}, {1176,4580}, {4027,5027}
X(4577) = X(i)-cross conjugate of X(j) for these (i,j): (6,4590), (99,689), (110,827), (826,2), (4576,6573), (4579,662), (4580,308), (5012,249), (5027,733)
X(4577) = X(i)-isoconjugate of X(j) for these {i,j}: {1,3005}, {2,2084}, {6,8061}, {31,826}, {38,512}, {39,661}, {42,2530}, {75,688}, {141,798}, {427,810}, {523,1964}, {649,3954}, {656,1843}, {669,1930}, {850,1923}, {876,4093}, {882,2236}, {1401,4041}, {1577,3051}, {1634,2643}, {1924,8024}, {1973,2525}, {2501,4020}, {2531,3112}, {3121,4568}, {3122,4553}, {3404,3569}, {3688,4017}
X(4577) = trilinear product X(i)*X(j) for these {i,j}: {2,4599}, {6,4593}, {31,689}, {75,827}, {82,99}, {83,662}, {110,3112}, {162,1799}, {163,308}, {251,799}, {274,4628}, {561,4630}, {811,1176}, {2966,3405}
X(4577) = barycentric product X(i)*X(j) for these {i,j}: {1,4593}, {6,689}, {75,4599}, {76,827}, {82,799}, {83,99}, {110,308}, {251,670}, {310,4628}, {648,1799}, {662,3112}, {733,880}, {1176,6331}, {1502,4630}, {6573,7760}
X(4577) = perspector of ABC and the tangential triangle, wrt the anticomplementary triangle, of the bianticevian conic of X(2) and X(6)
X(4578) lies on these lines: {55, 4152}, {100, 190}, {200, 2310}, {346, 480}, {644, 3939}, {646, 4076}, {651, 1026}, {677, 765}, {883, 934}, {1376, 4454}
X(4579) lies on these lines: {6, 82}, {9, 2112}, {59, 4552}, {100, 109}, {101, 932}, {110, 645}, {190, 692}, {660, 662}, {894, 2330}, {1083, 4473}, {1332, 3799}
X(4580) lies on these lines: {23, 385}, {83, 2394}, {250, 827}, {308, 2395}, {525, 3049}, {647, 3267}, {656, 4019}, {733, 2868}, {804, 2514}, {850, 2485}, {879, 1176}
X(4580) = isogonal conjugate of X(35325)
X(4580) = X(19)-isoconjugate of X(1634)
X(4581) lies on these lines: {422, 2501}, {513, 2517}, {522, 649}, {523, 1325}, {850, 4374}, {885, 2298}, {901, 4427}, {961, 2401}, {1169, 2395}, {4017, 4369}, {4160, 4404}
X(4582) lies on these lines: {8, 4542}, {88, 4358}, {106, 726}, {190, 514}, {522, 3699}, {646, 4391}, {901, 4427}, {1320, 3685}, {2397, 2401}, {2403, 2415}
X(4582) = isotomic conjugate of X(30725)
X(4583) lies on these lines: {190, 789}, {291, 350}, {334, 4013}, {335, 4358}, {514, 668}, {660, 799}, {693, 1978}, {889, 900}, {1921, 3263}, {3676, 4554}
X(4583) = isotomic conjugate of X(659)
X(4584) lies on these lines: {99, 813}, {292, 1931}, {334, 2196}, {335, 2161}, {660, 662}, {666, 2311}, {741, 898}, {805, 932}, {876, 3573}, {1020, 1275}
X(4584) = isogonal conjugate of X(21832)
X(4585) lies on these lines: {2, 6}, {110, 3909}, {163, 662}, {190, 644}, {320, 2323}, {513, 3573}, {692, 3888}, {765, 1026}, {1022, 1023}, {3799, 3908}
X(4586) lies on the Steiner circumellipse and these lines: {99, 163}, {101, 668}, {190, 692}, {290, 1910}, {662, 670}, {664, 1415}, {671, 923}, {870, 1438}, {985, 3227}, {1494, 2159}
X(4586) = isogonal conjugate of X(3250)
X(4586) = isotomic conjugate of X(824)
X(4586) = complement of X(39345)
X(4587) lies on these lines: {3, 1810}, {48, 1811}, {78, 1802}, {100, 101}, {283, 1808}, {906, 1331}, {1332, 1813}, {1415, 2284}, {1793, 2327}, {1809, 2289}
X(4588) lies on these lines: {31, 106}, {81, 759}, {89, 105}, {104, 1621}, {651, 2222}, {692, 901}, {1290, 1633}, {1914, 2384}, {2280, 2291}
X(4588) = isogonal conjugate of X(4777)
X(4588) = barycentric product of circumcircle intercepts of line X(2)X(44)
X(4588) = Ψ(X(1), X(89))
X(4588) = Ψ(X(2), X(44))
X(4589) lies on these lines: {80, 334}, {99, 813}, {291, 2669}, {660, 799}, {668, 2533}, {741, 3510}, {789, 805}, {874, 875}, {3570, 4444}
X(4589) = isogonal conjugate of X(4455)
X(4589) = isotomic conjugate of X(4010)
As P traces the line X(2)X(6), the line X(99)X(110) is the locus of the trilinear pole of the tangent at P to the hyperbola {{A,B,C,X(2),P}}. Theline X(99)X(110) is the radical axis of the circumcircle and anti-Artzt circle. (Randy Hutson, December 10, 2016)
X(4590) is the Brianchon point (perspector) of the inellipse that is the barycentric square of line X(2)X(6). The center of this inellipse is X(620). (Randy Hutson, October 15, 2018)
X(4590) lies on these lines: {99, 523}, {249, 524}, {250, 325}, {385, 3266}, {670, 2966}, {688, 805}, {757, 765}, {2396, 2407}
X(4590) = isogonal conjugate of X(3124)
X(4590) = isotomic conjugate of X(115)
X(4590) = cevapoint of X(2) and X(99)
X(4590) = X(2)-cross conjugate of X(99)
X(4590) = cevapoint of X(6189) and X(6190)
X(4590) = antitomic conjugate of X(35511)
X(4590) = pole wrt polar circle of trilinear polar of X(8754)
X(4590) = X(19)-isoconjugate of X(20975)
X(4590) = X(48)-isoconjugate (polar conjugate)-of-X(8754)
X(4590) = barycentric square of X(99)
X(4591) lies on these lines: {106, 691}, {110, 901}, {249, 4556}, {662, 1019}, {903, 2966}, {2226, 3285}, {2403, 2407}, {2421, 2441}
X(4591) = isogonal conjugate of X(4120)
X(4592) lies on these lines: {63, 3708}, {99, 109}, {110, 1310}, {162, 799}, {163, 662}, {255, 293}, {332, 1795}, {906, 1332}
X(4592) = isotomic conjugate of X(24006)
X(4592) = X(92)-isoconjugate of X(798)
X(4592) = trilinear product X(8115)*X(8116)
As the trilinear product of Steiner circumellipse antipodes, X(4593) lies on conic {{A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75). (Randy Hutson, July 11, 2019)
X(4593) lies on these lines: {83, 1509}, {101, 689}, {163, 799}, {668, 692}, {789, 827}, {923, 3112}, {1415, 4554}, {1928, 2172}
X(4593) = isogonal conjugate of X(2084)
X(4593) = isotomic conjugate of X(8061)
X(4593) = perspector of ABC and tangential triangle, wrt anticevian triangle of X(31), of bianticevian conic of X(1) and X(31)
X(4593) = perspector of ABC and tangential triangle, wrt anticevian triangle of X(75), of bianticevian conic of X(1) and X(75)
X(4593) = trilinear product of intercepts of Steiner circumellipse and line X(2)X(32)
X(4594) lies on these lines: {86, 256}, {190, 4079}, {274, 4459}, {645, 3570}, {789, 805}, {811, 877}, {874, 3903}, {1178, 3253}
X(4594) = isotomic conjugate of X(2533)
X(4595) lies on these lines: {99, 4482}, {100, 932}, {101, 1016}, {190, 646}, {644, 3570}, {664, 3669}, {3699, 3799}
X(4596) lies on these lines: {80, 1268}, {99, 3952}, {291, 1126}, {662, 1018}, {762, 2134}, {763, 3293}, {1414, 4551}
X(4596) = isogonal conjugate of X(4983)
X(4596) = isotomic conjugate of X(30591)
X(4597) lies on the Steiner circumellipse and these lines: {1, 903}, {89, 3227}, {100, 4378}, {190, 1023}, {1121, 4384}, {2163, 3226}, {2320, 2481}
X(4597) = reflection of X(35170) in X(2)
X(4597) = isogonal conjugate of X(4775)
X(4597) = isotomic conjugate of X(4777)
X(4597) = complement of X(39364)
X(4597) = Steiner-circumellipse-antipode of X(35170)
X(4598) lies on these lines: {2, 1977}, {88, 330}, {100, 932}, {649, 1978}, {651, 3570}, {660, 3699}, {673, 2319}
X(4598) = isogonal conjugate of X(20979)
X(4598) = isotomic conjugate of X(3835)
X(4598) = anticomplement of complementary conjugate of X(21191)
X(4598) = X(19)-isoconjugate of X(22090)
X(4599) lies on these lines: {82, 897}, {100, 827}, {110, 660}, {163, 799}, {251, 593}, {689, 825}, {1821, 3112}
X(4599) = isogonal conjugate of X(8061)
X(4599) = trilinear pole of line X(1)X(82)
X(4599) = trilinear product of circumcircle intercepts of line X(2)X(32)
X(4599) = perspector of ABC and tangential triangle, wrt excentral triangle, of bianticevian conic of X(1) and X(31)
Line X(99)X(101) is the locus of trilinear poles of tangents at P to conic {{A,B,C,X(10),P}}, as P moves on the Nagel line. (Randy Hutson, March 21, 2019)
X(4600) lies on these lines: {99, 901}, {190, 892}, {643, 799}, {1016, 1252}, {1331, 1978}, {2396, 2398}
X(4600) = isogonal conjugate of X(3122)
X(4600) = isotomic conjugate of X(3120)
X(4601) lies on these lines: {99, 889}, {645, 666}, {668, 892}, {799, 3257}, {880, 2284}, {2396, 2397}
X(4601) = isogonal conjugate of X(3121)
X(4601) = isotomic conjugate of X(3125)
X(4601) = cevapoint of X(86) and X(190)
As the trilinear product of Steiner circumellipse antipodes, X(4602) lies on conic {{A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75). (Randy Hutson, July 11, 2019)
X(4602) lies on these lines: {99, 789}, {304, 1928}, {310, 3840}, {668, 670}, {689, 831}, {1581, 1925}
X(4602) = isogonal conjugate of X(1924)X(4603) lies on these lines: {81, 893}, {100, 805}, {643, 3573}, {645, 3570}, {662, 2421}, {799, 880}
X(4604) lies on these lines: {6, 88}, {101, 3257}, {190, 1023}, {655, 664}, {673, 2364}, {1001, 1156}
X(4605) lies on these lines: {12, 4092}, {101, 653}, {514, 651}, {664, 1461}, {1018, 1020}, {1426, 2901}
X(4606) lies on these lines: {100, 4069}, {644, 662}, {646, 799}, {651, 1018}, {658, 4552}, {673, 2345}
X(4606) = isogonal conjugate of X(4790)
X(4606) = isotomic conjugate of X(4801)
X(4607) lies on these lines: {88, 239}, {100, 667}, {190, 649}, {660, 875}, {799, 1019}, {3257, 3570}
X(4607) = isogonal conjugate of X(3768)
X(4607) = isotomic conjugate of X(4728)
X(4608) lies on these lines: {423, 2501}, {514, 4024}, {523, 4467}, {693, 4036}, {1171, 2395}, {1268, 3004}
X(4608) = isogonal conjugate of X(35327)
X(4608) = isotomic conjugate of X(4427)
X(4608) = anticomplement of X(4988)
X(4608) = crossdifference of every pair of points on line X(2308)X(20970)
X(4608) = cevapoint of X(i) and X(j) for these {i,j}: {10, 24076}, {512, 6586}, {513, 14838}
X(4609) lies on these lines: {76, 3124}, {99, 689}, {110, 880}, {669, 3222}, {670, 888}
X(4609) = isotomic conjugate of X(669)
X(4609) = isogonal conjugate of X(9426)
X(4609) = cevapoint of Kiepert hyperbola intercepts of de Longchamps line
X(4609) = X(523)-cross conjugate of X(76)
X(4610) lies on these lines: {86, 3120}, {99, 110}, {310, 1790}, {662, 799}, {873, 2185}
X(4610) = isogonal conjugate of X(4079)
X(4610) = isotomic conjugate of X(4024)
X(4610) = anticomplement of X(6627)
Line X(22)X(206) is the tangent to hyperbola {{A,B,C,X(4),X(22)}} at X(22). (Randy Hutson, March 21, 2019)
X(4611) lies on these lines: {32, 620}, {99, 112}, {110, 827}, {127, 315}, {691, 3565}
X(4612) lies on these lines: {21, 4516}, {99, 112}, {110, 931}, {163, 662}, {314, 2193}
X(4612) = X(650)-cross conjugate of X(21)
X(4613) lies on these lines: {1, 335}, {100, 789}, {190, 692}, {660, 3903}, {825, 835}
X(4614) lies on these lines: {99, 3699}, {100, 1414}, {644, 662}, {1010, 4314}
X(4615) lies on these lines: {99, 901}, {799, 3257}, {892, 903}, {2396, 2403}
X(4615) = isogonal conjugate of X(14407)
X(4615) = isotomic conjugate of X(4120)
X(4616) lies on these lines: {99, 934}, {552, 1088}, {658, 662}, {1020, 1275}
X(4616) = isotomic conjugate of X(4524)
X(4617) lies on these lines: {109, 934}, {244, 269}, {479, 1407}, {651, 658}
X(4617) = isogonal conjugate of X(4130)
X(4618) lies on these lines: {1, 679}, {88, 2087}, {903, 4089}, {1022, 1023}
X(4618) = isogonal conjugate of X(3251)
X(4619) lies on these lines: {55, 59}, {57, 2149}, {394, 1252}, {1262, 1407}
X(4620) lies on these lines: {664, 892}, {1262, 1275}, {2396, 2406}
X(4621) lies on these lines: {645, 3807}, {646, 3570}, {3573, 3699}
X(4621) = isotomic conjugate of X(3776)
X(4622) lies on these lines: {88, 1931}, {99, 901}, {662, 1019}
X(4622) = isogonal conjugate of X(4730)
X(4623) lies on these lines: {99, 670}, {662, 799}, {757, 3253}
X(4623) = isotomic conjugate of X(4705)
X(4624) lies on these lines: {658, 4552}, {664, 3952}, {4033, 4554}
X(4625) lies on these lines: {99, 934}, {645, 651}, {664, 670}
X(4625) = isotomic conjugate of X(4041)
X(4626) lies on these lines: {279, 1086}, {651, 658}, {934, 1633}
X(4626) = isogonal conjugate of X(4105)
X(4626) = isotomic conjugate of X(4163)
X(4627) lies on these lines: {110, 3939}, {294, 2303}, {644, 662}
X(4627) = isogonal conjugate of X(4841)
X(4628) lies on these lines: {101, 827}, {292, 1333}, {689, 769}
X(4628) = isogonal conjugate of X(16892)
X(4629) lies on these lines: {101, 4556}, {662, 1018}, {1255, 2161}
X(4629) = isogonal conjugate of X(4988)
X(4630) lies on these lines: {51, 251}, {110, 827}, {1176, 1177}
X(4630) = isogonal conjugate of X(23285)
X(4631) lies on these lines: {99, 931}, {648, 670}, {662, 799}
X(4632) lies on these lines: {99, 3952}, {335, 1255}, {799, 4033}
X(4632) = isotomic conjugate of X(4988)
X(4633) lies on these lines: {99, 3699}, {646, 799}
X(4633) = isogonal conjugate of X(4832)
X(4633) = isotomic conjugate of X(4841)
X(4634) lies on these lines: {670, 4555}, {799, 3257}
X(4634) = isotomic conjugate of X(4730)
X(4635) lies on these lines: {99, 934}, {658, 799}
X(4635) = isotomic conjugate of X(4171)
X(4636) lies on these lines: {110, 351}, {162, 662}
X(4637) lies on these lines: {110, 934}, {658, 662}
X(4637) = isogonal conjugate of X(4171)
X(4638) lies on these lines: {6, 2226}, {1022, 1023}
X(4638) = isogonal conjugate of X(6544)
X(4639) lies on these lines: {660, 799}, {670, 4033}
X(4639) = isotomic conjugate of X(21832)
X(4640) lies on these lines: {1, 3052}, {2, 1155}, {3, 960}, {6, 1707}, {9, 165}, {10, 30}, {21, 65}, {31, 1386}, {35, 72}, {36, 392}, {37, 171}, {38, 902}, {40, 958}, {42, 896}, {43, 44}, {46, 405}, {55, 63}, {57, 1001}, {58, 3931}, {71, 1755}, {100, 210}, {109, 1214}, {190, 3967}, {192, 3769}, {200, 3929}, {227, 1935}, {238, 3752}, {283, 1399}, {284, 4047}, {306, 3712}, {321, 4427}, {333, 3696}, {345, 3416}, {354, 1621}, {442, 1770}, {452, 1788}, {484, 3753}, {516, 2886}, {517, 993}, {524, 4028}, {536, 4362}, {600,5507}, {614, 3246}, {692, 3955}, {901, 2752}, {940, 968}, {956, 3880}, {982, 1279}, {984, 3550}, {986, 1104}, {988, 1191}, {1005, 1776}, {1013, 1748}, {1150, 3706}, {1212, 3496}, {1319, 3877}, {1334, 4447}, {1385, 3878}, {1402, 3286}, {1427, 1758}, {1633, 4220}, {1762, 3198}, {1882, 3559}, {1889, 2355}, {2185, 2651}, {2245, 4199}, {2646, 3869}, {2975, 3057}, {3006, 4450}, {3011, 3782}, {3035, 3452}, {3242, 3749}, {3305, 4413}, {3306, 3848}, {3419, 4302}, {3555, 3746}, {3681, 3689}, {3703, 3977}, {3739, 3980}, {3748, 3873}, {3811, 3927}, {3816, 3911}, {3840, 4432}, {3971, 4434}, {4005, 4420}
X(4640) = complement of X(1836)
X(4640) = anticomplement of X(3838)
X(4640) = complementary conjugate of complement of the X(37741)
X(4641) lies on these lines: {1, 3683}, {2, 44}, {6, 63}, {9, 940}, {10, 540}, {31, 518}, {37, 81}, {38, 1386}, {42, 896}, {43, 1155}, {55, 1707}, {57, 1122}, {58, 72}, {65, 1046}, {78, 4252}, {101, 1412}, {141, 4001}, {171, 210}, {172, 1812}, {190, 1999}, {191, 3931}, {193, 345}, {209, 511}, {218, 222}, {228, 3286}, {238, 354}, {306, 524}, {333, 894}, {386, 3916}, {394, 2911}, {527, 3782}, {536, 3187}, {553, 3008}, {580, 1071}, {595, 3555}, {614, 3999}, {651, 1427}, {726, 3791}, {748, 3742}, {750, 3740}, {942, 1724}, {960, 1468}, {971, 1754}, {984, 3745}, {1104, 3868}, {1211, 4416}, {1214, 2003}, {1279, 3873}, {1333, 3998}, {1407, 1445}, {1790, 2174}, {1859, 1957}, {1864, 1936}, {1931, 2185}, {2194, 3955}, {2999, 3928}, {3052, 3870}, {3210, 3759}, {3218, 3752}, {3293, 3579}, {3550, 3689}, {3629, 3977}, {3687, 4274}, {3693, 3719}, {3696, 4418}, {3706, 3923}, {3712, 4028}, {3896, 4427}, {4090, 4434}
X(4641) = {X(37),X(81)}-harmonic conjugate of X(37595)
X(4642) lies on these lines: {1, 88}, {4, 2181}, {8, 38}, {10, 321}, {12, 3120}, {31, 40}, {37, 3698}, {39, 2170}, {42, 65}, {43, 3869}, {46, 1468}, {55, 3924}, {58, 484}, {72, 3214}, {145, 982}, {201, 1834}, {517, 1193}, {519, 3670}, {612, 1706}, {614, 1697}, {748, 1722}, {758, 3293}, {826, 4041}, {899, 960}, {902, 1104}, {958, 4414}, {978, 3877}, {984, 1278}, {988, 3872}, {1125, 1739}, {1201, 3057}, {1203, 3245}, {1220, 4418}, {1393, 2099}, {1500, 3125}, {1575, 3727}, {1828, 2347}, {1837, 2310}, {1953, 4261}, {1962, 3753}, {2092, 2171}, {2136, 3677}, {2276, 3959}, {3216, 3878}, {3241, 3976}, {3244, 3953}, {3621, 4392}, {3696, 3728}, {3702, 3831}, {3714, 4365}, {3720, 3812}, {3721, 3930}, {3743, 3918}, {3755, 3778}, {3893, 4003}, {3913, 3938}, {3949, 4016}
X(4643) lies on these lines: {1, 524}, {2, 44}, {6, 4357}, {7, 966}, {8, 536}, {9, 141}, {10, 527}, {37, 69}, {38, 3764}, {45, 599}, {63, 1211}, {72, 4259}, {75, 1654}, {144, 2345}, {190, 3661}, {192, 319}, {193, 1100}, {213, 4503}, {239, 4389}, {307, 965}, {333, 3772}, {344, 3620}, {391, 4000}, {513, 3789}, {545, 3679}, {594, 3729}, {742, 984}, {752, 4407}, {940, 4001}, {1086, 4384}, {1150, 4396}, {1376, 4104}, {1449, 3629}, {1698, 4472}, {1743, 3589}, {2321, 4445}, {3008, 3707}, {3187, 3578}, {3219, 4376}, {3242, 3883}, {3247, 3630}, {3617, 4454}, {3631, 3731}, {3663, 3686}, {3666, 4277}, {3751, 4026}, {3775, 3923}, {3914, 4042}, {4007, 4478}, {4034, 4399}, {4371, 4452}, {4517, 4553}
X(4643) = complement of X(4644)
X(4643) = anticomplement of X(4670)
X(4644) lies on these lines: {1, 527}, {2, 44}, {6, 7}, {8, 524}, {9, 3664}, {10, 4470}, {31, 3475}, {37, 144}, {42, 3000}, {57, 2183}, {58, 3487}, {69, 894}, {72, 4340}, {75, 193}, {77, 3553}, {86, 1778}, {142, 1743}, {145, 536}, {172, 348}, {239, 1992}, {273, 3087}, {278, 2003}, {329, 940}, {344, 4473}, {345, 4376}, {391, 3739}, {513, 1002}, {518, 4307}, {545, 3241}, {553, 2999}, {966, 4416}, {971, 3332}, {1014, 2178}, {1100, 3672}, {1386, 4310}, {1419, 3668}, {1423, 2260}, {1449, 3663}, {1468, 1935}, {2550, 3751}, {2650, 3486}, {3242, 4344}, {3616, 4364}, {3618, 3662}, {3629, 4361}, {3630, 4445}, {3729, 3879}, {3755, 4312}, {4393, 4440}
X(4644) = anticomplement of X(4643)
X(4645) lies on these lines: {1, 2896}, {2, 31}, {6, 4429}, {7, 8}, {10, 894}, {44, 966}, {57, 3705}, {71, 1761}, {78, 2263}, {79, 1089}, {86, 4026}, {100, 851}, {142, 3883}, {145, 4310}, {190, 3932}, {239, 1738}, {291, 2227}, {312, 1836}, {318, 1892}, {333, 3925}, {334, 1966}, {335, 740}, {345, 3474}, {354, 4514}, {511, 3888}, {513, 2517}, {516, 3685}, {527, 3717}, {660, 1821}, {726, 4440}, {1279, 3616}, {1281, 3509}, {1284, 4447}, {1376, 4417}, {1457, 1818}, {1621, 4450}, {1921, 2113}, {1961, 4425}, {1999, 3914}, {3006, 3218}, {3550, 3771}, {3729, 3790}, {3755, 3879}, {3769, 3772}, {3781, 3869}, {3872, 4327}, {4032, 4451}
X(4645) = isogonal conjugate of X(8852)
X(4645) = {X(7),X(8)}-harmonic conjugate of X(24349)
X(4646) lies on these lines: {1, 474}, {4, 4277}, {6, 40}, {8, 3666}, {10, 37}, {12, 3914}, {42, 65}, {43, 960}, {55, 1104}, {58, 3579}, {72, 3293}, {165, 4252}, {192, 341}, {210, 2292}, {386, 517}, {387, 1108}, {392, 3216}, {516, 4263}, {518, 986}, {536, 4385}, {614, 3303}, {756, 3983}, {976, 3689}, {1001, 1722}, {1155, 1468}, {1191, 1697}, {1193, 3057}, {1212, 2276}, {1279, 3295}, {1385, 4256}, {1418, 3339}, {1453, 3052}, {1465, 3340}, {1869, 1880}, {1902, 3192}, {2177, 3924}, {3085, 3772}, {3175, 3701}, {3240, 3869}, {3555, 3670}, {3698, 4433}, {3744, 3871}, {3889, 3999}
X(4647) lies on these lines: {1, 75}, {2, 3743}, {8, 79}, {10, 321}, {37, 4099}, {58, 4418}, {63, 1710}, {72, 3696}, {191, 333}, {239, 1203}, {312, 1698}, {442, 3704}, {519, 2650}, {523, 764}, {594, 3954}, {690, 3762}, {726, 3728}, {942, 3706}, {1109, 3626}, {1125, 1962}, {1215, 3293}, {1269, 3775}, {1441, 3671}, {1724, 3923}, {1739, 3831}, {1839, 3686}, {2294, 2321}, {2901, 4365}, {3120, 3454}, {3216, 3725}, {3244, 3902}, {3634, 4358}, {3647, 4427}, {3649, 4046}, {3670, 3741}, {3679, 4385}, {3695, 3925}, {3697, 3967}, {3714, 3753}, {3746, 3757}, {3842, 4043}, {3927, 4042}, {3952, 4015}
X(4647) = anticomplement of X(3743)
X(4647) = reflection of X(4065) in X(1125)
X(4647) = anticomplement of X(4065) wrt incentral triangle
X(4647) = QA-P5 (Isotomic Center) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/26-qa-p5.html)
X(4648) lies on these lines: {1, 142}, {2, 6}, {3, 3332}, {4, 991}, {7, 37}, {8, 3739}, {9, 3664}, {45, 144}, {57, 71}, {77, 948}, {145, 4361}, {226, 269}, {344, 894}, {346, 4363}, {354, 3779}, {388, 1458}, {405, 4340}, {497, 2293}, {519, 4371}, {527, 3731}, {612, 3475}, {750, 1253}, {942, 3781}, {968, 3474}, {975, 3487}, {1001, 4307}, {1038, 2263}, {1058, 2140}, {1086, 3672}, {1125, 4349}, {1279, 3616}, {1449, 3008}, {1834, 4208}, {2114, 4466}, {2345, 3912}, {3019, 3524}, {3241, 4402}, {3247, 3663}, {3617, 4445}, {3621, 4399}, {3623, 4395}, {3879, 4384}, {3943, 4461}
X(4648) = isotomic conjugate of X(32022)
X(4648) = complement of X(391)
X(4648) = anticomplement of X(17259)
X(4648) = {X(2),X(6)}-harmonic conjugate of X(37650)
X(4649) lies on these lines: {1, 6}, {2, 3775}, {10, 86}, {31, 3750}, {35, 3286}, {42, 81}, {43, 940}, {58, 1918}, {87, 2334}, {145, 4527}, {190, 3993}, {210, 1961}, {292, 3774}, {320, 3821}, {354, 1051}, {386, 2274}, {524, 4026}, {537, 4366}, {726, 4360}, {740, 894}, {750, 3240}, {757, 1326}, {985, 1002}, {1046, 3931}, {1206, 3051}, {1215, 1999}, {1308, 2711}, {1429, 1469}, {1468, 2209}, {1509, 2669}, {1621, 2308}, {1738, 3664}, {1962, 3219}, {3017, 3822}, {3744, 3979}, {3745, 3961}, {3757, 3791}, {3758, 3923}, {3792, 4260}, {3896, 4418}
X(4649) = isogonal conjugate of X(30571)
X(4649) = anticomplement of X(3775)
X(4650) lies on these lines: {1, 3052}, {2, 896}, {3, 1046}, {31, 982}, {36, 3185}, {43, 1155}, {55, 3979}, {57, 238}, {58, 986}, {63, 171}, {81, 4414}, {165, 1350}, {222, 1758}, {320, 3771}, {333, 3980}, {518, 3550}, {579, 1755}, {595, 3976}, {726, 3769}, {750, 3219}, {752, 3705}, {758, 4257}, {846, 940}, {902, 3873}, {910, 1743}, {920, 3075}, {968, 4038}, {1054, 4383}, {1150, 4418}, {1376, 1757}, {1430, 1748}, {1454, 1935}, {1724, 3336}, {1754, 1768}, {2163, 3899}, {2650, 4189}, {3210, 3791}, {3722, 4430}, {3784, 3792}
X(4651) lies on these lines: {1, 2}, {37, 3896}, {71, 391}, {75, 3681}, {100, 333}, {209, 2550}, {210, 321}, {310, 668}, {313, 4441}, {392, 3902}, {518, 4113}, {594, 2238}, {672, 3686}, {740, 756}, {850, 4524}, {956, 4191}, {1089, 4015}, {1150, 1376}, {1621, 3996}, {1757, 4418}, {1962, 3842}, {2886, 4023}, {2975, 4210}, {3219, 4427}, {3305, 3886}, {3695, 4204}, {3697, 3701}, {3706, 3740}, {3714, 3983}, {3891, 4361}, {3914, 4104}, {3925, 3936}, {3932, 3969}, {3956, 3992}, {3963, 4111}, {3971, 4365}, {3994, 4096}, {4415, 4442}
X(4651) = anticomplement of X(3720)
X(4652) lies on these lines: {1, 89}, {2, 4292}, {3, 63}, {8, 165}, {9, 404}, {10, 4190}, {21, 57}, {31, 988}, {35, 3870}, {40, 2975}, {46, 993}, {77, 283}, {84, 411}, {191, 997}, {329, 3523}, {405, 3306}, {474, 3305}, {550, 3419}, {631, 908}, {758, 3612}, {936, 3219}, {956, 3579}, {958, 1155}, {1193, 1707}, {1376, 3983}, {1420, 3877}, {1445, 1466}, {1621, 3333}, {1758, 4320}, {2478, 3911}, {3340, 3897}, {3361, 3616}, {3576, 3869}, {3601, 3868}, {3624, 3648}, {3666, 4252}, {3876, 3929}
Let A'B'C' be the 2nd circumperp triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B", C" cyclically. The lines AA", BB", CC" concur in X(4653). (Randy Hutson, June 27, 2018)
X(4653) lies on these lines: {1, 21}, {2, 4256}, {10, 1043}, {27, 4304}, {28, 33}, {29, 1785}, {35, 4225}, {36, 3720}, {37, 101}, {45, 3940}, {55, 859}, {56, 4278}, {86, 99}, {333, 519}, {386, 405}, {501, 2360}, {581, 3560}, {612, 4228}, {940, 4257}, {958, 4281}, {990, 3576}, {991, 1012}, {995, 1001}, {999, 3286}, {1010, 1125}, {1014, 4328}, {1054, 3833}, {1319, 1412}, {1333, 2242}, {1388, 1408}, {1449, 1778}, {1717, 3612}, {2074, 2299}, {2177, 3679}, {2287, 3731}, {2303, 3247}, {3295, 4267}
X(4653) = {X(1),X(21)}-harmonic conjugate of X(58)
X(4653) = trilinear pole, wrt 2nd circumperp triangle, of line X(36)X(238)
X(4654) lies on these lines: {1, 30}, {2, 7}, {12, 3339}, {46, 3584}, {55, 4312}, {56, 4355}, {65, 3679}, {85, 4102}, {196, 1855}, {223, 4328}, {278, 1419}, {354, 971}, {376, 3487}, {381, 942}, {388, 519}, {428, 1892}, {481, 1659}, {495, 2093}, {516, 3475}, {551, 1420}, {938, 3839}, {950, 3543}, {1086, 2999}, {1210, 3545}, {1432, 4496}, {1697, 4295}, {1698, 3715}, {1788, 3828}, {2078, 4428}, {2895, 4034}, {3175, 3970}, {3243, 3434}, {3256, 4421}, {3338, 3582}, {3586, 3830}, {3746, 4338}, {3951, 4197}
X(4654) = 2nd-extouch-to-intouch similarity image of X(2)
X(4655) lies on these lines: {1, 320}, {2, 896}, {3, 2792}, {6, 3821}, {8, 537}, {9, 3836}, {10, 527}, {57, 3847}, {63, 2887}, {69, 740}, {141, 3923}, {238, 3662}, {516, 1350}, {599, 2796}, {726, 3416}, {758, 4259}, {940, 4425}, {982, 4388}, {986, 1330}, {1125, 4252}, {1150, 3120}, {1211, 3980}, {1281, 3314}, {1738, 4416}, {1757, 4429}, {1836, 3741}, {2245, 3454}, {2835, 3878}, {3670, 3764}, {3729, 3773}, {3751, 4085}, {3775, 4312}, {3782, 4362}, {3792, 3869}, {3914, 4001}, {3936, 4414}, {3938, 4450}
Let A'B'C' be the Danneels-Bevan homothetic triangle (see X(2999)). Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(4656). (Randy Hutson, June 7, 2019)
X(4656) lies on these lines: {1, 329}, {2, 2415}, {10, 321}, {37, 226}, {38, 1736}, {42, 4356}, {45, 3772}, {57, 4419}, {92, 1785}, {142, 3782}, {192, 3687}, {210, 3755}, {306, 3950}, {312, 4357}, {516, 612}, {527, 940}, {614, 4353}, {740, 4061}, {975, 4292}, {976, 4314}, {1125, 4011}, {1211, 2321}, {1999, 4416}, {2887, 4078}, {2999, 3672}, {3008, 3305}, {3452, 3666}, {3465, 4304}, {3817, 3989}, {3936, 4098}, {3946, 4383}, {3967, 4026}, {3969, 4072}, {3993, 4028}, {4029, 4035}, {4085, 4096}
X(4657) lies on these lines: {1, 141}, {2, 37}, {3, 142}, {6, 4357}, {9, 3589}, {10, 3946}, {44, 3618}, {57, 1761}, {63, 583}, {69, 1100}, {86, 1333}, {319, 4393}, {348, 1418}, {519, 4445}, {524, 1449}, {594, 3875}, {597, 1743}, {599, 3879}, {894, 4389}, {940, 2214}, {1211, 4272}, {1213, 4384}, {1279, 3616}, {1654, 3759}, {1698, 3932}, {2321, 4021}, {3617, 4371}, {3619, 3723}, {3632, 4478}, {3634, 4078}, {3661, 4360}, {3663, 4363}, {3679, 4399}, {3731, 4422}, {3763, 3912}, {4419, 4488}
X(4657) = complement of X(2345)
X(4657) = anticomplement of X(17385)
X(4657) = {X(2),X(75)}-harmonic conjugate of X(17303)
X(4657) = {X(17301),X(17303)}-harmonic conjugate of X(75)
X(4657) = {X(17399),X(17400)}-harmonic conjugate of X(75)
X(4657) = perspector of Gemini triangle 22 and cross-triangle of Gemini triangles 21 and 22
X(4658) lies on these lines: {1, 21}, {10, 86}, {55, 4278}, {56, 4276}, {65, 1412}, {106, 931}, {284, 501}, {333, 1125}, {354, 2360}, {386, 474}, {387, 3945}, {442, 3017}, {519, 1010}, {524, 4205}, {859, 3304}, {894, 2901}, {986, 1326}, {995, 4281}, {999, 4267}, {1014, 3339}, {1043, 3244}, {1051, 3833}, {1203, 3720}, {1323, 1434}, {1333, 2241}, {1408, 2099}, {1449, 2303}, {1778, 3247}, {1961, 3678}, {2299, 2906}, {3286, 3295}, {3736, 3913}, {3746, 4184}, {4188, 4256}
X(4659) lies on these lines: {1, 536}, {2, 1266}, {7, 2321}, {8, 527}, {9, 75}, {10, 4419}, {57, 321}, {69, 4007}, {76, 4494}, {86, 3644}, {142, 346}, {144, 3686}, {192, 3247}, {391, 4488}, {524, 3632}, {545, 3679}, {594, 4409}, {742, 3751}, {894, 1278}, {1125, 4470}, {1698, 4364}, {1706, 4385}, {1743, 4361}, {1909, 4050}, {2345, 3663}, {3243, 3886}, {3339, 3714}, {3416, 4312}, {3624, 4472}, {3661, 4440}, {3731, 3739}, {3749, 4376}, {3946, 4452}, {4034, 4416}
X(4660) lies on these lines: {1, 2896}, {2, 902}, {4, 9}, {6, 752}, {8, 726}, {31, 4450}, {43, 4388}, {55, 2887}, {69, 519}, {141, 528}, {148, 1654}, {238, 4429}, {329, 4090}, {497, 3840}, {535, 996}, {612, 4425}, {740, 3416}, {758, 3779}, {982, 4514}, {1001, 3836}, {1215, 1836}, {1376, 3847}, {1479, 3831}, {1738, 3883}, {2177, 3936}, {3006, 4414}, {3244, 4353}, {3434, 3741}, {3781, 3878}, {3782, 4030}, {3827, 4523}, {3914, 4362}, {3915, 4202}, {3974, 4135}
X(4661) lies on these lines: {1, 4134}, {2, 210}, {8, 79}, {9, 3957}, {10, 3894}, {38, 3240}, {43, 4392}, {63, 3158}, {72, 145}, {149, 329}, {200, 3218}, {517, 3543}, {519, 3899}, {942, 3921}, {960, 3623}, {984, 1962}, {1757, 3938}, {1898, 3621}, {2810, 2979}, {3219, 3870}, {3241, 3898}, {3243, 3305}, {3555, 3622}, {3616, 3678}, {3617, 3753}, {3626, 3901}, {3632, 4067}, {3635, 4537}, {3679, 3919}, {3751, 3920}, {3811, 4189}, {3871, 3927}, {3874, 3956}, {4188, 4420}
X(4661) = anticomplement of X(3873)
X(4662) lies on these lines: {1, 3697}, {2, 3983}, {8, 210}, {9, 3913}, {10, 141}, {21, 3689}, {65, 3617}, {72, 3679}, {78, 3711}, {200, 958}, {392, 3632}, {517, 546}, {519, 4015}, {1104, 3961}, {1125, 3956}, {1698, 3555}, {1722, 3242}, {2099, 3984}, {2646, 4420}, {2802, 4547}, {3214, 3666}, {3303, 3305}, {3452, 3813}, {3634, 3848}, {3683, 3871}, {3686, 4095}, {3691, 3693}, {3696, 4385}, {3698, 3868}, {3704, 3717}, {3710, 4126}, {3828, 3881}, {3869, 4005}, {3901, 4004}
X(4662) = complement of X(34791)
X(4663) lies on these lines: {1, 6}, {8, 1992}, {10, 524}, {42, 896}, {58, 3939}, {65, 651}, {69, 3844}, {81, 210}, {141, 3634}, {193, 3416}, {511, 3579}, {517, 576}, {519, 4527}, {575, 1385}, {597, 1125}, {599, 1698}, {749, 1469}, {894, 3696}, {940, 3740}, {1002, 2348}, {1126, 3931}, {1155, 3240}, {1999, 3967}, {2234, 3214}, {2235, 3780}, {2308, 3744}, {3244, 4432}, {3626, 3629}, {3664, 3826}, {3681, 3745}, {3742, 4383}, {3879, 3932}, {4026, 4416}
X(4664) lies on these lines: {1, 190}, {2, 37}, {9, 3759}, {44, 4393}, {45, 239}, {85, 4552}, {86, 3247}, {142, 4398}, {320, 4419}, {335, 545}, {341, 3931}, {518, 1992}, {519, 751}, {551, 726}, {597, 4370}, {599, 742}, {714, 1962}, {740, 3679}, {846, 3769}, {872, 4096}, {1500, 4033}, {2177, 3570}, {3161, 3618}, {3661, 3943}, {3663, 4098}, {3717, 4356}, {3731, 3875}, {3743, 4385}, {3790, 4026}, {3912, 4029}, {3950, 4357}, {4078, 4429}
X(4664) = reflection of X(2) in X(37)
X(4664) = reflection of X(4688) in X(4755)
X(4664) = isotomic conjugate of X(36871)
X(4664) = complement of X(4740)
X(4664) = anticomplement of X(4688)
X(4665) lies on these lines: {1, 4472}, {2, 3943}, {6, 4399}, {7, 3631}, {8, 524}, {10, 536}, {37, 4431}, {69, 4478}, {75, 141}, {142, 4058}, {145, 4470}, {192, 1213}, {239, 597}, {319, 3630}, {321, 3264}, {519, 4499}, {523, 4444}, {527, 3626}, {545, 3679}, {726, 4407}, {742, 3696}, {894, 3629}, {966, 4461}, {1125, 4527}, {2321, 3739}, {2345, 3589}, {3617, 4419}, {3664, 4060}, {3773, 3826}, {3846, 4078}, {4030, 4376}, {4384, 4422}
X(4665) = complement of X(17318)
X(4666) lies on these lines: {1, 2}, {9, 3873}, {21, 3333}, {55, 3306}, {57, 1621}, {63, 354}, {86, 2191}, {142, 3434}, {461, 1870}, {518, 3305}, {748, 3751}, {750, 3749}, {908, 3475}, {940, 1279}, {968, 982}, {1004, 2646}, {1005, 1420}, {1013, 1435}, {1088, 4350}, {1155, 4428}, {1376, 3748}, {3218, 4512}, {3243, 3681}, {3303, 3812}, {3315, 3677}, {3340, 3890}, {3361, 4189}, {3753, 3895}, {3848, 4413}, {3874, 3951}, {3886, 4359}, {4190, 4314}
X(4667) lies on these lines: {1, 527}, {2, 3707}, {6, 142}, {7, 1419}, {9, 3945}, {10, 524}, {57, 4266}, {81, 226}, {86, 4416}, {144, 3247}, {171, 3939}, {190, 4029}, {193, 3686}, {354, 3271}, {518, 4349}, {519, 4363}, {536, 3244}, {551, 3246}, {597, 3834}, {894, 2321}, {940, 3452}, {1100, 3663}, {1266, 4393}, {1278, 4464}, {1992, 4384}, {2284, 3997}, {3241, 4454}, {3243, 4344}, {3629, 3739}, {3679, 4470}, {3758, 3912}
X(4668) lies on these lines: {1, 2}, {40, 1657}, {165, 548}, {355, 3627}, {388, 4114}, {391, 4058}, {517, 3843}, {518, 4004}, {594, 1743}, {966, 4029}, {1482, 3711}, {1706, 3336}, {2802, 3876}, {3245, 3927}, {3294, 4050}, {3416, 3630}, {3421, 3585}, {3550, 4042}, {3678, 3899}, {3681, 4127}, {3686, 3973}, {3697, 3880}, {3731, 3943}, {3754, 3894}, {3834, 4445}, {3869, 3988}, {3873, 3918}, {3877, 4015}, {3878, 4547}, {3886, 4535}, {3898, 4540}
X(4668) = {X(8),X(10)}-harmonic conjugate of X(3632)
X(4669) lies on these lines: {1, 2}, {6, 4545}, {142, 4478}, {210, 2802}, {354, 3968}, {355, 3830}, {381, 4301}, {392, 3956}, {515, 3534}, {517, 3845}, {518, 3919}, {537, 3696}, {966, 4098}, {993, 4421}, {3036, 3452}, {3057, 4015}, {3555, 3918}, {3656, 3817}, {3681, 4525}, {3686, 4058}, {3697, 3884}, {3698, 3881}, {3707, 4370}, {3740, 3898}, {3814, 3829}, {3869, 4537}, {3902, 3992}, {3950, 4060}, {4007, 4072}, {4090, 4113}
X(4669) = {X(2),X(8)}-harmonic conjugate of X(4677)
X(4669) = {X(8),X(10)}-harmonic conjugate of X(3625)
X(4669) = center of inverse-in-incircle-of-Spieker-circle
X(4670) lies on these lines: {1, 536}, {2, 44}, {6, 3739}, {8, 4470}, {10, 524}, {37, 86}, {42, 2234}, {75, 1100}, {76, 4410}, {141, 3664}, {142, 3589}, {192, 3723}, {513, 875}, {519, 4499}, {527, 1125}, {545, 551}, {584, 1958}, {594, 3879}, {597, 3008}, {742, 1386}, {1212, 1944}, {1213, 4416}, {1449, 4361}, {1909, 4377}, {1961, 3967}, {2345, 3945}, {3616, 4419}, {3622, 4454}, {3629, 3686}, {3666, 4376}
X(4670) = complement of X(4643)
X(4670) = anticomplement of X(4708)
X(4671) lies on these lines: {1, 4066}, {2, 37}, {8, 80}, {43, 4365}, {76, 1978}, {145, 3702}, {190, 1150}, {329, 2893}, {518, 4519}, {726, 4392}, {740, 3240}, {908, 2321}, {984, 3994}, {1621, 4387}, {2476, 3695}, {3006, 3790}, {3218, 3729}, {3434, 3974}, {3617, 3701}, {3661, 4044}, {3679, 4125}, {3681, 3706}, {3696, 4009}, {3714, 3869}, {3741, 4135}, {3886, 3935}, {3912, 4054}, {3969, 4417}, {4429, 4442}
X(4671) = isotomic conjugate of X(89)
X(4671) = anticomplement of X(4850)
X(4671) = complement of polar conjugate of isogonal conjugate of X(23170)
X(4671) = trilinear pole of line X(4770)X(4777) (the perspectrix of ABC and Gemini triangle 20)
X(4672) lies on these lines: {1, 190}, {2, 896}, {5, 2792}, {6, 740}, {9, 3842}, {10, 44}, {31, 1215}, {213, 2235}, {238, 894}, {256, 4283}, {321, 2308}, {516, 4085}, {519, 4527}, {527, 1125}, {575, 2783}, {597, 2796}, {612, 4096}, {726, 1386}, {730, 3997}, {940, 4011}, {1100, 3993}, {1281, 3329}, {2234, 3216}, {2835, 3754}, {3589, 3821}, {3745, 3971}, {3775, 4416}, {3980, 4383}, {4078, 4349}
X(4673) lies on these lines: {1, 75}, {8, 210}, {69, 962}, {72, 4043}, {76, 4301}, {78, 3996}, {85, 4441}, {145, 321}, {239, 1191}, {320, 4295}, {346, 1212}, {519, 4066}, {937, 1222}, {946, 4417}, {958, 3685}, {1089, 3632}, {1219, 4461}, {1441, 4323}, {1616, 4361}, {1621, 1792}, {2321, 3061}, {3303, 3757}, {3596, 4342}, {3617, 4358}, {3622, 4359}, {3704, 3705}, {3718, 3883}, {4050, 4095}
Let A'B'C' be the cevian triangle, and let A"B"C" be the circumcevian triangle of X(1). The four points of intersection of the circumcircles of BA'I, CA'I with the side lines A"C", A"B", respectively, lie on a circle Oa; define Ob and Oc cyclically. Then the triangles ABC and OaObOc are orthologic, with orthology centers the X(3) and X(4674). (Angel Montesdeoca, April 14, 2019)
X(4674) lies on these lines: {1, 88}, {8, 596}, {10, 3120}, {19, 1743}, {37, 1018}, {40, 2218}, {42, 3919}, {43, 994}, {46, 2217}, {57, 1417}, {65, 3293}, {75, 537}, {80, 900}, {191, 1247}, {238, 3245}, {267, 1046}, {291, 876}, {484, 759}, {517, 1739}, {756, 3968}, {897, 1757}, {1086, 1145}, {1449, 2214}, {2153, 3179}, {2292, 3918}, {2640, 2948}, {3214, 4084}, {3922, 3931}
X(4674) = isotomic conjugate of X(30939)
X(4675) lies on these lines: {1, 528}, {2, 44}, {6, 142}, {7, 37}, {10, 599}, {45, 527}, {57, 2245}, {69, 3739}, {86, 1333}, {141, 4472}, {226, 1407}, {244, 3764}, {524, 4384}, {603, 4466}, {940, 3772}, {942, 4259}, {1100, 3945}, {1104, 4340}, {1125, 4252}, {1279, 4307}, {1836, 3000}, {2999, 4285}, {3672, 3723}, {3751, 3826}, {3752, 4277}, {3879, 4361}, {3912, 4363}
X(4676) lies on these lines: {1, 190}, {2, 1155}, {6, 3685}, {8, 44}, {10, 598}, {31, 312}, {75, 238}, {171, 4011}, {192, 1386}, {344, 4307}, {404, 1633}, {516, 4429}, {595, 4385}, {740, 3759}, {748, 4418}, {894, 1001}, {960, 4195}, {978, 2234}, {997, 4234}, {1125, 4389}, {1265, 4339}, {1743, 3886}, {1999, 4387}, {2176, 2235}, {3161, 4344}, {3185, 4203}, {3616, 4419}
X(4677) lies on these lines: {1, 2}, {40, 3534}, {165, 952}, {319, 4398}, {355, 3845}, {484, 3928}, {514, 4543}, {517, 3830}, {668, 4479}, {956, 4421}, {966, 4545}, {1483, 3653}, {1706, 3337}, {1743, 4007}, {2321, 3973}, {2802, 3681}, {3421, 3583}, {3678, 3885}, {3731, 4034}, {3839, 4301}, {3869, 4536}, {3880, 4539}, {3889, 3918}, {3890, 4015}, {3902, 4487}, {3919, 4430}, {4006, 4051}
X(4677) = {X(2),X(8)}-harmonic conjugate of X(4669)
X(4677) = center of inverse-in-Spieker-circle-of-incircle
X(4678) lies on these lines: {1, 2}, {149, 2551}, {355, 3146}, {391, 594}, {517, 3832}, {518, 3922}, {952, 3523}, {956, 4188}, {966, 3943}, {1278, 3696}, {1449, 4545}, {1483, 3525}, {1654, 4461}, {1706, 3218}, {2136, 3305}, {2475, 3421}, {3161, 4058}, {3681, 3962}, {3697, 3877}, {3698, 3873}, {3740, 3890}, {3868, 4004}, {3869, 4005}, {3880, 3983}, {4007, 4029}, {4389, 4452}
X(4678) = {X(8),X(10)}-harmonic conjugate of X(145)
X(4679) lies on these lines: {2, 1155}, {5, 40}, {8, 210}, {9, 11}, {55, 3452}, {56, 226}, {63, 3816}, {200, 3058}, {329, 354}, {390, 3689}, {452, 2646}, {516, 4413}, {614, 4415}, {748, 3772}, {908, 1001}, {938, 3962}, {962, 3698}, {1985, 2183}, {2886, 3305}, {3416, 4358}, {3434, 3740}, {3475, 3622}, {3687, 4387}, {3847, 4011}, {3886, 4023}, {4003, 4419}
X(4680) lies on these lines: {1, 977}, {4, 1089}, {8, 79}, {10, 31}, {30, 3703}, {36, 3705}, {41, 4153}, {69, 1111}, {312, 3583}, {315, 1930}, {319, 3761}, {320, 3894}, {345, 4302}, {495, 4030}, {519, 3891}, {626, 4372}, {674, 3416}, {744, 984}, {752, 1757}, {754, 4376}, {976, 3454}, {993, 3006}, {1759, 4136}, {1839, 2321}, {2835, 3421}, {3585, 4385}
X(4681) lies on these lines: {2, 37}, {44, 4360}, {45, 3875}, {141, 3950}, {142, 4098}, {190, 1100}, {518, 3244}, {545, 3664}, {726, 3636}, {740, 3626}, {742, 3631}, {894, 3723}, {984, 3632}, {1213, 4431}, {2321, 4364}, {2325, 3589}, {2667, 4117}, {3247, 4363}, {3663, 3834}, {3706, 3989}, {3731, 4361}, {3823, 4078}, {3943, 4357}, {3946, 4422}, {3982, 4032}
X(4681) = complement of X(4686)
X(4681) = anticomplement of X(4739)
X(4682) lies on these lines: {1, 474}, {2, 1386}, {6, 3740}, {37, 171}, {43, 1100}, {45, 1707}, {81, 210}, {165, 3247}, {354, 3920}, {518, 612}, {524, 4104}, {536, 3980}, {614, 3848}, {750, 3666}, {894, 3967}, {960, 975}, {1010, 3714}, {1999, 3696}, {3175, 4418}, {3246, 4423}, {3452, 4349}, {3579, 3743}, {3720, 3744}, {3739, 4362}, {3961, 4038}
X(4683) lies on these lines: {1, 540}, {2, 896}, {10, 79}, {38, 256}, {63, 1756}, {81, 4425}, {191, 3454}, {319, 4365}, {320, 3720}, {333, 3120}, {599, 4387}, {740, 2895}, {748, 3662}, {752, 3920}, {846, 3936}, {1150, 3944}, {1211, 4418}, {1330, 2292}, {2792, 4220}, {2887, 3219}, {3218, 3847}, {3578, 4442}, {3914, 4416}, {3961, 4450}, {4414, 4417}
X(4684) lies on these lines: {1, 69}, {7, 3886}, {8, 142}, {145, 3662}, {306, 3873}, {320, 516}, {354, 3687}, {391, 1449}, {518, 3717}, {519, 1738}, {524, 1279}, {527, 3685}, {579, 3692}, {740, 1266}, {982, 4028}, {1001, 4416}, {1043, 4298}, {1621, 4001}, {3244, 3821}, {3555, 4260}, {3705, 4035}, {3875, 4310}, {4353, 4360}, {4356, 4389}
X(4685) lies on these lines: {1, 2}, {209, 752}, {210, 740}, {238, 3996}, {313, 4479}, {321, 4090}, {553, 1469}, {726, 3681}, {756, 3896}, {1211, 4085}, {1215, 3696}, {1575, 4541}, {2238, 2321}, {2276, 3686}, {2901, 4015}, {3666, 4113}, {3751, 3980}, {3755, 4104}, {3773, 4046}, {3847, 4023}, {3886, 4011}, {3952, 4135}, {4082, 4133}, {4126, 4439}
X(4686) lies on these lines: {2, 37}, {44, 3729}, {141, 1266}, {142, 3943}, {144, 4371}, {244, 4519}, {319, 4440}, {354, 4365}, {518, 3632}, {545, 4399}, {594, 3663}, {599, 4007}, {726, 3626}, {740, 3244}, {742, 3629}, {984, 3987}, {1086, 2321}, {1100, 3875}, {1269, 4377}, {3661, 4398}, {3790, 3823}, {4031, 4032}, {4405, 4480}
X(4686) = complement of X(3644)
X(4686) = anticomplement of X(4681)
X(4687) lies on these lines: {1, 872}, {2, 37}, {9, 86}, {43, 2667}, {45, 894}, {142, 4389}, {190, 3731}, {319, 966}, {335, 4422}, {518, 3616}, {662, 2268}, {740, 1698}, {742, 3763}, {749, 984}, {1213, 3661}, {1255, 3187}, {1961, 3769}, {3247, 4360}, {3634, 3993}, {3662, 4364}, {3723, 4393}, {3986, 4357}, {4029, 4431}
X(4687) = complement of X(4699)
X(4687) = anticomplement of X(31238)
X(4687) = {X(2),X(37)}-harmonic conjugate of X(75)
X(4688) lies on these lines: {2, 37}, {10, 537}, {44, 4363}, {142, 594}, {274, 670}, {518, 599}, {519, 3696}, {551, 740}, {597, 742}, {726, 3828}, {984, 1739}, {1100, 4361}, {1213, 3663}, {1266, 4364}, {1418, 1441}, {1730, 3929}, {3264, 4377}, {3661, 3834}, {3723, 3875}, {3765, 4410}, {3879, 4399}, {3945, 4371}, {4395, 4472}
X(4688) = reflection of X(37) in X(2)
X(4688) = reflection of X(4664) in X(4755)
X(4688) = complement of X(4664)
X(4688) = anticomplement of X(4755)
X(4689) lies on these lines: {1, 3}, {10, 3712}, {37, 100}, {42, 896}, {43, 3683}, {44, 751}, {45, 3693}, {210, 846}, {345, 3617}, {392, 4256}, {518, 2177}, {612, 4421}, {614, 4428}, {901, 2721}, {902, 1386}, {968, 1376}, {984, 3689}, {1500, 1908}, {1621, 3752}, {3625, 4030}, {3626, 3703}, {3993, 4434}, {4383, 4512}
X(4690) lies on these lines: {8, 536}, {9, 4445}, {10, 524}, {37, 319}, {44, 3661}, {69, 3739}, {75, 4410}, {141, 3008}, {142, 3631}, {210, 4553}, {519, 4364}, {527, 3626}, {594, 4416}, {599, 3834}, {668, 4377}, {1213, 3879}, {1386, 3775}, {2321, 4478}, {3630, 3664}, {3663, 4399}, {3679, 4363}, {3707, 4422}, {4034, 4361}
Recalling that triangle centers are functions, at (a,b,c) = (6,9,13), the values of X(4691) and X(21267) are equal.
X(4691) lies on these lines: {1, 2}, {341, 4066}, {355, 1657}, {515, 548}, {516, 3627}, {517, 3850}, {518, 3918}, {594, 2325}, {942, 3968}, {960, 3956}, {966, 4058}, {1100, 4545}, {1213, 4060}, {1654, 4480}, {2802, 4540}, {3057, 3921}, {3681, 4084}, {3697, 3878}, {3698, 3874}, {3740, 3884}, {3893, 3898}, {3946, 4405}, {3986, 4007}
X(4691) = complement of X(3635)
X(4691) = {X(8),X(10)}-harmonic conjugate of X(1125)
X(4692) lies on these lines: {1, 312}, {2, 3992}, {8, 79}, {10, 38}, {30, 4030}, {75, 537}, {321, 519}, {341, 1698}, {392, 3967}, {495, 3703}, {551, 4125}, {714, 984}, {726, 4424}, {1056, 3974}, {1125, 3701}, {1219, 3086}, {1269, 4085}, {1909, 1930}, {3006, 3822}, {3244, 3702}, {3555, 3714}, {3583, 4514}, {3831, 3953}
X(4693) lies on these lines: {1, 536}, {8, 4439}, {36, 4436}, {43, 4387}, {45, 3679}, {190, 519}, {238, 239}, {320, 2796}, {321, 3750}, {350, 874}, {528, 3943}, {846, 3706}, {984, 3886}, {1621, 4365}, {3175, 3961}, {3684, 4037}, {3758, 3923}, {3783, 4465}, {3883, 4133}, {3935, 3994}, {3971, 3996}, {4038, 4418}, {4775, 4777}
X(4694) lies on these lines: {1, 3}, {10, 4487}, {38, 551}, {42, 3892}, {106, 4511}, {145, 3987}, {244, 519}, {386, 3889}, {596, 3702}, {756, 1125}, {758, 1149}, {995, 3873}, {1015, 3726}, {1100, 2251}, {1193, 3881}, {1201, 3874}, {1647, 3814}, {2275, 3970}, {2292, 3636}, {3216, 3555}, {3667, 4017}, {3957, 4256}
X(4695) lies on these lines: {1, 3833}, {8, 982}, {10, 321}, {38, 3679}, {42, 3753}, {244, 519}, {484, 896}, {517, 899}, {523, 656}, {986, 3617}, {1149, 3880}, {1575, 2170}, {1722, 3915}, {1962, 3968}, {2650, 3293}, {3125, 3930}, {3621, 3976}, {3625, 3953}, {3626, 3670}, {3698, 4433}, {3840, 3902}, {3931, 4002}
X(4696) lies on these lines: {1, 996}, {2, 341}, {4, 8}, {10, 38}, {12, 3006}, {75, 3617}, {145, 312}, {519, 1089}, {672, 4095}, {960, 3952}, {1043, 3935}, {1109, 3626}, {1125, 3992}, {1220, 3920}, {1230, 1834}, {1909, 3263}, {2475, 3770}, {3057, 3967}, {3244, 4125}, {3625, 4066}, {3632, 3902}, {3717, 3963}
X(4697) lies on these lines: {1, 4234}, {2, 896}, {6, 3980}, {10, 540}, {43, 2234}, {75, 3791}, {81, 740}, {86, 846}, {171, 385}, {190, 1961}, {537, 3920}, {553, 1125}, {726, 3745}, {940, 3923}, {1010, 1046}, {1962, 4427}, {2308, 4359}, {3219, 3842}, {3685, 4038}, {3720, 4432}, {3775, 4001}, {4362, 4363}
X(4698) lies on these lines: {2, 37}, {44, 86}, {142, 3986}, {239, 3723}, {518, 1125}, {740, 3634}, {756, 4022}, {872, 3720}, {899, 2667}, {984, 3624}, {1213, 3912}, {1698, 3696}, {2805, 3035}, {3161, 4470}, {3247, 4361}, {3629, 3707}, {3731, 4363}, {3823, 4026}, {3834, 4357}, {4021, 4395}
X(4698) = centroid of ABCX(37)
X(4698) = Kosnita(X(37),X(2)) point
X(4698) = complement of X(3739)
X(4699) lies on these lines: {2, 37}, {7, 1654}, {10, 3662}, {86, 4361}, {142, 3661}, {145, 3696}, {239, 1449}, {274, 330}, {518, 3617}, {726, 1698}, {740, 3616}, {742, 3618}, {894, 1743}, {982, 3728}, {1213, 4389}, {3314, 3925}, {3624, 3993}, {3912, 4058}, {4072, 4431}, {4364, 4398}
X(4699) = complement of X(4704)
X(4699) = anticomplement of X(4687)
X(4700) lies on these lines: {1, 3707}, {6, 10}, {9, 145}, {37, 3635}, {44, 519}, {45, 3244}, {142, 193}, {239, 527}, {374, 3555}, {391, 1449}, {524, 3008}, {594, 4545}, {966, 3624}, {1100, 3636}, {1404, 3911}, {1743, 2321}, {1992, 4384}, {2262, 4018}, {3629, 3664}, {3759, 3946}
X(4701) lies on these lines: {1, 2}, {37, 4545}, {319, 1266}, {391, 4072}, {517, 3853}, {960, 4547}, {1089, 4487}, {2802, 3988}, {3057, 4533}, {3555, 3922}, {3678, 3880}, {3686, 3943}, {3697, 3898}, {3698, 3892}, {3834, 4399}, {3874, 4004}, {3878, 3893}, {3899, 4537}, {3946, 4478}, {3950, 4034}
X(4701) = {X(8),X(10)}-harmonic conjugate of X(4746)
X(4702) lies on these lines: {1, 536}, {44, 519}, {190, 518}, {239, 3246}, {306, 3058}, {321, 3748}, {390, 3416}, {528, 3912}, {740, 1279}, {1001, 3696}, {1149, 2234}, {1386, 3797}, {1621, 3706}, {3175, 3938}, {3241, 3758}, {3295, 3714}, {3689, 4358}, {3740, 3996}, {3870, 3967}, {3935, 4009}
X(4703) lies on these lines: {2, 896}, {6, 4425}, {9, 2887}, {10, 1836}, {63, 3847}, {141, 4011}, {329, 1215}, {333, 3944}, {516, 4104}, {612, 752}, {726, 3966}, {846, 4417}, {984, 4388}, {1211, 3923}, {3219, 4438}, {3305, 3836}, {3416, 3971}, {3683, 3771}, {3821, 4383}, {4362, 4415}
X(4704) lies on these lines: {2, 37}, {8, 3993}, {9, 4393}, {45, 4360}, {144, 1959}, {145, 984}, {239, 3731}, {518, 3623}, {726, 3616}, {740, 3617}, {742, 3620}, {894, 3247}, {3618, 4473}, {3661, 3950}, {3707, 4464}, {3723, 3758}, {3912, 4098}, {3986, 4431}, {4029, 4357}
X(4704) = complement of X(4821)
X(4704) = anticomplement of X(4699)
X(4705) lies on these lines: {10, 514}, {42, 663}, {100, 691}, {213, 2422}, {484, 513}, {512, 661}, {523, 1577}, {650, 667}, {659, 830}, {784, 4391}, {826, 4088}, {882, 3954}, {905, 4378}, {2489, 3709}, {3293, 4040}, {4010, 4129}, {4061, 4546}, {4079, 4155}, {4160, 4367}
X(4705) = isotomic conjugate of X(4623)
X(4705) = trilinear product of extraversions of X(513)
X(4706) lies on these lines: {2, 4519}, {8, 4003}, {10, 3666}, {38, 4113}, {145, 354}, {210, 3210}, {239, 1155}, {519, 3999}, {536, 899}, {982, 3632}, {3008, 3712}, {3011, 4395}, {3290, 3943}, {3663, 4023}, {3706, 3752}, {3742, 3896}, {3834, 4062}, {3875, 4413}
X(4707) lies on these lines: {1, 2785}, {10, 4088}, {148, 150}, {239, 514}, {512, 3801}, {525, 1577}, {690, 4010}, {826, 2533}, {918, 1086}, {1018, 1020}, {1845, 3738}, {2457, 4064}, {2774, 3868}, {3566, 4170}, {3904, 3960}, {3906, 4122}, {4040, 4142}, {4049, 4080}
X(4708) lies on these lines: {2, 44}, {10, 536}, {37, 3661}, {274, 4410}, {319, 3723}, {518, 4407}, {524, 1125}, {527, 3634}, {545, 3828}, {597, 3707}, {742, 3842}, {756, 2228}, {1086, 1213}, {1100, 1654}, {1698, 4363}, {3247, 4445}, {3821, 3846}, {4021, 4399}
X(4708) = complement of X(4670)
X(4709) lies on these lines: {8, 726}, {10, 37}, {75, 519}, {192, 3679}, {210, 4135}, {321, 4090}, {518, 3625}, {528, 4399}, {551, 3739}, {984, 3626}, {1743, 3923}, {2796, 4416}, {2887, 4046}, {3706, 3840}, {3728, 4424}, {3971, 4365}, {4043, 4125}
X(4710) lies on these lines: {1, 3596}, {10, 3728}, {75, 3775}, {313, 740}, {321, 1109}, {700, 984}, {1089, 1826}, {1230, 4365}, {3696, 4377}, {3751, 4494}, {3765, 3923}, {3773, 4033}, {3948, 3993}, {3992, 4078}, {4043, 4527}, {4044, 4133}, {4110, 4385}
X(4711) lies on these lines: {8, 210}, {10, 3742}, {145, 3983}, {517, 3845}, {518, 599}, {519, 3740}, {758, 3626}, {942, 3968}, {958, 3158}, {3617, 3812}, {3625, 3898}, {3632, 3697}, {3635, 4540}, {3711, 3872}, {3715, 3895}, {3899, 4539}, {3913, 4512}
X(4712) lies on the inellipse cdentered at X(10) and on these lines: {1, 644}, {2, 38}, {8, 3177}, {9, 105}, {10, 1111}, {31, 1331}, {63, 100}, {144, 4073}, {190, 2310}, {518, 672}, {678, 896}, {918, 2254}, {1018, 2809}, {1145, 2826}, {3041, 3119}, {3263, 3717}, {3942, 4553}
X(4712) = antipode of X(1111) in inellipse centered at X(10)
X(4712) = reflection of X(1111) in X(10)
X(4712) = trilinear square of X(518)
X(4713) lies on these lines: {2, 45}, {6, 350}, {31, 4396}, {43, 536}, {76, 2176}, {101, 3734}, {213, 3760}, {239, 4479}, {312, 742}, {527, 3840}, {538, 995}, {1001, 4368}, {1575, 3729}, {2238, 4361}, {3230, 3761}, {3730, 3934}, {3915, 4400}
X(4713) = anticomplement of X(25350)
X(4714) lies on these lines: {1, 3996}, {8, 2891}, {10, 321}, {40, 1746}, {75, 537}, {333, 484}, {519, 4359}, {551, 3902}, {594, 3125}, {1739, 3741}, {3626, 4487}, {3634, 3702}, {3696, 3753}, {3714, 4002}, {3828, 4358}, {3921, 3967}, {3952, 3956}
X(4715) lies on these lines: {2, 44}, {10, 4499}, {30, 511}, {239, 903}, {551, 3246}, {668, 4506}, {751, 4392}, {1757, 3823}, {2321, 3630}, {3241, 4419}, {3257, 4396}, {3629, 3663}, {3679, 4363}, {3739, 4416}, {3828, 4472}, {3912, 4370}, {3943, 4480}
X(4715) = isogonal conjugate of X(28317)
X(4715) = isotomic conjugate of X(35170)
X(4715) = X(2)-Ceva conjugate of X(35124)
X(4716) lies on these lines: {1, 3696}, {8, 3775}, {10, 4360}, {171, 3187}, {238, 239}, {319, 3821}, {519, 1738}, {536, 1757}, {984, 3875}, {1698, 4007}, {3242, 3632}, {3723, 3846}, {3750, 3896}, {3759, 3923}, {4026, 4399}, {4038, 4359}
X(4717) lies on these lines: {1, 4365}, {8, 3583}, {10, 312}, {75, 551}, {321, 519}, {758, 3706}, {1089, 3626}, {1125, 1962}, {1227, 2796}, {3625, 4385}, {3678, 4113}, {3679, 4125}, {3686, 4115}, {3753, 4519}, {3828, 4358}, {3956, 4009}
X(4718) lies on these lines: {2, 37}, {44, 3875}, {518, 3633}, {545, 3879}, {726, 3635}, {740, 3625}, {742, 3630}, {1086, 3950}, {1100, 3729}, {1418, 4552}, {3629, 4464}, {3663, 3943}, {3723, 4363}, {3834, 4398}, {4032, 4114}, {4364, 4431}
X(4718) = complement of X(4764)
X(4718) = anticomplement of X(4726)
Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define OB and OC cyclically. Let LA be the external tangent to circles OB and OC that is the farther of the two from OA. Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines A'A'', B'B'', C'C'' concur in X(4719). See the reference at X(1001).
X(4719) lies on these lines: {1, 474}, {2, 3714}, {3, 1386}, {6, 988}, {37, 978}, {386, 518}, {404, 3745}, {740, 1125}, {958, 2999}, {960, 1193}, {995, 3931}, {1203, 3916}, {1449, 3361}, {3216, 3740}, {3755, 3813}, {3868, 4003}
X(4720) lies on these lines: {8, 21}, {30, 2895}, {58, 3632}, {81, 519}, {86, 3241}, {145, 1010}, {284, 4007}, {314, 1320}, {644, 2287}, {952, 4221}, {956, 4184}, {1014, 3476}, {2177, 3679}, {3706, 4511}, {3714, 4420}, {3786, 3877}
X(4721) lies on these lines: {6, 3760}, {41, 3734}, {58, 4396}, {76, 213}, {384, 2251}, {527, 3831}, {536, 3293}, {538, 1193}, {595, 4400}, {672, 3934}, {742, 1089}, {1125, 4465}, {1698, 4363}, {1909, 3230}, {2176, 3761}, {2300, 3770}
X(4722) lies on these lines: {1, 4127}, {6, 38}, {10, 3578}, {31, 1331}, {42, 896}, {44, 3720}, {81, 756}, {191, 1126}, {518, 2308}, {748, 1743}, {997, 1468}, {1100, 3989}, {1707, 2177}, {1962, 3219}, {1999, 3994}, {3629, 3703}
X(4723) lies on these lines: {8, 210}, {10, 38}, {21, 1261}, {100, 2757}, {321, 3679}, {517, 3952}, {519, 3992}, {522, 3717}, {668, 3263}, {1089, 3626}, {1227, 3264}, {1339, 2415}, {3617, 4385}, {3685, 4076}, {3691, 4095}, {3699, 4511}
X(4723) = isogonal conjugate of X(1417)
X(4724) lies on these lines: {1, 514}, {44, 513}, {512, 4498}, {522, 3935}, {676, 1459}, {693, 3716}, {1019, 4401}, {1443, 1447}, {1960, 4378}, {2499, 2978}, {3309, 4041}, {3617, 4147}, {3762, 4474}, {3907, 4462}, {4010, 4382}, {4379, 4448}
X(4724) = isogonal conjugate of X(37138)
X(4724) = anticomplement of X(24720)
X(4724) = crosspoint of X(i) and X(j) for these {i,j}: {1, 37138}, {190, 870}
X(4724) = crosssum of X(i) and X(j) for these {i,j}: {1, 4724}, {514, 30949}, {649, 869}, {3661, 25259}
X(4724) = crossdifference of every pair of points on line X(1)X(672)
X(4725) lies on these lines: {2, 319}, {8, 4470}, {30, 511}, {44, 4473}, {239, 3834}, {551, 3775}, {1449, 4445}, {1654, 3723}, {2321, 3629}, {3244, 4364}, {3626, 4472}, {3630, 3663}, {3631, 3946}, {3632, 4363}, {3664, 4399}, {3739, 3879}
X(4726) lies on these lines: {2, 37}, {141, 4058}, {142, 4072}, {518, 3625}, {527, 4399}, {545, 3686}, {594, 1266}, {740, 3635}, {1086, 4431}, {1449, 4363}, {1743, 4361}, {2321, 3834}, {3631, 4060}, {4021, 4472}, {4371, 4454}
X(4726) = complement of X(4718)
X(4727) lies on these lines: {6, 3633}, {8, 37}, {44, 519}, {45, 3632}, {320, 536}, {524, 4480}, {594, 1125}, {1100, 2321}, {1213, 4060}, {1698, 4007}, {2345, 3623}, {3589, 4464}, {3625, 4029}, {3707, 3950}, {3912, 4395}
Let Oa be the circle centered at A with radius ka (for some constant k>0), and define Ob and Oc cyclically. Let A' be the exsimilicenter of Ob and Oc, and define B' and C' cyclically. The centroid of (degenerate) triangle A'B'C' = X(4728). This is independent of the choice of k. (Randy Hutson, October 15, 2018)
Let A2B2C2 and A30B30C30 be Gemini triangles 2 and 30, resp. Let A' be the crosspoint of A2 and A30, and define B' and C' cyclically. Triangle A'B'C' is degenerate, on line X(514)X(661), and X(4728) is the centroid of A'B'C'. (Randy Hutson, January 15, 2019)
X(4728) lies on these lines: {1, 4508}, {2, 812}, {11, 244}, {335, 4080}, {513, 4379}, {514, 661}, {649, 4106}, {650, 4382}, {804, 1962}, {918, 4120}, {1699, 2820}, {1978, 3807}, {2786, 4453}, {3004, 4024}, {3768, 4465}
X(4728) = isogonal conjugate of X(34075)
X(4728) = isotomic conjugate of X(4607)
X(4728) = anticomplement of X(4763)
X(4728) = crossdifference of every pair of points on line X(31)X(101)
X(4728) = tripolar centroid of X(75)
X(4729) lies on these lines: {10, 4170}, {100, 2705}, {145, 4504}, {512, 661}, {649, 3900}, {656, 4132}, {663, 1635}, {798, 4171}, {2254, 3777}, {3309, 4498}, {3566, 4088}, {3667, 4404}, {3887, 4063}, {4017, 4139}, {4162, 4394}
X(4729) = isogonal conjugate of isotomic conjugate of X(4404)
X(4730) lies on these lines: {8, 2787}, {10, 4010}, {512, 661}, {513, 3245}, {656, 4139}, {659, 3887}, {667, 3900}, {678, 1635}, {690, 4088}, {764, 891}, {900, 1145}, {1018, 4069}, {1734, 2530}, {2533, 4151}, {2642, 2643}
X(4730) = isogonal conjugate of X(4622)
X(4730) = isotomic conjugate of X(4634)
X(4730) = crossdifference of every pair of points on line X(81)X(88)
X(4731) lies on these lines: {2, 3880}, {8, 3742}, {10, 12}, {354, 3679}, {392, 3828}, {1125, 3893}, {1319, 4413}, {1698, 3057}, {1706, 4512}, {1739, 4003}, {2093, 3715}, {3241, 3848}, {3617, 3812}, {3626, 3892}, {3634, 3898}
X(4732) lies on these lines: {10, 37}, {42, 4457}, {75, 537}, {321, 4096}, {518, 3626}, {519, 3739}, {752, 3686}, {984, 1278}, {1125, 3846}, {1738, 3775}, {1739, 4022}, {3663, 4407}, {3980, 4042}, {3992, 4043}, {4431, 4439}
X(4733) lies on these lines: {1, 4399}, {2, 4046}, {8, 86}, {10, 37}, {144, 1654}, {518, 4111}, {524, 3416}, {1001, 4433}, {1086, 3775}, {1211, 3120}, {3030, 3740}, {3661, 3826}, {3741, 3756}, {3844, 4437}, {3846, 3912}
X(4734) lies on these lines: {2, 740}, {8, 3666}, {42, 3210}, {43, 192}, {145, 982}, {171, 4393}, {354, 3241}, {1215, 1278}, {1376, 4360}, {1403, 3212}, {2999, 3685}, {3644, 3967}, {3662, 4028}, {3705, 3755}
X(4735) lies on these lines: {1, 4484}, {10, 714}, {37, 3122}, {38, 2228}, {39, 674}, {42, 2245}, {44, 3764}, {536, 4443}, {982, 3834}, {1386, 4283}, {1631, 2273}, {2223, 4286}, {3739, 4446}, {3779, 4261}
X(4736) lies on the inellipse centered at X(10) and on these lines: {1, 643}, {10, 1109}, {36, 214}, {191, 1098}, {201, 1089}, {244, 1125}, {523, 1145}, {662, 2948}, {756, 4013}, {849, 1046}, {1018, 3708}, {1111, 4357}, {2611, 2802}, {3778, 4424}
X(4736) = antipode of X(1109) in inellipse centered at X(10)
X(4736) = reflection of X(1109) in X(10)
X(4736) = trilinear square of X(758)
X(4737) lies on these lines: {1, 341}, {4, 8}, {10, 982}, {75, 537}, {145, 3701}, {312, 519}, {997, 3699}, {1058, 2899}, {1089, 3632}, {3241, 4358}, {3621, 3702}, {3729, 3732}, {3877, 3952}, {3880, 3967}
X(4738) lies on the inellipse centered at X(10) and the inellipse centered at X(24003), and also on these lines: {1, 1120}, {8, 80}, {10, 244}, {75, 537}, {121, 1647}, {341, 3632}, {519, 3992}, {900, 1145}, {1109, 3626}, {1317, 4152}, {2170, 4103}, {2835, 3421}, {3625, 3701}, {3902, 4125}
X(4738) = isotomic conjugate of X(679)
X(4738) = antipode of X(244) in inellipse centered at X(10)
X(4738) = antipode of X(1) in inellipse centered at X(24003)
X(4738) = reflection of X(244) in X(10)
X(4738) = reflection of X(1) in X(24003)
X(4738) = trilinear square of X(519)
X(4739) lies on these lines: {2, 37}, {142, 4058}, {518, 3626}, {594, 3834}, {726, 3846}, {740, 3636}, {1213, 1266}, {1449, 4361}, {1743, 4363}, {3632, 3696}, {3664, 4399}, {3946, 4472}, {4402, 4470}
X(4739) = complement of X(4681)
X(4740) lies on these lines: {2, 37}, {8, 537}, {330, 3227}, {594, 4398}, {599, 903}, {726, 3679}, {740, 3241}, {742, 1992}, {1266, 3661}, {3620, 4373}, {3662, 4431}, {3729, 3973}, {4363, 4393}
X(4740) = anticomplement of X(4664)
X(4741) lies on these lines: {2, 44}, {7, 1654}, {8, 537}, {69, 192}, {75, 4410}, {144, 3620}, {190, 599}, {319, 1278}, {524, 4389}, {527, 3661}, {2895, 3210}, {3008, 3662}, {3764, 4392}
X(4741) = anticomplement of X(3758)
X(4742) lies on these lines: {1, 321}, {2, 3902}, {8, 3740}, {145, 3701}, {312, 3241}, {519, 3992}, {551, 4359}, {740, 1149}, {995, 3896}, {1089, 3635}, {3244, 4125}, {3623, 4385}
X(4743) lies on these lines: {8, 4407}, {10, 37}, {42, 4442}, {145, 4310}, {319, 3632}, {1086, 3244}, {2887, 3896}, {3122, 3987}, {3624, 3886}, {3626, 4364}, {3636, 3946}, {3962, 4523}
X(4744) lies on these lines: {10, 12}, {517, 3892}, {519, 3894}, {551, 3877}, {942, 3898}, {993, 1159}, {1125, 3899}, {3244, 3873}, {3625, 3868}, {3626, 3901}, {3742, 3878}, {3874, 3880}
X(4745) lies on these lines: {1, 2}, {355, 3534}, {516, 3654}, {517, 3956}, {518, 3968}, {960, 4540}, {2802, 3740}, {3545, 4301}, {3681, 3919}, {3820, 3829}, {3874, 4002}, {3878, 3983}
X(4745) = {X(8),X(10)}-harmonic conjugate of X(3634)
X(4746) lies on these lines: {1, 2}, {517, 3861}, {594, 4545}, {2802, 4547}, {3834, 4478}, {3874, 3922}, {3878, 4533}, {3880, 4015}, {3892, 4002}, {3898, 3983}, {3943, 4060}, {4034, 4058}
X(4746) = {X(8),X(10)}-harmonic conjugate of X(4701)
X(4747) lies on these lines: {1, 4454}, {2, 44}, {86, 144}, {145, 4363}, {346, 894}, {524, 3617}, {527, 3616}, {536, 3623}, {1100, 4452}, {3247, 4488}, {3622, 4419}, {3672, 4366}
X(4747) = anticomplement of X(4748)
X(4748) lies on these lines: {2, 44}, {7, 1213}, {8, 4364}, {10, 4419}, {190, 2345}, {524, 3616}, {527, 1698}, {536, 3617}, {966, 4000}, {1654, 4393}, {3672, 4371}, {4021, 4034}
X(4748) = complement of X(4747)
X(4748) = anticomplement of X(4798)
X(4749) lies on these lines: {1, 524}, {6, 692}, {31, 2245}, {32, 4471}, {42, 44}, {55, 4277}, {213, 674}, {583, 3778}, {1104, 1245}, {1918, 4271}, {2277, 3941}, {4253, 4484}
Let Oa be the circle centered at A with radius k(b + c) (for arbitrary constant k>0), and define Ob and Oc cyclically. Let A' be the exsimilicenter of Ob and Oc, and define B' and C' cyclically. Then X(4750) = centroid of (degenerate) triangle A'B'C'. This result is independent of the choice of k. (Randy Hutson, December 10, 2016)
X(4750) lies on these lines: {2, 2786}, {11, 244}, {239, 514}, {351, 690}, {522, 4379}, {812, 4453}, {918, 1635}, {2487, 3700}, {3676, 4382}, {3776, 4380}, {4024, 4369}
X(4750) = tripolar centroid of X(86)
X(4750) = centroid of degenerate cross-triangle of excentral and anticomplementary triangles
X(4750) = intersection of tangents to Steiner inellipse at X(1086) and X(2482)
X(4750) = crosspoint wrt medial triangle of X(1086) and X(2482)
X(4750) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 2
X(4751) lies on these lines: {2, 37}, {8, 3846}, {86, 1449}, {320, 966}, {518, 3619}, {740, 3624}, {984, 3634}, {1213, 3662}, {1266, 3986}, {1743, 3758}, {3616, 3696}
X(4752) lies on these lines: {9, 2802}, {100, 101}, {190, 514}, {956, 3730}, {996, 1000}, {2284, 3908}, {2316, 4370}, {3208, 4251}, {3309, 4553}, {3699, 4169}, {3900, 3939}
X(4753) lies on these lines: {10, 524}, {44, 519}, {190, 740}, {239, 537}, {984, 4393}, {1999, 4096}, {3679, 3758}, {3681, 3791}, {3751, 4384}, {4085, 4416}, {4418, 4457}
X(4754) lies on these lines: {1, 538}, {8, 524}, {75, 3780}, {86, 1655}, {171, 4400}, {274, 2238}, {732, 894}, {1125, 4465}, {1215, 4447}, {3691, 3739}, {4418, 4433}
X(4755) lies on these lines: {2, 37}, {141, 3986}, {518, 551}, {519, 3842}, {537, 1125}, {740, 3828}, {1107, 3227}, {2325, 4472}, {3834, 4364}, {3846, 3993}
X(4755) = midpoint of X(2) and X(37)
X(4755) = midpoint of X(4664) and X(4688)
X(4755) = complement of X(4688)
X(4756) lies on these lines: {81, 3971}, {100, 190}, {537, 3315}, {644, 4115}, {1757, 3994}, {2832, 3799}, {3218, 4009}, {3219, 3967}, {3681, 3886}, {4096, 4418}
X(4757) lies on these lines: {10, 12}, {145, 2802}, {214, 3336}, {517, 548}, {942, 3636}, {960, 3833}, {1484, 2800}, {3616, 3878}, {3624, 3869}, {3632, 3868}}
X(4758) lies on these lines: {1, 4470}, {2, 3707}, {86, 3912}, {142, 610}, {519, 4472}, {524, 3634}, {527, 1125}, {536, 3636}, {551, 4363}, {3616, 4454}
X(4759) lies on these lines: {10, 598}, {44, 519}, {190, 238}, {527, 1125}, {537, 3246}, {551, 3758}, {752, 4422}, {2796, 3008}, {3923, 4384}, {3993, 4393}
X(4760) lies on these lines: {55, 4363}, {239, 1914}, {385, 4037}, {524, 896}, {659, 812}, {742, 902}, {2243, 3912}, {2246, 4422}, {2276, 3758}, {3685, 4396}
X(4760) = anticomplement of X(25383)
X(4761) lies on these lines: {1, 4369}, {8, 4160}, {10, 661}, {65, 4077}, {512, 1577}, {513, 3762}, {514, 1734}, {690, 4122}, {1019, 3907}, {1499, 3700}
X(4762) lies on these lines: {2, 650}, {30, 511}, {661, 4106}, {798, 4498}, {1635, 4379}, {3175, 4024}, {3700, 4468}, {4130, 4391}, {4140, 4462}, {4369, 4394}
X(4762) = isogonal conjugate of X(8693)Let Oa, Ob, Oc be the circles with the segments BC, CA, AB, resp. as diameters. X(4763) is the centroid of the exsimilicenters of Ob and Oc, Oc and Oa, Oa and Ob, which are collinear on the Gergonne line. (Randy Hutson, July 31 2018)
X(4763) lies on these lines: {2, 812}, {241, 514}, {351, 740}, {900, 3035}, {1639, 2786}, {2977, 4458}, {3798, 4521}, {3835, 4394}, {4147, 4504}
X(4763) = complement of X(4728)X(4764) lies on these lines: {2, 37}, {190, 3973}, {726, 3625}, {740, 3633}, {2321, 4398}, {3630, 4409}, {3729, 3759}, {3758, 3875}, {4389, 4431}
X(4764) = anticomplement of X(4718)X(4765) lies on these lines: {239, 514}, {522, 650}, {523, 2527}, {647, 4151}, {661, 3667}, {1024, 2339}, {2488, 3900}, {4041, 4546}, {4467, 4468}, {4700, 4706}
X(4766) lies on these lines: {2, 31}, {41, 315}, {213, 626}, {325, 672}, {514, 661}, {742, 4144}, {754, 2251}, {1930, 4153}, {3263, 4071}
X(4766) = isotomic conjugate of X(37208)
X(4767) lies on these lines: {2, 4126}, {81, 4090}, {88, 537}, {100, 190}, {528, 4152}, {644, 3239}, {668, 693}, {3240, 4465}, {3935, 4009}
X(4768) lies on these lines: {8, 3738}, {9, 4148}, {10, 1769}, {11, 123}, {513, 4404}, {522, 3717}, {523, 2530}, {646, 3699}, {900, 1145}
X(4768) = intersection of perspectrices of [ABC and Gemini triangle 13] and [ABC and Gemini triangle 14]
X(4769) lies on these lines: {1, 626}, {8, 315}, {10, 32}, {40, 2794}, {355, 511}, {746, 984}, {754, 3679}, {1760, 4178}, {2175, 4150}
X(4770) lies on these lines: {10, 523}, {512, 661}, {650, 1960}, {784, 4147}, {891, 1491}, {1500, 3709}, {1734, 4490}, {3906, 4088}, {4777, 4791}
X(4771) lies on these lines: {6, 3980}, {239, 385}, {519, 3290}, {594, 4457}, {740, 2238}, {3214, 4095}, {3666, 3686}, {3791, 4386}, {4700, 4706}
X(4772) lies on these lines: {2, 37}, {518, 3922}, {740, 3622}, {966, 4440}, {1213, 4398}, {3621, 3696}, {3728, 4392}, {3973, 4384}
X(4773) lies on these lines: {523, 649}, {650, 3667}, {654, 1768}, {812, 1638}, {900, 1635}, {2487, 4382}, {2527, 4024}, {3700, 4394}, {4700, 4706}
X(4774) lies on these lines: {8, 523}, {513, 4474}, {814, 4380}, {1909, 4374}, {2295, 3287}, {2533, 3907}, {2785, 4122}, {4083, 4382}
X(4775) lies on these lines: {1, 513}, {187, 237}, {213, 3063}, {788, 3768}, {814, 4170}, {1491, 3887}, {2530, 3309}, {4040, 4083}, {4693, 4777}
X(4775) = isogonal conjugate of X(4597)X(4776) lies on these lines: {2, 513}, {321, 4079}, {514, 661}, {650, 4380}, {824, 4120}, {900, 1491}, {3797, 4010}, {4359, 4502}
X(4776) = isotomic conjugate of X(37209)
X(4777) lies on these lines: {30, 511}, {37, 650}, {75, 693}, {649, 2529}, {1769, 4041}, {3739, 4500}, {3797, 4010}, {4036, 4397}, {4693, 4775}, {4770, 4791}
X(4777) = isogonal conjugate of X(4588)X(4778) lies on these lines: {30, 511}, {661, 4521}, {1443, 1447}, {1459, 4040}, {2490, 2529}, {2517, 3762}, {3733, 4401}, {4086, 4462}, {4700, 4706}
X(4778) = isogonal conjugate of X(8694)X(4779) lies on these lines: {1, 4454}, {8, 9}, {86, 3445}, {145, 190}, {192, 3623}, {497, 3712}, {1279, 4452}, {3243, 4488}
X(4780) lies on these lines: {10, 37}, {42, 4054}, {69, 519}, {551, 3946}, {1125, 3886}, {3896, 3914}, {4061, 4425}, {4067, 4523}
X(4781) lies on these lines: {2, 4432}, {89, 3241}, {99, 901}, {100, 190}, {537, 678}, {649, 1018}, {835, 1293}, {2796, 4080}
X(4782) lies on these lines: {1, 667}, {44, 513}, {512, 4401}, {669, 4132}, {900, 4522}, {3309, 3579}, {4010, 4380}, {4367, 4498}
X(4783) lies on these lines: {10, 3122}, {519, 3264}, {740, 3948}, {900, 1145}, {3696, 4377}, {3701, 4527}, {3943, 3992}, {4010, 4155}
X(4784) lies on these lines: {1, 512}, {44, 513}, {523, 4467}, {669, 2106}, {2786, 4122}, {3126, 3647}, {3634, 4129}, {4010, 4369}
X(4785) lies on these lines: {2, 649}, {30, 511}, {553, 3676}, {598, 4049}, {661, 4380}, {798, 4129}, {4106, 4369}, {4401, 4455}
X(4785) = crossdifference of every pair of points of line X(6)X(3009)
X(4786) lies on these lines: {2, 3667}, {110, 2740}, {239, 514}, {1790, 1919}, {2487, 4106}, {3676, 4380}, {4394, 4468}
X(4787) lies on these lines: {42, 4266}, {674, 869}, {982, 3888}, {1405, 2177}, {2275, 3056}, {3248, 4484}, {3882, 3938}
X(4788) lies on these lines: {2, 37}, {145, 726}, {239, 3973}, {740, 3621}, {3622, 3993}, {3729, 4393}, {3943, 4398}
X(4788) = anticomplement of X(1278)
X(4788) = isotomic conjugate of isogonal conjugate of X(36647)
X(4789) lies on these lines: {2, 523}, {81, 3287}, {321, 4374}, {514, 661}, {649, 4500}, {824, 4379}, {4024, 4369}
X(4789) = isotomic conjugate of X(37210)
X(4790) lies on these lines: {44, 513}, {512, 4162}, {665, 3803}, {1019, 1429}, {3004, 3798}, {3667, 3700}, {4106, 4369}, {4700, 4706}
X(4790) = isogonal conjugate of X(4606)
X(4790) = anticomplement of X(4940)
X(4791) lies on these lines: {1, 4474}, {10, 522}, {76, 3261}, {514, 661}, {814, 4401}, {2517, 3667}, {4147, 4151}, {4770, 4777}
X(4792) lies on these lines: {1, 88}, {8, 4080}, {44, 517}, {320, 519}, {901, 3245}, {1022, 2254}, {3626, 4013}
X(4793) lies on these lines: {8, 3585}, {10, 3702}, {75, 519}, {551, 3902}, {2802, 3696}, {3626, 4066}, {3679, 4125}
X(4794) lies on these lines: {1, 514}, {512, 4401}, {513, 1960}, {650, 3887}, {928, 2488}, {3626, 4147}
X(4795) lies on these lines: {1, 545}, {2, 44}, {519, 4363}, {524, 3416}, {527, 551}, {536, 3241}, {750, 1644}
X(4796) lies on these lines: {2, 44}, {524, 3626}, {527, 3636}, {536, 3244}, {1100, 4440}, {3632, 4363}, {3664, 4422}
X(4797) lies on these lines: {2, 2243}, {10, 754}, {31, 742}, {306, 524}, {1918, 2236}, {3052, 3757}, {3683, 4364}
X(4797) = complement of X(4799)
X(4798) lies on these lines: {1, 4472}, {2, 44}, {86, 3661}, {524, 1698}, {536, 3616}, {1125, 4363}, {3624, 4364}
X(4798) = complement of X(4748)
X(4799) lies on these lines: {1, 754}, {2, 2243}, {315, 3721}, {524, 3187}, {626, 1759}, {2887, 4376}, {3944, 4396}
anticomplement of X(4797)X(4800) lies on these lines: {2, 900}, {350, 3766}, {513, 4379}, {659, 812}, {2238, 4435}, {2276, 4526}
X(4801) lies on these lines: {514, 661}, {523, 3777}, {764, 784}, {1019, 4380}, {3837, 4490}, {4369, 4498}
X(4801) = isogonal conjugate of X(34074)
X(4801) = isotomic conjugate of X(4606)
X(4802) lies on these lines: {30, 511}, {693, 4036}, {2529, 4394}, {2605, 4449}, {3733, 4378}, {4106, 4122}
X(4802) = isogonal conjugate of X(8652)X(4803) lies on these lines: {8, 58}, {86, 519}, {1010, 3625}, {1043, 3626}, {2177, 3679}, {2802, 3786}
X(4804) lies on these lines: {10, 1577}, {145, 3907}, {513, 4382}, {514, 4170}, {522, 693}, {523, 661}
X(4804) = anticomplement of X(4913)
X(4805) lies on these lines: {2, 2251}, {8, 712}, {10, 4376}, {31, 754}, {41, 626}, {213, 315}
X(4806) lies on these lines: {10, 512}, {513, 3716}, {523, 661}, {900, 1491}, {1019, 3624}, {3616, 4367}
X(4807) lies on these lines: {8, 1019}, {10, 512}, {514, 1734}, {519, 4367}, {2533, 4151}, {3667, 4086}
X(4808) lies on these lines: {10, 3801}, {512, 4088}, {523, 1577}, {826, 4041}, {3159, 4122}, {4064, 4139}
X(4809) lies on these lines: {11, 244}, {513, 4453}, {514, 659}, {523, 1635}, {918, 4448}
X(4810) lies on these lines: {149, 900}, {513, 4382}, {659, 812}, {1491, 4106}, {4083, 4474}
X(4811) lies on these lines: {320, 350}, {522, 3717}, {523, 4462}, {900, 2517}, {1577, 3667}
X(4812) lies on these lines: {2, 37}, {239, 2273}, {740, 976}, {2344, 2997}, {3096, 3662}
X(4813) lies on these lines: {2, 4932}, {44, 513}, {514, 4024}, {663, 2520}, {3835, 4379}, {4502, 4526}
X(4813) = isogonal conjugate of X(37211)
X(4813) = anticomplement of X(4932)
X(4813) = crosspoint of X(1) and X(37211)
X(4813) = crosssum of X(i) and X(j) for these {i,j}: {1, 4813}, {386, 649}, {514, 27186}
X(4813) = crossdifference of every pair of points of line X(1)X(2308)
X(4814) lies on these lines: {8, 522}, {650, 663}, {657, 1334}, {1734, 3960}, {3700, 4528}
X(4815) lies on these lines: {513, 4170}, {522, 693}, {523, 1577}, {656, 4151}, {2533, 4139}
X(4816) lies on these lines: {1, 2}, {44, 4007}, {45, 4034}, {355, 3861}, {1743, 4060}
X(4816) = complement of X(4394)
X(4817) lies on these lines: {190, 789}, {513, 875}, {514, 659}, {649, 693}, {825, 927}
X(4818) lies on these lines: {522, 2526}, {523, 3776}, {824, 1491}, {3837, 4500}, {4700, 4706}
X(4819) lies on these lines: {42, 594}, {210, 3950}, {214, 519}, {3925, 4028}
X(4820) lies on these lines: {513, 4024}, {522, 650}, {824, 4106}, {2786, 4500}
X(4820) = barycentric product X(8)*X(1698)X(4821) lies on these lines: {2, 37}, {726, 3617}, {740, 3623}, {903, 4445}
X(4821) = anticomplement of X(4704)
X(4822) lies on these lines: {512, 661}, {513, 663}, {514, 4170}, {3800, 4088}
X(4823) lies on these lines: {514, 661}, {784, 3837}, {1019, 4379}, {4063, 4382}
X(4823) = isotomic conjugate of X(37211)
X(4824) lies on these lines: {10, 514}, {513, 4380}, {523, 661}, {693, 4036}
X(4825) lies on these lines: {513, 3245}, {650, 3251}, {667, 1635}
X(4826) lies on these lines: {512, 798}, {661, 4132}, {834, 3768}
X(4827) lies on these lines: {521, 650}, {657, 3900}, {3709, 4162}, {4700, 4706}
X(4828) lies on these lines: {75, 693}, {514, 1921}, {900, 4374}
X(4829) lies on these lines: {740, 1284}, {2292, 4046}, {3896, 4393}
X(4830) lies on these lines: {513, 4507}, {659, 812}, {3907, 4498}, {4700, 4706}
X(4831) lies on these lines: {10, 540}, {524, 896}, {3629, 4414}, {4700, 4706}
X(4832) lies on these lines: {512, 798}, {649, 854}, {665, 834}, {4700, 4706}
X(4832) = isogonal conjugate of X(4633)
X(4832) = crossdifference of every pair of points of line X(8)X(86)
X(4833) lies on these lines: {6, 661}, {36, 238}, {834, 4498}, {4693, 4775}
X(4834) lies on these lines: {187, 237}, {484, 513}, {1019, 4083}
X(4835) lies on these lines: {8, 192}, {86, 1431}, {1213, 3863}
X(4836) lies on these lines: {2, 2244}, {32, 744}, {2205, 2237}
X(4836) = complement of X(4837)
X(4837) lies on these lines: {2, 2244}, {315, 744}, {626, 4381}
X(4837) = anticomplement of X(4836)X(4838) lies on these lines: {523, 661}, {649, 2529}
X(4839) lies on these lines: {812, 4107}, {4139, 4140}
X(4840) lies on these lines: {36, 238}, {523, 4467}
X(4840) = intersection of perspectrices of [ABC and Gemini triangle 23] and [ABC and Gemini triangle 24]X(4841) lies on these lines: {241, 514}, {523, 661}, {4700, 4706}
X(4841) = isogonal conjugate of X(4627)
X(4841) = isotomic conjugate of X(4633)
X(4841) = crossdifference of every pair of points of line X(55)X(58)
X(4842) lies on these lines: {3835, 4411}, {4369, 4374}
X(4843) lies on these lines: {30, 511}, {3700, 4041}
X(4843) = isogonal conjugate of X(5545)
X(4843) = crossdifference of every pair of points of line X(6)X(1412)
X(4844) lies on these lines: {1, 4379}, {30, 511}
X(4844) = isogonal conjugate of X(8695)
X(4845) lies on these lines: {1, 651}, {33, 1783}, {36, 103}, {55, 101}, {200, 644}, {220, 3939}, {497, 544}, {519, 666}, {645, 1043}, {926, 2316}, {999, 2808}
X(4845) = isogonal conjugate of X(1323)
X(4845) = trilinear pole of line X(55)X(657)
X(4846) = 3(J2 - 3)*X(2) - 2J2*X(74)
X(4846) is the Segovia point of the circumcevian triangle of the circumcenter. (Antreas Hatzipolakis, January 23 2012; Hyacinthos #20737; see Segovia Point, continued.)
Let A'B'C' be the translation of ABC by the vector X(4)X(3). Let MA be the midpoint of the (possibly nonreal) points in which the circumcircle of ABC meets the line B'C', and define MB and MC cyclically. The triangle MAMBMC is perspective to ABC, and the perspector is X(4846). If you have GeoGebra, you can view X(4846). (Angel Montesdeoca, May 20, 2018)
X(4846) lies on these lines: {2, 74}, {3, 4549}, {5, 64}, {6, 30}, {20, 54}, {65, 1479}, {66, 3818}, {67, 1352}, {68, 185}, {73, 1062}, {140, 3532}, {182, 1177}, {265, 974}, {376, 3431}, {381, 1514}, {382, 3527}, {621, 2993}, {622, 2992}, {1173, 3146}, {1204, 3549}, {1480, 3058}, {3309, 3657}, {3531, 3830}
X(4846) = isogonal conjugate of X(378)
X(4846) = isotomic conjugate of polar conjugate of X(34288)
X(4846) = anticomplement of X(4550)
X(4846) = antigonal conjugate of X(34802)
X(4846) = Jerabek hyperbola antipode of X(34802)
X(4846) = X(19)-isoconjugate of X(15066)
X(4847) lies on these lines: {1, 2}, {9, 497}, {11, 210}, {21, 4314}, {38, 3663}, {57, 2550}, {63, 516}, {72, 946}, {75, 1088}, {81, 4349}, {142, 354}, {149, 3219}, {219, 3686}, {226, 518}, {278, 1861}, {312, 3717}, {318, 1838}, {321, 4712}, {329, 1699}, {333, 643}, {345, 3886}, {377, 4298}, {390, 4512}, {442, 3555}, {443, 3333}, {480, 4423}, {515, 956}, {527, 1836}, {528, 4640}, {594, 1108}, {908, 3681}, {950, 958}, {960, 3813}, {982, 1738}, {984, 4656}, {993, 4304}, {1001, 1260}, {1004, 4311}, {1150, 1754}, {1214, 2968}, {1329, 4662}, {1376, 1617}, {1659, 3641}, {1706, 1788}, {2257, 2345}, {2321, 3693}, {2325, 4387}, {2476, 3947}, {2975, 4297}, {3058, 3683}, {3189, 3601}, {3242, 3772}, {3243, 3475}, {3474, 3928}, {3666, 3755}, {3671, 3868}, {3677, 4000}, {3697, 4187}, {3702, 3710}, {3715, 4679}, {3740, 3816}, {3742, 3826}, {3825, 4015}, {3841, 3881}, {3847, 4104}, {3869, 4301}, {3877, 4342}, {3889, 4197}, {4009, 4126}, {4023, 4113}, {4073, 4357}, {4388, 4416}
X(4847) = complement of X(3870)
X(4847) = excentral-to-ABC barycentric image of X(55)
X(4847) = {X(2),X(8)}-harmonic conjugate of X(200)
X(4848) lies on these lines: {1, 631}, {2, 3340}, {4, 2093}, {7, 3617}, {8, 57}, {10, 12}, {11, 4301}, {21, 3256}, {40, 950}, {46, 515}, {56, 519}, {73, 3293}, {80, 1770}, {145, 1420}, {165, 3486}, {201, 4424}, {225, 4674}, {355, 4292}, {388, 553}, {484, 4324}, {496, 517}, {516, 1837}, {527, 3436}, {938, 1697}, {946, 1737}, {952, 4311}, {956, 1466}, {1042, 3214}, {1125, 2099}, {1145, 3555}, {1148, 1785}, {1155, 4297}, {1214, 4646}, {1254, 4695}, {1284, 4078}, {1319, 3244}, {1388, 3635}, {1393, 1739}, {1400, 2321}, {1402, 4433}, {1423, 3717}, {1457, 3216}, {1617, 3913}, {1698, 3485}, {1828, 2835}, {2078, 3871}, {2285, 3686}, {3295, 3654}, {3361, 3476}, {3452, 3869}, {3579, 4304}, {3621, 4308}, {3625, 4315}, {3626, 4031}, {3696, 4032}, {3755, 3778}, {4114, 4691}
X(4849) lies on these lines: {1, 3697}, {6, 200}, {10, 4035}, {31, 3689}, {37, 42}, {43, 518}, {44, 55}, {72, 3293}, {100, 4641}, {145, 4487}, {171, 4663}, {209, 3198}, {213, 4515}, {218, 2900}, {354, 899}, {594, 4061}, {612, 1100}, {740, 3967}, {748, 3748}, {968, 3715}, {1104, 3811}, {1108, 3190}, {1215, 3696}, {1279, 3870}, {1376, 3751}, {1386, 3961}, {1402, 4557}, {1427, 4551}, {1743, 3052}, {1757, 4640}, {1999, 3699}, {2177, 3683}, {2187, 3204}, {2292, 4005}, {2340, 3780}, {2650, 3698}, {2999, 3242}, {3175, 3896}, {3216, 3555}, {3219, 4689}, {3240, 3666}, {3678, 3931}, {3744, 3935}, {3755, 4415}, {3932, 4028}, {3943, 4082}, {3962, 4642}, {3987, 4018}, {3993, 4096}, {3999, 4430}, {4011, 4702}, {4026, 4104}, {4649, 4682}
X(4850) lies on these lines: {1, 88}, {2, 37}, {6, 2243}, {7, 1465}, {38, 43}, {42, 982}, {57, 77}, {63, 1743}, {65, 4719}, {145, 4646}, {227, 3600}, {238, 4414}, {239, 980}, {320, 4277}, {386, 3670}, {513, 751}, {518, 3240}, {614, 1621}, {748, 846}, {899, 984}, {908, 3663}, {978, 2292}, {986, 1193}, {988, 2975}, {995, 3877}, {1104, 4189}, {1155, 1386}, {1201, 3890}, {1266, 4054}, {1279, 4689}, {1376, 3920}, {1453, 4652}, {1466, 4296}, {1471, 1758}, {2092, 3662}, {2275, 3121}, {2308, 4650}, {2352, 4210}, {3006, 4429}, {3216, 3876}, {3219, 4383}, {3242, 3935}, {3310, 4453}, {3616, 3931}, {3624, 3743}, {3677, 3870}, {3696, 4706}, {3741, 4709}, {3889, 3953}, {3896, 4734}, {3911, 3946}, {4031, 4667}, {4427, 4676}
X(4850) = complement of X(4671)
X(4850) = anticomplement of X(30818)
X(4850) = polar conjugate of isogonal conjugate of X(23206)
X(4851) lies on these lines: {1, 141}, {2, 319}, {6, 3879}, {7, 536}, {8, 3739}, {9, 524}, {10, 4445}, {37, 69}, {44, 193}, {45, 4416}, {75, 4675}, {86, 3661}, {142, 519}, {144, 4715}, {145, 3834}, {192, 320}, {306, 940}, {312, 3770}, {346, 4644}, {518, 3781}, {527, 3950}, {599, 4357}, {750, 4062}, {894, 4795}, {966, 4690}, {1086, 3875}, {1376, 4028}, {1449, 3589}, {1654, 4687}, {1743, 3629}, {1999, 3772}, {2321, 3664}, {2345, 3945}, {3242, 4684}, {3244, 3946}, {3247, 3631}, {3620, 3723}, {3621, 4371}, {3630, 3731}, {3632, 4399}, {3633, 4395}, {3644, 4440}, {3662, 4360}, {3679, 4478}, {3720, 3966}, {3729, 3943}, {3751, 3932}, {3779, 4553}, {3993, 4655}, {4007, 4665}, {4419, 4681}, {4686, 4727}, {4704, 4741}
X(4851) = complement of X(5839)
X(4851) = anticomplement of X(17348)
X(4852) lies on these lines: {1, 3696}, {2, 3723}, {6, 536}, {8, 4657}, {9, 4681}, {10, 4399}, {37, 239}, {44, 192}, {69, 4725}, {75, 1100}, {86, 4688}, {141, 519}, {142, 3244}, {145, 3834}, {190, 4718}, {193, 4715}, {518, 4523}, {524, 3663}, {527, 3629}, {528, 4780}, {664, 1418}, {726, 4663}, {740, 1386}, {750, 4706}, {894, 4686}, {1086, 3879}, {1125, 4405}, {1150, 3187}, {1278, 3758}, {1449, 4363}, {1999, 3752}, {2321, 3589}, {3241, 4402}, {3247, 4755}, {3625, 4478}, {3632, 4445}, {3672, 4643}, {3686, 4021}, {3744, 3896}, {3769, 4734}, {3791, 4640}, {3912, 4464}, {3950, 4422}, {4107, 4145}, {4357, 4690}, {4384, 4698}, {4452, 4644}
X(4852) = complement of X(17299)
X(4852) = anticomplement of X(17229)
X(4853) lies on these lines: {1, 2}, {9, 3057}, {21, 3895}, {40, 956}, {55, 2136}, {56, 1706}, {69, 4328}, {75, 269}, {165, 2975}, {210, 2098}, {318, 4692}, {329, 4301}, {517, 3927}, {518, 3340}, {594, 3554}, {728, 1212}, {937, 996}, {946, 3421}, {958, 1697}, {1038, 2968}, {1040, 4030}, {1219, 4320}, {1259, 3746}, {1260, 3303}, {1320, 3876}, {1376, 1420}, {1449, 3713}, {1467, 3476}, {1490, 3419}, {1699, 3436}, {2082, 4390}, {2297, 2345}, {2324, 2654}, {2550, 4321}, {2646, 3158}, {3059, 3243}, {3177, 3729}, {3247, 3965}, {3304, 3698}, {3305, 3890}, {3333, 3753}, {3486, 4326}, {3601, 3913}, {4449, 4546}
X(4853) = {X(1),X(8)}-harmonic conjugate of X(200)
X(4854) lies on these lines: {1, 30}, {2, 3712}, {10, 3175}, {11, 114}, {12, 3931}, {37, 3914}, {42, 4415}, {55, 1284}, {56, 4221}, {57, 2941}, {192, 3703}, {210, 3755}, {226, 4356}, {321, 4026}, {354, 3663}, {442, 3743}, {497, 3672}, {516, 3745}, {524, 4683}, {528, 3920}, {594, 4365}, {740, 1211}, {968, 3772}, {1086, 3720}, {1213, 4037}, {1365, 3021}, {1834, 2292}, {1962, 3120}, {2486, 3136}, {2796, 4697}, {2887, 3993}, {2999, 4679}, {3318, 3320}, {3454, 4065}, {3706, 4357}, {3875, 3966}, {3896, 4819}, {3932, 3995}, {3971, 4085}, {4000, 4423}, {4104, 4780}, {4205, 4647}, {4360, 4388}, {4685, 4743}
X(4854) = X(97)-of-intouch-triangle
X(4855) lies on these lines: {1, 88}, {2, 950}, {3, 63}, {8, 3523}, {9, 4189}, {10, 3612}, {20, 908}, {21, 936}, {35, 997}, {36, 3811}, {40, 4511}, {46, 4084}, {48, 3692}, {56, 3870}, {57, 4188}, {140, 3419}, {145, 1420}, {165, 3869}, {200, 2975}, {208, 4242}, {226, 4190}, {271, 1809}, {329, 3522}, {603, 1331}, {958, 3983}, {976, 988}, {993, 4015}, {1125, 3434}, {1201, 3749}, {1265, 3977}, {1319, 3913}, {1376, 2646}, {1385, 3872}, {1388, 3880}, {1790, 1792}, {1837, 3035}, {2478, 4304}, {3057, 4421}, {3207, 3693}, {3361, 3873}, {3436, 4297}, {3586, 4193}, {3624, 3841}
X(4856) lies on these lines: {1, 391}, {6, 519}, {10, 1449}, {37, 3635}, {145, 1743}, {190, 4464}, {193, 3663}, {239, 3664}, {346, 3633}, {516, 4780}, {524, 3946}, {527, 3629}, {551, 966}, {594, 4701}, {1100, 1125}, {1992, 3875}, {2262, 3874}, {2345, 3625}, {3008, 3759}, {3169, 4253}, {3187, 4054}, {3241, 3731}, {3589, 4725}, {3626, 4545}, {3632, 4058}, {3941, 4097}, {3958, 4065}, {4021, 4393}, {4361, 4667}
X(4857) lies on these lines: {1, 4}, {2, 4309}, {3, 3582}, {5, 3058}, {8, 4125}, {10, 149}, {11, 35}, {12, 3850}, {18, 1250}, {30, 4325}, {36, 496}, {55, 1656}, {56, 1657}, {79, 354}, {80, 3057}, {100, 3825}, {381, 3303}, {382, 3304}, {390, 498}, {484, 1210}, {495, 3858}, {499, 3523}, {516, 3336}, {528, 4187}, {551, 2475}, {1089, 4514}, {1698, 3434}, {1770, 3337}, {1936, 2964}, {2478, 3679}, {2551, 4668}, {3086, 3522}, {3146, 4317}, {3219, 3467}, {3295, 3851}, {3361, 4333}, {3436, 3633}, {3814, 3871}
X(4857) = {X(1),X(4)}-harmonic conjugate of X(5270)
X(4858) lies on these lines: {2, 2006}, {4, 2823}, {8, 3254}, {9, 75}, {10, 1072}, {11, 123}, {57, 92}, {85, 1121}, {116, 2973}, {124, 3120}, {142, 1441}, {219, 4361}, {238, 1733}, {239, 1944}, {242, 3220}, {244, 1109}, {281, 4000}, {312, 646}, {321, 3452}, {338, 1577}, {514, 3942}, {522, 1090}, {604, 2995}, {918, 1086}, {960, 4647}, {1089, 1329}, {1212, 4688}, {1229, 2321}, {1930, 3061}, {2170, 3904}, {2289, 2997}, {3036, 4738}, {3262, 3912}, {3271, 4124}, {4086, 4092}
X(4858) = isogonal conjugate of X(2149)
X(4858) = isotomic conjugate of X(4564)
X(4858) = complement of X(4552)
X(4858) = anticomplement of X(16578)
X(4858) = center of hyperbola {A,B,C,X(75),X(92)}, which is the locus of trilinear poles of lines passing through X(1577)
X(4858) = perspector of circumconic centered at X(1577)
X(4858) = X(2)-Ceva conjugate of X(1577)
X(4858) = pole wrt polar circle of trilinear polar of X(7012) (line X(101)X(108))
X(4858) = polar conjugate of X(7012)
X(4859) lies on these lines: {1, 142}, {2, 2415}, {7, 1743}, {9, 1086}, {10, 4310}, {57, 1723}, {75, 646}, {87, 1027}, {141, 3679}, {238, 4312}, {344, 1266}, {527, 3973}, {599, 4034}, {978, 2140}, {984, 1698}, {1125, 4779}, {1449, 4675}, {1654, 3662}, {1699, 1721}, {3624, 4657}, {3632, 3834}, {3633, 4395}, {3677, 3925}, {3763, 4688}, {3950, 4452}, {4371, 4677}, {4383, 4654}, {4399, 4816}
X(4859) = complement of X(3161)
X(4859) = polar conjugate of X(37756)
X(4859) = perspector of Gemini triangle 7 and vertex-triangle of Gemini triangles 5 and 7
X(4860) lies on these lines: {1, 3}, {2, 3715}, {6, 244}, {7, 11}, {45, 672}, {63, 3742}, {88, 1002}, {89, 105}, {474, 3874}, {516, 4031}, {518, 3306}, {527, 4679}, {553, 1836}, {750, 3242}, {1001, 3218}, {1376, 3873}, {1435, 1859}, {1471, 2361}, {1837, 4298}, {2280, 4289}, {3058, 3474}, {3085, 3296}, {3086, 3649}, {3243, 3689}, {3305, 3848}, {3624, 3927}, {3683, 3928}, {3817, 3982}, {3889, 3913}, {3894, 3940}, {3957, 4421}, {4519, 4659}, {4640, 4666}
X(4860) = {X(1),X(5708)}-harmonic conjugate of X(5221)
X(4860) = {X(1),X(1159)}-harmonic conjugate of X(2099)
X(4861) lies on these lines: {1, 2}, {9, 1392}, {21, 643}, {35, 2802}, {41, 4051}, {55, 3885}, {72, 1173}, {75, 1442}, {100, 1385}, {140, 1145}, {318, 1870}, {377, 3476}, {404, 1319}, {405, 3890}, {517, 2975}, {644, 1212}, {944, 3434}, {956, 1482}, {958, 2098}, {1043, 3902}, {1222, 1411}, {1259, 3303}, {1376, 1388}, {1387, 4187}, {2099, 3868}, {2170, 2329}, {2320, 3601}, {2646, 3871}, {3061, 4390}, {3219, 3878}, {3337, 3919}, {3723, 3965}
X(4862) lies on these lines: {1, 7}, {2, 4488}, {8, 4373}, {9, 1086}, {57, 3782}, {69, 1266}, {75, 537}, {141, 4659}, {142, 3731}, {144, 3008}, {319, 4816}, {320, 3633}, {517, 1122}, {519, 4452}, {527, 1743}, {599, 4007}, {982, 1699}, {1119, 1785}, {1698, 4357}, {1836, 3677}, {2481, 3551}, {3056, 4014}, {3247, 4675}, {3620, 4431}, {3624, 4389}, {3662, 3729}, {3666, 4654}, {3772, 3928}, {3946, 4644}, {4445, 4726}
X(4862) = complement of X(4488)
X(4862) = isotomic conjugate of the isogonal conjugate of X(32577)
X(4863) lies on these lines: {1, 3925}, {2, 3689}, {8, 210}, {9, 3058}, {10, 3303}, {11, 200}, {63, 528}, {78, 3813}, {80, 4677}, {145, 3475}, {149, 3681}, {226, 519}, {319, 4479}, {354, 2550}, {355, 546}, {390, 3683}, {518, 1836}, {938, 3698}, {962, 3962}, {1953, 3930}, {2646, 3189}, {2886, 3870}, {3242, 3914}, {3452, 3711}, {3703, 3886}, {3705, 3996}, {3717, 4387}, {3772, 3938}, {3826, 4666}, {3883, 4042}
X(4864) lies on these lines: {1, 6}, {8, 3823}, {38, 3748}, {100, 3999}, {145, 3834}, {244, 3689}, {320, 3623}, {354, 750}, {513, 4162}, {519, 3836}, {726, 4702}, {752, 4356}, {910, 3726}, {990, 1482}, {1155, 3722}, {1280, 3693}, {2098, 4319}, {2099, 4327}, {2177, 4003}, {3241, 4645}, {3244, 3755}, {3315, 3935}, {3635, 4353}, {3666, 3957}, {3742, 3961}, {3744, 3873}, {3752, 3870}, {3886, 4686}, {4392, 4689}, {4430, 4641}
X(4865) lies on these lines: {1, 977}, {2, 4434}, {6, 4071}, {8, 1215}, {10, 3966}, {31, 3006}, {38, 4655}, {63, 752}, {171, 3705}, {226, 519}, {612, 3847}, {614, 3836}, {726, 1836}, {740, 3434}, {982, 4645}, {984, 4388}, {2886, 4362}, {3120, 3891}, {3178, 3295}, {3244, 4138}, {3416, 3741}, {3666, 4660}, {3703, 3923}, {3744, 3771}, {3932, 4011}, {3936, 3938}, {3961, 4417}, {4307, 4697}, {4414, 4450}
X(4866) lies on these lines: {1, 210}, {4, 3679}, {7, 10}, {8, 4082}, {9, 3913}, {21, 200}, {40, 3062}, {57, 3983}, {78, 2320}, {79, 2093}, {80, 4668}, {84, 165}, {314, 341}, {474, 3361}, {936, 1476}, {941, 3731}, {960, 3680}, {1000, 3632}, {1156, 4606}, {1320, 3876}, {1392, 3872}, {1697, 3715}, {1698, 3296}, {1743, 2298}, {2136, 4711}, {2481, 4385}, {3340, 4005}, {3601, 3711}
X(4866) = isogonal conjugate of X(3361)
X(4867) lies on these lines: {1, 6}, {35, 3869}, {36, 214}, {55, 3899}, {56, 3901}, {80, 519}, {100, 3245}, {191, 2646}, {320, 4089}, {404, 4084}, {517, 3689}, {993, 2320}, {997, 3306}, {999, 3894}, {1159, 4413}, {1385, 3962}, {1482, 3632}, {2098, 3633}, {2099, 3679}, {2802, 3935}, {2975, 4067}, {3336, 4018}, {3746, 3878}, {3811, 3895}, {3898, 3957}, {4693, 4775}, {4717, 4720}
X(4868) lies on these lines: {1, 88}, {2, 4714}, {10, 37}, {40, 572}, {42, 758}, {81, 484}, {227, 3671}, {386, 3878}, {519, 3666}, {551, 3752}, {756, 3956}, {982, 3892}, {986, 3874}, {995, 3898}, {1046, 1126}, {1193, 3884}, {1739, 3720}, {1962, 3968}, {2292, 3293}, {2650, 4757}, {3175, 4125}, {3214, 4015}, {3644, 4385}, {3670, 3881}, {3822, 3914}, {3918, 3987}, {3992, 3995}
X(4869) lies on these lines: {2, 6}, {7, 346}, {8, 142}, {57, 3692}, {85, 1229}, {144, 320}, {145, 3834}, {192, 4346}, {390, 4645}, {516, 4779}, {527, 3161}, {536, 4373}, {1086, 4452}, {1418, 3693}, {2094, 3977}, {2325, 4488}, {2345, 4675}, {3241, 3946}, {3616, 3883}, {3617, 3739}, {3621, 4361}, {3622, 4657}, {3662, 3672}, {3664, 4747}, {4445, 4678}, {4698, 4748}
X(4869) = anticomplement of X(37650)
X(4870) lies on these lines: {1, 381}, {2, 65}, {3, 4338}, {12, 519}, {30, 2646}, {56, 4355}, {226, 535}, {354, 912}, {376, 1836}, {498, 3654}, {517, 3584}, {547, 1737}, {549, 1155}, {553, 1125}, {942, 3582}, {946, 3058}, {1464, 3720}, {1478, 3655}, {1837, 3545}, {1864, 3487}, {2099, 3679}, {3057, 3656}, {3486, 3839}, {3524, 4295}, {3534, 3612}, {3838, 4511}
X(4871) lies on these lines: {1, 2}, {11, 3836}, {57, 4011}, {244, 726}, {350, 1266}, {513, 3716}, {527, 4465}, {537, 3999}, {740, 4706}, {896, 4759}, {982, 3971}, {1054, 3685}, {1155, 4432}, {1215, 3742}, {1279, 4434}, {1284, 3911}, {1575, 3943}, {2238, 4700}, {2276, 4029}, {2887, 3816}, {3306, 3923}, {3756, 3932}, {3873, 4090}, {3992, 4694}, {4655, 4679}, {4695, 4742}
X(4871) = complement of X(899)
X(4872) lies on these lines: {1, 4056}, {2, 910}, {4, 85}, {7, 354}, {11, 1447}, {20, 348}, {27, 86}, {30, 1565}, {33, 77}, {69, 189}, {75, 1370}, {103, 516}, {150, 517}, {152, 971}, {279, 3146}, {304, 315}, {319, 3681}, {320, 350}, {326, 4123}, {515, 664}, {950, 3674}, {1111, 3583}, {1434, 4292}, {1479, 3673}, {1837, 3212}, {3664, 4038}
X(4873) lies on these lines: {1, 3943}, {2, 4029}, {6, 3633}, {8, 9}, {37, 1698}, {44, 3632}, {45, 3679}, {69, 4480}, {142, 4461}, {200, 4069}, {312, 646}, {320, 3729}, {344, 4431}, {594, 3731}, {966, 4058}, {1125, 2345}, {1449, 3244}, {2323, 4513}, {3158, 3974}, {3621, 4700}, {3693, 4519}, {3763, 4718}, {3912, 4659}, {4370, 4677}, {4405, 4422}
X(4874) lies on these lines: {2, 1491}, {230, 231}, {513, 3716}, {514, 1125}, {649, 4010}, {659, 693}, {663, 2533}, {667, 814}, {812, 4782}, {900, 4786}, {1201, 4449}, {1635, 4804}, {2787, 4791}, {3762, 4378}, {4122, 4809}, {4132, 4507}, {4170, 4834}, {4367, 4391}, {4379, 4448}, {4380, 4810}, {4401, 4823}, {4486, 4817}, {4761, 4775}, {4763, 4777}, {4784, 4800}
X(4875) lies on these lines: {8, 1212}, {9, 3057}, {37, 145}, {75, 3177}, {169, 956}, {210, 3061}, {220, 3872}, {239, 257}, {241, 4384}, {910, 2975}, {958, 2082}, {960, 2170}, {965, 3554}, {966, 1108}, {1100, 2287}, {1334, 3880}, {1475, 3812}, {1841, 2322}, {2329, 2348}, {2646, 3684}, {3208, 3893}, {3632, 3991}, {3686, 3965}, {3694, 4034}, {3753, 4253}
X(4876) lies on these lines: {1, 39}, {4, 1840}, {6, 983}, {7, 192}, {8, 2170}, {9, 3056}, {21, 644}, {37, 256}, {55, 2344}, {79, 2795}, {84, 295}, {104, 813}, {294, 2340}, {314, 646}, {334, 350}, {511, 3508}, {660, 1156}, {885, 3716}, {987, 1922}, {1432, 1916}, {1911, 2298}, {1959, 3930}, {2238, 3507}, {3509, 4447}, {3680, 4050}
X(4876) = isogonal conjugate of X(1429)
X(4876) = trilinear pole of line X(210)X(650)
X(4877) lies on these lines: {1, 1778}, {6, 4653}, {9, 21}, {19, 2960}, {30, 1213}, {37, 58}, {45, 1333}, {55, 4111}, {81, 3247}, {86, 527}, {191, 2294}, {333, 2321}, {405, 579}, {573, 3560}, {1006, 1765}, {1043, 3686}, {1474, 2074}, {1761, 3647}, {2178, 4278}, {2303, 3731}, {2328, 2361}, {3730, 4269}, {3927, 4658}, {4034, 4720}
X(4878) lies on these lines: {6, 480}, {12, 3214}, {37, 42}, {43, 4000}, {44, 2293}, {45, 4343}, {200, 2345}, {344, 3870}, {674, 2347}, {1018, 4097}, {1026, 3879}, {1055, 4497}, {1193, 3242}, {1253, 2911}, {1400, 4557}, {1486, 3217}, {1818, 3751}, {2183, 3779}, {2257, 3190}, {3240, 3672}, {3293, 3755}, {3668, 4551}, {3950, 4069}, {4006, 4538}
X(4879) lies on these lines: {1, 512}, {145, 4806}, {513, 4162}, {514, 4775}, {519, 4129}, {523, 4833}, {659, 663}, {814, 4810}, {891, 4040}, {1125, 4807}, {1491, 3900}, {1577, 4774}, {1960, 4063}, {2530, 3887}, {2605, 4132}, {2785, 3801}, {2787, 4170}, {3250, 4435}, {3309, 3777}, {3737, 4139}, {3907, 4010}, {4391, 4800}, {4823, 4844}
X(4880) lies on these lines: {1, 3052}, {3, 3901}, {35, 3868}, {36, 214}, {46, 200}, {55, 3894}, {72, 3336}, {191, 942}, {404, 4067}, {484, 518}, {517, 1768}, {519, 3245}, {527, 1737}, {960, 3337}, {999, 3899}, {1046, 1203}, {1698, 3715}, {1699, 2095}, {1739, 1757}, {1789, 3193}, {2975, 4084}, {3746, 3874}, {4716, 4802}
X(4881) lies on these lines: {1, 1392}, {2, 515}, {3, 3877}, {20, 1519}, {35, 3898}, {36, 214}, {56, 3873}, {78, 4661}, {100, 1319}, {145, 1420}, {210, 2975}, {404, 1385}, {474, 3897}, {997, 3219}, {999, 3957}, {1125, 2475}, {1870, 4242}, {2646, 3742}, {3336, 4744}, {3601, 3622}, {3612, 3616}, {3794, 4225}, {4189, 4512}
Consider any triangle ABC. Let A-excircle ωA of ABC is tangent to BC at A'. Similarly define ωB, ωC, B', C'. Consider two tangent lines ℓa, ℓa' from A' to ωB, ωC which are different from BC. Similarly define ℓb, ℓb', ℓc, ℓc'. Then ℓa, ℓa', ℓb, ℓb', ℓc, ℓc' form a hexagon which has an incircle. The center of the incircle of the hexagon is X(4882). (Angel Montesdeoca, October 2, 2018)
X(4882) lies on these lines: {1, 2}, {9, 3913}, {40, 971}, {72, 1750}, {165, 3916}, {210, 1697}, {341, 3886}, {480, 3295}, {518, 1706}, {958, 3158}, {960, 2136}, {1260, 3746}, {1376, 3361}, {1743, 3713}, {2324, 4007}, {3057, 3711}, {3243, 3812}, {3303, 3983}, {3601, 3689}, {3731, 3965}, {3871, 4512}, {3876, 3895}, {4040, 4546}
X(4882) = X(18909)-of-excentral-triangle
X(4882) = AC-incircle-inverse of X(38475)
X(4882) = orthologic center of these triangles: excentral to Andromeda
X(4883) lies on these lines: {1, 3}, {6, 4666}, {37, 3873}, {42, 3742}, {81, 1279}, {518, 756}, {519, 4457}, {748, 4663}, {899, 3848}, {1001, 4641}, {1100, 3290}, {1149, 3725}, {1211, 4684}, {1722, 2334}, {2308, 3246}, {3434, 4675}, {3635, 4706}, {3689, 3979}, {3751, 4423}, {3812, 4695}, {3938, 4682}, {4418, 4702}
X(4884) lies on these lines: {38, 141}, {145, 3052}, {345, 3242}, {496, 3159}, {518, 4028}, {519, 4640}, {545, 1836}, {614, 4422}, {726, 2886}, {899, 4126}, {982, 3932}, {1401, 4553}, {3006, 3782}, {3705, 4415}, {3712, 3938}, {3717, 3752}, {3742, 4078}, {3744, 3977}, {3816, 3971}, {3840, 4439}, {4030, 4414}, {4042, 4399}
X(4885) lies on these lines: {2, 650}, {37, 4411}, {513, 3716}, {514, 4521}, {522, 676}, {523,5159}, {649, 4106}, {661, 4379}, {812, 4394}, {900, 2487}, {905, 1577}, {918, 3239}, {960, 1938}, {1635, 4382}, {1638, 3700}, {1639, 4468}, {2516, 4763}, {3669, 4391}, {3739, 4500}, {3960, 4791}, {4458, 4522}, {4467, 4820}
X(4885) = isotomic conjugate of X(30610)
X(4885) = complement of X(650)
X(4885) = anticomplement of X(31287)
X(4885) = vertex conjugate of X(56) and X(57)
X(4885) = polar conjugate of isogonal conjugate of X(22091)
X(4886) lies on these lines: {2, 319}, {8, 210}, {239, 1211}, {261, 284}, {320, 2895}, {345, 391}, {1654, 3666}, {3210, 4643}, {3218, 3578}, {3219, 4271}, {3661, 4383}, {3685, 4046}, {3696, 4388}, {3705, 4042}, {3715, 3790}, {3752, 4690}, {3883, 3996}, {4060, 4102}, {4399, 4415}, {4425, 4716}
X(4887) lies on these lines: {1, 7}, {2, 4480}, {44, 527}, {45, 142}, {69, 3625}, {75, 3626}, {88, 908}, {320, 519}, {545, 2325}, {553, 3782}, {940, 4114}, {1125, 4389}, {3620, 4058}, {3631, 4060}, {3634, 4357}, {3666, 3982}, {3667, 3676}, {3879, 4398}, {3912, 4440}, {4395, 4700}
X(4887) = incircle-inverse of X(38941)
X(4888) lies on these lines: {1, 7}, {2, 3973}, {9, 4675}, {69, 3679}, {75, 3632}, {86, 2163}, {142, 1743}, {320, 1698}, {527, 3731}, {940, 4654}, {942, 1122}, {1086, 1449}, {1756, 3338}, {3241, 4373}, {3244, 4452}, {3589, 4795}, {3624, 4357}, {3633, 3879}, {3950, 4454}, {4000, 4667}, {4034, 4688}
X(4889) lies on these lines: {1, 4445}, {37, 4725}, {141, 3244}, {145, 3834}, {192, 4715}, {319, 3723}, {519, 3739}, {524, 4681}, {536, 3879}, {894, 4727}, {1086, 4464}, {1125, 4478}, {3241, 4657}, {3629, 3950}, {3631, 4021}, {3633, 4361}, {3664, 4726}, {3686, 4755}, {4060, 4472}, {4461, 4795}
X(4890) lies on these lines: {1, 256}, {57, 2938}, {65, 4356}, {210, 3986}, {354, 3663}, {674, 3723}, {1015, 2309}, {1100, 3271}, {1213, 4111}, {1500, 3778}, {1963, 3110}, {2092, 2667}, {2223, 4343}, {2245, 4068}, {2294, 4516}, {2772, 3022}, {3247, 3779}, {3664, 4014}, {3753, 4780}
X(4890) = X(95)-of-intouch-triangle
X(4891) lies on these lines: {1, 3714}, {38, 4681}, {145, 4487}, {171, 4702}, {354, 536}, {518, 3971}, {519, 3740}, {740, 3742}, {1279, 1999}, {2667, 3706}, {3175, 3873}, {3246, 3791}, {3752, 4734}, {3816, 4028}, {3834, 3914}, {4011, 4663}, {4365, 4726}, {4415, 4684}
X(4892) lies on these lines: {2, 896}, {10, 12}, {306, 4527}, {321, 4535}, {513, 3716}, {537, 3006}, {740, 3120}, {752, 3011}, {908, 3836}, {1836, 3771}, {2796, 3712}, {3741, 3838}, {3772, 3791}, {3773, 4054}, {3914, 4743}, {3943, 4071}, {3994, 4080}, {4434, 4645}
X(4892) = isotomic conjugate of isogonal conjugate of X(20984)
X(4892) = polar conjugate of isogonal conjugate of X(22098)
X(4892) = complement of X(896)
X(4892) = complementary conjugate of X(16597)
X(4893) lies on these lines: {2, 514}, {42, 663}, {43, 4040}, {44, 513}, {522, 4120}, {523, 1639}, {812, 4776}, {891, 3250}, {3239, 4024}, {3240, 4794}, {3310, 3709}, {3679, 4844}, {3720, 4449}, {3835, 4382}, {4728, 4762}, {4770, 4775}, {4777, 4800}
X(4894) lies on these lines: {1, 977}, {4, 4692}, {5, 4030}, {8, 80}, {10, 748}, {35, 3705}, {75, 4056}, {319, 3760}, {345, 4309}, {519, 4101}, {2280, 4153}, {2478, 3992}, {3058, 3695}, {3434, 4647}, {3454, 3938}, {3583, 4385}, {3670, 4660}
X(4895) lies on these lines: {1, 2254}, {8, 3716}, {519, 3762}, {522, 3904}, {644, 3939}, {650, 663}, {661, 4775}, {667, 4729}, {678, 1635}, {690, 2650}, {900, 1317}, {926, 2170}, {1320, 3738}, {1639, 4152}, {3063, 4171}, {3309, 4449}, {4813, 4822}
X(4896) lies on these lines: {1, 7}, {10, 320}, {44, 142}, {45, 527}, {69, 3626}, {75, 3625}, {354, 4014}, {545, 4029}, {551, 4389}, {597, 4796}, {940, 3982}, {1086, 4667}, {1266, 3244}, {3008, 4644}, {3630, 4739}, {3666, 4114}, {3707, 4715}
X(4897) lies on these lines: {1, 1499}, {2, 2487}, {513, 3004}, {514, 4790}, {523, 4467}, {525, 1019}, {649, 918}, {650, 3798}, {661, 4750}, {693, 900}, {1638, 3835}, {2786, 3700}, {3265, 3733}, {3566, 4367}, {3667, 3676}, {3776, 4785}, {4394, 4468}
X(4897) = pole wrt incircle of line X(2)X(6)
X(4897) = center of cevian circle of X(190)
X(4898) lies on these lines: {1, 2321}, {8, 3986}, {9, 3633}, {37, 3632}, {145, 1743}, {344, 4464}, {346, 3244}, {391, 519}, {594, 3624}, {966, 4545}, {1213, 3247}, {1449, 3943}, {1698, 4007}, {3008, 4460}, {3616, 4058}, {3635, 4072}
X(4899) lies on these lines: {8, 144}, {10, 3662}, {145, 1743}, {306, 4661}, {344, 3243}, {354, 4126}, {518, 3717}, {519, 1757}, {537, 1738}, {1026, 1458}, {2078, 4571}, {3632, 4133}, {3667, 4404}, {3681, 3687}, {3699, 3911}, {3935, 3977}
X(4900) lies on these lines: {1, 3689}, {4, 3632}, {7, 519}, {8, 4342}, {9, 3880}, {21, 3895}, {78, 1392}, {80, 4677}, {104, 165}, {200, 1320}, {517, 3062}, {1000, 3679}, {1156, 2802}, {1476, 3361}, {2320, 3872}, {3296, 3633}
X(4901) lies on these lines: {1, 3589}, {8, 9}, {10, 4000}, {200, 1040}, {238, 3632}, {345, 3158}, {497, 4082}, {536, 984}, {1441, 4696}, {1699, 3967}, {2257, 3610}, {2550, 4659}, {3243, 3912}, {3617, 4357}, {3966, 4126}, {4030, 4512}
X(4902) lies on these lines: {1, 7}, {69, 4677}, {75, 4668}, {519, 4373}, {527, 3973}, {903, 3875}, {1086, 1743}, {3633, 4452}
X(4903) lies on these lines: {2, 726}, {8, 210}, {145, 4090}, {1215, 3616}, {1278, 4135}, {3158, 3685}, {3161, 3985}, {3175, 4734}, {3210, 3994}, {3452, 3790}, {3699, 4387}, {3705, 4082}, {3742, 3967}, {3873, 4358}, {3952, 4661}
Let La, Lb, Lc be the lines through A, B, C, resp. parallel to line X(1)X(6). Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in line X(1)X(6). The triangle A"B"C" is homothetic to ABC, with center of homothety X(4904); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)
X(4904) lies on these lines: {2, 644}, {7, 3732}, {8, 277}, {11, 116}, {12, 2140}, {141, 3679}, {142, 1145}, {150, 673}, {244, 2968}, {514, 1358}, {905, 1015}, {918, 1086}, {1375, 1429}, {1462, 1783}, {1565, 2170}
X(4904) = complement of X(644)
X(4905) lies on these lines: {1, 3309}, {10, 4462}, {36, 238}, {46, 4063}, {512, 3777}, {514, 1734}, {651, 1110}, {656, 4778}, {663, 3960}, {764, 4083}, {2832, 4498}, {3667, 4017}, {3810, 4707}, {3887, 4449}, {4151, 4801}
X(4906) is the perspector of the 1st Zaniah triangle and the tangential triangle, wrt the 1st Zaniah triangle, of the {1st Zaniah, 2nd Zaniah}-circumconic. (Randy Hutson, November 30, 2018)
X(4906) lies on these lines: {1, 474}, {31, 3999}, {63, 3246}, {81, 105}, {244, 3744}, {518, 614}, {612, 3848}, {982, 1279}, {1001, 3677}, {1104, 3976}, {1621, 4003}, {3210, 4702}, {3242, 3740}, {3683, 4392}, {3873, 4663}
X(4906) = complement of X(30615)
X(4907) lies on these lines: {1, 971}, {9, 294}, {33, 1395}, {37, 4326}, {40, 1736}, {55, 3731}, {57, 1721}, {241, 2951}, {497, 3663}, {728, 4073}, {984, 1697}, {1449, 4336}, {1723, 2361}, {2293, 3247}, {2810, 3022}
X(4908) lies on these lines: {2, 37}, {9, 4677}, {44, 519}, {45, 3679}, {190, 4715}, {545, 3912}, {551, 4029}, {594, 4745}, {903, 3834}, {1100, 3950}, {1766, 3534}, {2321, 4669}, {3762, 4762}, {4777, 4800}
X(4909) lies on these lines: {1, 7}, {69, 551}, {75, 3635}, {86, 519}, {193, 3986}, {319, 3634}, {527, 3723}, {594, 4758}, {1014, 3746}, {1100, 3008}, {1125, 3879}, {3636, 4357}, {4060, 4472}, {4698, 4700}
X(4910) lies on these lines: {1, 4399}, {6, 4464}, {141, 3633}, {145, 3834}, {519, 4445}, {536, 4460}, {1992, 4718}, {3241, 3739}, {3244, 4361}, {3618, 4727}, {3623, 4371}, {3672, 4725}, {4360, 4643}, {4686, 4795}
X(4911) lies on these lines: {1, 4056}, {4, 7}, {5, 1447}, {65, 150}, {69, 4385}, {75, 315}, {76, 320}, {79, 2481}, {85, 1478}, {348, 4293}, {355, 3212}, {515, 3674}, {1111, 3585}, {3091, 3598}
X(4912) lies on these lines: {1, 4796}, {30, 511}, {44, 4440}, {190, 3834}, {597, 3663}, {599, 3729}, {1086, 4480}, {1266, 4409}, {1698, 4363}, {3616, 4419}, {3617, 4454}, {3623, 4644}, {4416, 4726}, {4659, 4668}
X(4912) = isogonal conjugate of X(8696)
X(4913) lies on these lines: {2, 4804}, {8, 3907}, {513, 4507}, {514, 4818}, {522, 650}, {523, 2487}, {812, 1491}, {1125, 4151}, {1577, 1698}, {2787, 4770}, {4088, 4467}, {4753, 4784}, {4763, 4777}
X(4913) = complement of X(4804)
X(4914) lies on these lines: {8, 210}, {306, 3748}, {354, 3416}, {518, 2895}, {519, 1211}, {2321, 3058}, {2348, 3686}, {2550, 3646}, {3625, 4104}, {3679, 4383}, {3683, 3703}, {3687, 3689}, {3969, 4702}
X(4915) lies on these lines: {1, 2}, {9, 3880}, {165, 956}, {515, 2951}, {958, 2136}, {960, 3680}, {1697, 3683}, {1699, 3421}, {1706, 3361}, {1750, 3419}, {2324, 4034}, {3057, 3715}, {3895, 4512}
X(4916) lies on these lines: {141, 3241}, {142, 3633}, {145, 3834}, {391, 4725}, {519, 4371}, {1086, 4460}, {3161, 3629}, {3616, 4445}, {3621, 3739}, {3623, 4657}, {3729, 3879}, {4007, 4470}, {4461, 4727}
X(4917) lies on these lines: {1, 3833}, {63, 3871}, {65, 224}, {100, 3361}, {145, 1420}, {519, 3612}, {1697, 3935}, {1706, 3957}, {3295, 3305}, {3434, 3947}, {3601, 3621}, {3811, 3895}
X(4918) lies on these lines: {1, 3712}, {10, 3175}, {145, 3052}, {519, 3647}, {540, 3650}, {1211, 2292}, {3178, 3649}, {3695, 4424}, {3698, 4078}, {3710, 4646}, {3932, 4642}, {3962, 4028}
X(4919) lies on these lines: {1, 39}, {9, 644}, {41, 3885}, {78, 4050}, {101, 2802}, {220, 4051}, {294, 3680}, {517, 3509}, {2098, 3061}, {2329, 3057}, {3684, 3880}, {4514, 4544}
X(4920) lies on these lines: {1, 4056}, {7, 987}, {315, 3905}, {349, 1111}, {626, 712}, {726, 3933}, {946, 3663}, {1930, 2887}, {3665, 3782}, {3673, 3944}, {4168, 4805}, {4372, 4799}
X(4921) lies on these lines: {2, 6}, {8, 4234}, {21, 519}, {58, 3679}, {100, 4685}, {3175, 3219}, {3654, 4221}, {3681, 3794}, {4184, 4421}, {4496, 4603}, {4677, 4720}
X(4922) lies on these lines: {1, 2787}, {519, 4730}, {804, 3023}, {814, 4382}, {900, 1317}, {1960, 3762}, {2533, 3907}, {2789, 4458}, {4083, 4380}, {4160, 4824}, {4164, 4579}
X(4923) lies on these lines: {1, 4371}, {8, 9}, {10, 4698}, {142, 3696}, {740, 4407}, {3057, 4111}, {3452, 3706}, {3625, 4663}, {3626, 4078}, {3775, 4780}, {4082, 4113}
X(4924) lies on these lines: {8, 3664}, {10, 4684}, {145, 1743}, {238, 3244}, {518, 3663}, {519, 1992}, {3008, 3243}, {3621, 4431}, {3632, 4307}, {4656, 4661}
X(4925) lies on these lines: {10, 2826}, {513, 2977}, {522, 676}, {523, 3776}, {659, 1376}, {900, 3035}, {918, 2254}, {960, 2821}, {1734, 3910}, {2886, 3837}, {2976, 3667}
X(4926) lies on these lines: {6, 4501}, {30, 511}, {75, 4408}, {649, 4820}, {650, 4120}, {2516, 3239}, {3700, 4394}, {4024, 4790}, {4036, 4811}, {4106, 4467}
X(4926) = isogonal conjugate of X(8697)
X(4926) = crossdifference of every pair of points on line X(6)X(5563)
Let A' be the pole with respect to the incircle of the A-median, and define B' and C' cyclically. Then A', B', C' are collinear on the line X(3667)X(3676), which is the polar of X(2) with respect to the incircle, and X(4927) is the centroid of {A', B', C'}. (Randy Hutson, September 29, 2014)
X(4927) lies on these lines: {11, 1111}, {325, 523}, {514, 1639}, {812, 1638}, {900, 903}, {918, 4120}, {2487, 4380}, {3667, 3676}, {3700, 3776}, {3716, 4778}
X(4927) = isotomic conjugate of X(6079)
X(4928) lies on these lines: {2, 812}, {11, 116}, {513, 3716}, {514, 1639}, {1022, 3762}, {1638, 2786}, {2820, 3817}, {3239, 3776}, {4120, 4453}, {4379, 4776}
X(4928) = complement of X(1635)
X(4928) = isotomic conjugate of isogonal conjugate of X(23650)
X(4928) = polar conjugate of isogonal conjugate of X(22437)
X(4929) lies on these lines: {1, 344}, {8, 3663}, {145, 1743}, {390, 519}, {537, 4312}, {740, 3632}, {1757, 3633}, {3621, 4416}, {3679, 3775}
X(4930) lies on these lines: {1, 3683}, {329, 3241}, {355, 381}, {517, 3158}, {527, 3655}, {997, 1159}, {1385, 3928}, {1388, 3901}, {2099, 3679}, {3878, 4428}
X(4931) lies on these lines: {321, 1577}, {522, 1635}, {523, 661}, {649, 4820}, {756, 4041}, {824, 4728}, {2786, 4789}, {3971, 4151}, {4777, 4800}
X(4932) lies on these lines: {2, 4813}, {239, 514}, {513, 3716}, {522, 4784}, {659, 4778}, {693, 4785}, {812, 4790}, {900, 4500}, {3676, 4817}
X(4932) = complement of X(4813)
X(4933) lies on these lines: {2, 740}, {518, 4141}, {519, 3722}, {524, 896}, {599, 4414}, {2177, 3679}, {2796, 3936}, {4028, 4722}, {4777, 4800}
X(4934) lies on these lines: {1, 542}, {7, 4616}, {11, 1365}, {125, 2611}, {523, 4092}, {1367, 3022}, {1565, 4014}, {2310, 4017}, {4466, 4516}
X(4935) lies on these lines: {8, 443}, {145, 4487}, {321, 3632}, {519, 3701}, {3621, 4671}, {3625, 4647}, {3633, 4723}, {3902, 4066}
X(4936) lies on these lines: {1, 644}, {9, 3057}, {145, 1743}, {169, 4752}, {200, 220}, {1334, 4512}, {2136, 2348}, {2256, 2297}
X(4937) lies on these lines: {2, 726}, {519, 3952}, {536, 899}, {537, 4358}, {3679, 4125}, {3720, 3967}, {4693, 4767}, {4777, 4800}
X(4938) lies on these lines: {8, 2650}, {306, 4722}, {519, 4442}, {524, 896}, {1698, 4658}, {1962, 2895}, {3244, 4425}, {4727, 4802}
X(4939) lies on these lines: {4, 2840}, {11, 123}, {244, 522}, {341, 3680}, {1089, 3813}, {1365, 4815}, {1421, 1897}, {3880, 3992}
X(4940) lies on these lines: {2, 4790}, {513, 3716}, {650, 4380}, {661, 4106}, {4394, 4785}, {4500, 4802}, {4728, 4813}
X(4940) = complement of X(4790)
X(4941) lies on these lines: {1, 1463}, {192, 3123}, {256, 3672}, {982, 3663}, {984, 4085}, {2228, 3644}, {3122, 4398}
X(4942) lies on these lines: {321, 4042}, {940, 3994}, {1376, 3729}, {3715, 4756}, {3740, 4659}, {3927, 4066}, {3971, 4363}
X(4943) lies on these lines: {30, 511}, {661, 4773}, {2527, 4790}, {2976, 4806}
X(4943) = isogonal conjugate of X(8698)
X(4944) lies on these lines: {210, 3900}, {312, 4391}, {513, 4120}, {522, 650}, {661, 4802}, {4777, 4800}
X(4945) lies on these lines: {2, 45}, {80, 519}, {679, 3834}, {1022, 4728}, {3257, 4396}, {3679, 4767}
X(4946) lies on these lines: {1, 2}, {2238, 4029}, {2276, 4700}, {3773, 4819}, {3896, 3994}, {4689, 4753}
X(4947) lies on these lines: {1, 4014}, {256, 2481}, {291, 3123}, {903, 3122}, {982, 2310}, {3056, 3551}
X(4947) = areal center of cevian triangles of PU(34)
X(4948) lies on these lines: {2, 523}, {1491, 4762}, {3679, 4770}, {4753, 4784}, {4777, 4800}, {4825, 4844}
X(4949) lies on these lines: {241,512}, {513, 3700}, {900, 4765}, {2527, 3239}, {2976, 3667}, {4120, 4790}, {4727, 4802}
X(4950) lies on these lines: {1, 626}, {315, 3721}, {712, 4056}, {754, 1759}, {4136, 4376}
X(4951) lies on these lines: {522, 3971}, {523, 4776}, {824, 1491}, {4125, 4791}, {4777, 4800}
X(4952) lies on these lines: {8, 3772}, {145, 4487}, {518, 3784}, {519, 3452}, {4030, 4643}
X(4953) lies on these lines: {522, 1086}, {1146, 2310}, {2643, 4843}, {3271, 3900}, {3939, 4370}
X(4954) lies on these lines: {2, 3996}, {36, 100}, {536, 3935}, {2177, 3679}, {4693, 4767}
X(4955) lies on these lines: {7, 8}, {57, 3294}, {241, 553}, {1319, 1434}, {3665, 3671}
X(4956) lies on these lines: {2, 968}, {149, 519}, {2796, 3218}, {3679, 4125}
X(4957) lies on these lines: {11, 523}, {75, 545}, {918, 1086}, {3262, 3943}
X(4957) = isotomic conjugate of X(5385)
X(4958) lies on these lines: {522, 661}, {900, 1635}, {2786, 4453}, {4727, 4802}
X(4959) lies on these lines: {650, 663}, {3887, 4449}, {4163, 4543}
X(4960) lies on these lines: {239, 514}, {4716, 4802}, {4813, 4823}
X(4961) lies on these lines: {30, 511}, {4170, 4380}, {4810, 4823}
X(4962) lies on these lines: {30, 511}, {1635, 3239}, {4791, 4811}
X(4962) = isogonal conjugate of X(8699)
X(4963) lies on these lines: {514, 4122}, {4727, 4802}, {4753, 4784}
X(4964) lies on these lines: {30, 511}, {145, 4504}, {3632, 4170}
X(4965) lies on these lines: {3020, 4778}, {3248, 3737}, {3271, 4124}
X(4966) lies on these lines: {1, 141}, {2, 4023}, {8, 3826}, {11, 3936}, {69, 1001}, {142, 3696}, {145, 4429}, {238, 524}, {244, 4062}, {306, 354}, {320, 3685}, {516, 4702}, {518, 3717}, {519, 3836}, {528, 4645}, {726, 3943}, {740, 1086}, {942, 3704}, {1100, 1125}, {1211, 3720}, {1269, 3649}, {1386, 3879}, {1738, 3834}, {1757, 4422}, {2550, 4869}, {3218, 3712}, {3244, 4085}, {3589, 4649}, {3683, 4001}, {3687, 3742}, {3695, 3874}, {3703, 3873}, {3739, 4733}, {3752, 4028}, {3816, 4417}, {3957, 4030}, {3966, 4666}, {3976, 4446}, {4046, 4359}, {4054, 4519}, {4126, 4661}, {4133, 4686}, {4395, 4716}
X(4967) lies on these lines: {1, 4464}, {2, 2321}, {7, 3617}, {8, 3879}, {10, 75}, {37, 4431}, {69, 3679}, {86, 519}, {141, 4688}, {142, 3661}, {145, 4923}, {193, 4034}, {319, 3626}, {320, 4691}, {518, 4111}, {527, 1654}, {536, 1213}, {594, 3739}, {894, 3686}, {966, 3729}, {1086, 4739}, {1100, 4399}, {1125, 4360}, {1268, 3634}, {1698, 4078}, {2345, 4384}, {2663, 4489}, {3619, 4859}, {3622, 4460}, {3629, 4499}, {3662, 4772}, {3672, 3790}, {3846, 3932}, {3943, 4698}, {3950, 4687}, {3963, 4359}, {3986, 4664}, {4058, 4751}, {4363, 4416}, {4364, 4686}, {4445, 4675}, {4708, 4726}
X(4967) = complement of X(17319)
X(4968) lies on these lines: {1, 321}, {2, 3701}, {7, 8}, {10, 38}, {21, 3757}, {92, 4198}, {145, 3902}, {244, 3831}, {274, 1390}, {278, 318}, {312, 3616}, {341, 4003}, {354, 3714}, {442, 3006}, {519, 2650}, {551, 4066}, {726, 2292}, {742, 4754}, {946, 4054}, {956, 4185}, {1010, 3920}, {1089, 1125}, {1193, 1215}, {1230, 4205}, {1269, 4026}, {1468, 4362}, {1697, 4659}, {1770, 4450}, {2476, 3705}, {3241, 4673}, {3617, 4737}, {3622, 4671}, {3626, 4487}, {3634, 3992}, {3635, 4717}, {3915, 3923}, {4142, 4391}, {4677, 4935}, {4691, 4738}
X(4969) lies on these lines: {1, 4285}, {6, 8}, {9, 3633}, {37, 3244}, {44, 519}, {45, 145}, {75, 3629}, {141, 3759}, {193, 4361}, {239, 320}, {319, 3589}, {391, 3623}, {527, 4409}, {536, 4480}, {597, 3661}, {894, 4399}, {902, 4819}, {966, 3622}, {1100, 1125}, {1146, 2323}, {1266, 4715}, {1449, 1698}, {1743, 4873}, {1914, 3712}, {1992, 4363}, {2170, 4053}, {2308, 4046}, {2968, 3284}, {3187, 4415}, {3618, 4445}, {3630, 3662}, {3758, 4665}, {3912, 4725}, {4364, 4393}, {4416, 4852}, {4464, 4681}, {4667, 4688}
X(4969) = {X(6),X(8)}-harmonic conjugate of X(17369)
X(4970) lies on these lines: {1, 3210}, {2, 3993}, {38, 519}, {42, 726}, {43, 192}, {171, 4360}, {239, 846}, {306, 3821}, {333, 4716}, {536, 1215}, {740, 3666}, {899, 3995}, {984, 4685}, {1100, 4697}, {1125, 1962}, {1655, 4489}, {2177, 3891}, {2308, 4427}, {2901, 3831}, {2999, 4011}, {3187, 4414}, {3240, 4090}, {3244, 3873}, {3663, 4028}, {3687, 4425}, {3703, 4085}, {3740, 4681}, {3752, 4871}, {3775, 4046}, {3791, 4640}, {3840, 4850}, {3847, 4854}, {3875, 4362}, {3967, 4718}, {3989, 4651}, {4780, 4847}
X(4970) = complement of X(4365)
X(4971) lies on these lines: {1, 4472}, {2, 594}, {8, 4364}, {30, 511}, {37, 4399}, {141, 3875}, {145, 4363}, {239, 3943}, {1100, 4431}, {1386, 4133}, {1449, 4910}, {2321, 3589}, {2345, 4460}, {3241, 4499}, {3244, 4670}, {3621, 4419}, {3623, 4470}, {3625, 4690}, {3626, 4708}, {3629, 3729}, {3631, 3663}, {3632, 4643}, {3633, 4659}, {3664, 4726}, {3672, 4445}, {3679, 4026}, {3686, 4681}, {3879, 4686}, {3912, 4395}, {3932, 4716}, {4007, 4657}, {4021, 4060}, {4357, 4478}, {4366, 4370}, {4384, 4405}, {4416, 4718}
X(4971) = isogonal conjugate of X(8700)
X(4972) lies on these lines: {1, 4202}, {2, 11}, {8, 3891}, {10, 321}, {31, 4450}, {38, 3821}, {42, 2887}, {65, 4463}, {81, 4645}, {306, 3755}, {740, 3969}, {752, 2308}, {860, 1824}, {899, 3847}, {1211, 4651}, {1215, 3120}, {1220, 2475}, {1738, 4359}, {1757, 4683}, {2177, 3771}, {2975, 4201}, {3006, 3666}, {3187, 3416}, {3240, 4417}, {3293, 3454}, {3662, 3873}, {3705, 4850}, {3706, 3844}, {3720, 3836}, {3773, 4365}, {3834, 4883}, {3932, 3995}, {3952, 4415}, {4414, 4438}
X(4973) lies on these lines: {1, 4757}, {3, 3874}, {10, 529}, {21, 3337}, {35, 3881}, {36, 214}, {46, 3872}, {55, 3892}, {56, 3878}, {57, 993}, {404, 3678}, {474, 3715}, {484, 2802}, {519, 1155}, {535, 1737}, {551, 4640}, {553, 1125}, {902, 4694}, {982, 4257}, {995, 4650}, {997, 3928}, {999, 3898}, {1385, 4084}, {2392, 3937}, {2975, 3336}, {3244, 3579}, {3338, 4652}, {3814, 3911}
X(4974) lies on these lines: {1, 872}, {2, 3791}, {10, 1386}, {44, 726}, {75, 4672}, {238, 239}, {518, 4753}, {519, 1279}, {537, 1757}, {748, 3187}, {752, 1738}, {899, 4434}, {1100, 1125}, {1914, 4771}, {1964, 3216}, {2308, 4359}, {3008, 3836}, {3744, 4685}, {3883, 4085}, {3923, 4361}, {3993, 4852}, {4000, 4655}, {4362, 4383}
X(4974) = complement of X(32846)
X(4975) lies on these lines: {1, 312}, {2, 4714}, {8, 3898}, {10, 3902}, {36, 3685}, {39, 4099}, {321, 551}, {341, 3633}, {350, 1111}, {497, 4680}, {519, 3992}, {726, 4694}, {999, 4387}, {1015, 4037}, {1125, 1962}, {1201, 2901}, {1227, 4432}, {1698, 4673}, {1739, 4871}, {3244, 3701}, {3635, 4696}, {3840, 4424}
X(4976) lies on these lines: {513, 4841}, {514, 4790}, {522, 650}, {523, 649}, {654, 1021}, {661, 900}, {663, 4843}, {665, 4151}, {676, 4804}, {693, 1638}, {812, 3004}, {918, 4467}, {1635, 4024}, {2487, 4379}, {2527, 4838}, {2977, 4122}, {3904, 3910}, {4025, 4762}, {4041, 4528}, {4394, 4777}
X(4977) lies on these lines: {30, 511}, {649, 4841}, {650, 2523}, {659, 3004}, {661, 1639}, {676, 1459}, {693, 4806}, {1220, 4581}, {1491, 2977}, {2254, 4824}, {2505, 2526}, {2517, 4462}, {2605, 4040}, {3700, 4813}, {3737, 4960}, {3762, 4036}, {4024, 4958}, {4057, 4367}, {4560, 4840}, {4782, 4932}, {4969, 4976}
X(4977) = isogonal conjugate of X(8701)
X(4977) = isotomic conjugate of X(6540)
X(4977) = X(2)-Ceva conjugate of X(35076)
X(4977) = crossdifference of every pair of points on line X(6)X(595)
X(4978) lies on these lines: {513, 4170}, {514, 661}, {522, 4905}, {523, 2530}, {784, 3777}, {812, 1019}, {814, 4378}, {891, 2533}, {905, 4762}, {2254, 4151}, {2517, 4404}, {3669, 4077}, {3837, 4705}, {3910, 4707}, {3960, 4560}, {4063, 4369}, {4083, 4761}, {4086, 4802}, {4379, 4498}
X(4978) = isotomic conjugate of X(37212)
X(4979) lies on these lines: {44, 513}, {512, 4895}, {514, 4380}, {522, 4838}, {667, 4822}, {693, 4785}, {900, 4024}, {1019, 3960}, {1639, 2527}, {3004, 4750}, {3667, 4931}, {3700, 4958}, {4041, 4834}, {4106, 4379}, {4120, 4943}, {4369, 4728}, {4729, 4814}, {4944, 4949}, {4969, 4976}
X(4979) = isogonal conjugate of X(37212)
X(4979) = cevapoint of X(4977) and X(4992)
X(4979) = crosspoint of X(i) and X(j) for these {i,j}: {1, 37212}, {513, 1019}
X(4979) = crosssum of X(i) and X(j) for these {i,j}: {1, 4979}, {10, 24089}, {100, 1018}, {513, 3723}, {661, 3743}
X(4979) = crossdifference of every pair of points on line X(1)X(748)
X(4980) lies on these lines: {2, 37}, {8, 4018}, {519, 2650}, {527, 3578}, {551, 3702}, {1089, 3828}, {1150, 3928}, {1230, 3264}, {1441, 4654}, {3187, 4363}, {3679, 4696}, {3782, 4665}, {3929, 4659}, {3969, 4431}, {3982, 4060}, {4487, 4669}, {4714, 4723}, {4717, 4742}
X(4981) lies on these lines: {2, 210}, {8, 3896}, {10, 38}, {321, 984}, {333, 3920}, {519, 1962}, {612, 1150}, {740, 3989}, {756, 3741}, {1211, 3006}, {1213, 3726}, {2887, 4407}, {3187, 4042}, {3210, 3617}, {3666, 4651}, {3706, 3995}, {3720, 3842}, {4698, 4883}
X(4982) lies on these lines: {1, 3707}, {6, 2325}, {8, 1449}, {9, 3623}, {44, 3635}, {145, 4873}, {320, 3946}, {527, 4393}, {1100, 1125}, {2321, 3633}, {3241, 4029}, {3629, 4021}, {3664, 4395}, {4360, 4480}, {4405, 4670}
X(4983) is the bicentric difference of PU(113); i.e., of the points P(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b) (trilinears) and U(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a). See PU(113) at Bicentric Pairs.
X(4983) lies on these lines: {36, 238}, {512, 661}, {514, 4010}, {523, 4170}, {650, 4834}, {663, 2520}, {838, 2978}, {1577, 4806}, {2084, 4079}, {2533, 4129}, {3800, 4808}, {4151, 4824}, {4160, 4879}, {4162, 4775}
X(4983) = midpoint of X(663) and X(4813)
X(4983) = reflection of X(i) in X(j) for these {i,j}: {4705,661}, {4730,4705}, {4729,4770}, {4834,650}, {1577,4806}, {2533,4129}, {4790,6050}, {4978,4992}
X(4983) = X(i)-cevaconjugate of X(j) for these {i,j}: {1,3125}, {513,4979}, {756,3122}, {4977,4988}
X(4983) = crosspoint of X(i) and X(j) for these (i,j): {513,661}, {1125,4115}, {4052,4552}, {4977,4979}
X(4983) = crosssum of X(100) and X(662)
X(4983) = crossdifference of every pair of points on the line X(37)X(81)
X(4983) = X(i)-isoconjugate of X(j) for these {i,j}: {1,4596}, {2,4629}, {6,4632}, {99,1126}, {110,1268}, {190,1171}, {648,1796}, {662,1255}, {4102,4565}, {4570,4608}
X(4983) = isogonal conjugate of X(4596)
X(4983) = PU(113)-harmonic conjugate of X(6155)
X(4984) lies on these lines: {522, 649}, {650, 4949}, {812, 4453}, {900, 1635}, {3667, 4893}, {3798, 4382}, {4115, 4427}, {4379, 4786}, {4394, 4944}, {4543, 4730}, {4765, 4813}, {4790, 4802}, {4969, 4976}
X(4984) = tripolar centroid of X(1125)
X(4984) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 12
X(4985) lies on these lines: {513, 1577}, {514, 4815}, {522, 3717}, {523, 3762}, {693, 4778}, {834, 4170}, {900, 4036}, {2517, 3667}, {3716, 3737}, {3766, 4509}, {4140, 4526}, {4404, 4777}
X(4986) lies on these lines: {1, 4561}, {8, 150}, {75, 537}, {304, 3632}, {321, 4115}, {341, 3760}, {350, 3992}, {519, 3263}, {2170, 4568}, {2481, 4385}, {2786, 3762}, {4437, 4904}
X(4987) lies on these lines: {65, 4544}, {230, 4892}, {320, 3684}, {518, 4541}, {524, 4771}, {527, 3263}, {553, 3578}, {752, 3290}, {1213, 4697}, {1281, 3509}, {2796, 4037}
X(4988) lies on these lines: {2, 4608}, {239, 514}, {522, 4813}, {523, 661}, {650, 4802}, {1635, 2527}, {3005, 4705}, {4500, 4776}, {4521, 4893}, {4969, 4976}, {4777, 4949}
X(4988) = isogonal conjugate of X(4629)
X(4988) = isotomic conjugate of X(4632)
X(4988) = complement of X(4608)
X(4988) = complementary conjugate of complement of X(35327)
X(4988) = intersection of perspectrices of [ABC and Gemini triangle 11] and [ABC and Gemini triangle 12]
X(4989) lies on these lines: {1, 4878}, {44, 4353}, {238, 3946}, {390, 3755}, {553, 2308}, {1100, 1125}, {1191, 4342}, {1386, 3008}, {1453, 4293}, {4000, 4312}
X(4990) lies on these lines: {525, 676}, {650, 4843}, {663, 3700}, {667, 900}, {1639, 4041}, {2527, 4834}, {3239, 3900}, {3566, 4874}, {3716, 3910}, {4162, 4944}
X(4991) lies on these lines: {6, 726}, {10, 3759}, {519, 597}, {1051, 3757}, {1100, 1125}, {1279, 3635}, {2308, 4427}, {3244, 3717}, {3993, 4393}, {4672, 4852}
X(4992) lies on these lines: {512, 3837}, {514, 4806}, {814, 4106}, {891, 4129}, {900, 2530}, {2533, 4728}, {3835, 4083}, {4490, 4776}, {4560, 4810}
Let H=X(4), the orthocenter of triangle ABC, let A'B'C' be the orthic triangle, let A''=AA'∪B'C', and define B'' and C'' cyclically. Let LA be the perpendicular bisector of AA'', and define LB and LC cyclically. Let UA=LB∩LC., and define UB and UC cyclically. The lines AUA, BUB, CUC concur in X(4993). (César E. Lozada, Hyacinthos #20787, February 6, 2012)
X(4993) lies on these lines: {2, 95}, {5, 49}
Continuing from X(4993), let AA be the perpendicular bisector of HA'', and define VB and VC cyclically. Let WA=VB∩VC., and define WB and WC cyclically. The lines AWA, BWB, CWC concur in X(4994). (César E. Lozada, Hyacinthos #20787, February 6, 2012)
X(4994) lies on these lines: {4, 54}, {5, 97}, {50, 252}, {95, 3090}
X(4995) = 4r*X(2) + (R - 2r)*X(11) (Peter
Moses, February 28, 2012)
X(4995) = R*X(1) + r*X(2) + r*X(3) (Peter Moses, April
2, 2013)
X(4995) lies on these lines: {1, 549}, {2, 11}, {3, 4317}, {12, 30}, {56, 3524}, {140, 3582}, {165, 4654}, {376, 3085}, {381, 498}, {396, 1250}, {519, 2646}, {524, 2330}, {546, 4330}, {551, 3057}, {553, 1155}, {597, 3056}, {631, 3303}, {950, 3828}, {1478, 3534}, {1656, 4309}, {2482, 3023}, {3304, 3523}, {3545, 4294}, {3579, 3649}, {3601, 3679}, {3612, 3655}, {3628, 4857}, {3748, 3911}, {3830, 4302}, {4126, 4152}, {4689, 4854}
X(4996) lies on these lines: {3, 8}, {7, 1470}, {11, 21}, {12, 404}, {35, 2802}, {36, 214}, {55, 1320}, {80, 993}, {149, 4189}, {160, 1631}, {318, 3520}, {411, 2829}, {1387, 1621}, {1437, 3045}, {1768, 4652}, {1811, 3433}, {2771, 3916}, {3032, 4276}, {4188, 4293}
X(4996) = midpoint of X(100) and X(2975)
X(4996) = harmonic center of circumcircle and AC-incircle
X(4997) lies on these lines: {2, 45}, {8, 11}, {106, 1125}, {189, 1997}, {312, 646}, {320, 908}, {333, 645}, {901, 1311}, {1022, 3762}
X(4997) = isogonal conjugate of X(1404)
X(4997) = complement of X(30577)
X(4997) = trilinear pole of line X(8)X(522)
X(4997) = pole wrt polar circle of trilinear polar of X(1877)
X(4997) = X(48)-isoconjugate (polar conjugate) of X(1877)
X(4998) lies on these lines: {2, 1252}, {59, 4600}, {99, 2222}, {100, 693}, {190, 3239}, {320, 765}, {650, 666}, {651, 4607}, {664, 3676}, {1016, 1429}, {1025, 3570}, {1447, 3263}, {4417, 4619}, {4551, 4573}
X(4998) = isogonal conjugate of X(3271)Let A' be the perpendicular projection of X(3) on the line AX(21), and define B' and C' cyclically. The triangle A'B'C' is persepctive to the medial triangle of ABC, and the perspector is X(4999). (Angel Montesdeoca, March 5, 2019)
X(4999) lies on these lines: {2, 12}, {3, 2886}, {5, 993}, {9, 583}, {10, 140}, {11, 21}, {35, 528}, {36, 442}, {55, 3813}, {230, 1107}, {404, 3925}, {405, 499}, {474, 3826}, {498, 956}, {631, 1376}, {632, 3820}, {758, 942}, {946, 4640}, {988, 3772}, {1001, 3086}, {1058, 4428}, {1212, 3039}, {1387, 3884}, {1479, 3829}, {1836, 4652}, {2550, 3523}, {3218, 3649}, {3419, 3612}, {3628, 3814}, {3702, 3712}, {3812, 3911}, {3815, 4426}, {3838, 4292}
X(4999) = isogonal conjugate of X(18772)
X(4999) = isotomic conjugate of isogonal conjugate of X(20959)
X(4999) = complement of X(12)
X(4999) = anticomplement of X(6668)
X(4999) = polar conjugate of isogonal conjugate of X(22056)
X(4999) = complementary conjugate of X(34829)
Walsmith Points, X(5000) and X(5001)
Early in 2012, Russell Walsmith introduced this point and conjectured that it lies on the Euler line. Peter Moses found barycentrics for the point and its inverse-in-the-circumcircle, X(5001), and proved that the two points lie on the Euler line. To summarize from Walsmith's paper, (not available), in a three dimensional cartesian coordinate system, let A=(u,0,0), B=(0,v,0), C=(0,0,w), and let D be the point such that the distances |AD|, |BC|, |CD| are proportional to u,v,w; that is, D is the point having homogeneous tripolar coordinates u : v : w. The equations
a2 = v2 + w2, b2 = w2 + u2, c2 = u2 + v2
are equivalent to
u2 = (b2 + c2 - a2)/2, v2 = (c2 + a2 - b2)/2, w2 = (a2 + b2 - c2)/2.
The Euler line of ABC is given by the tripolar equation
(b2 - c2)x2 + (c2 - a2)y2 + (a2 - b2)z2 = 0,
and it is easy to see that this equation is satisfied by putting (x2, y2, z2) = (u2, v2, w2).
As a point on the Euler line, X(5000) has Shinagawa coefficients (1 -k2, -1 + 3k2).
The position of the point D = X(5000) on the Euler line is shown by distance ratios involving the familiar triangle centers G=X(2), O=X(3), and H=X(4) on the Euler line, as follows:
|OD|/|OG| = 3k2
|OD|/|OH| = k2
|GD|/|DH| = (1 - 3k2)/(3k2 - 3).
See the preamble just before X(5000).
If the reference triangle ABC is obtuse, then the barycentrics of X(5000) are a nonreal complex numbers. Nevertheless, for all triangles ABC, the midpoint of X(5000) and X(5001) is X(468). The circles mentioned just below are members of a coaxal system of which X(5000) and X(5001) are the so-called point-circles or limiting points, so that the circle with diameter X(5000)X(5001) is orthogonal to the circles in the system.
X(5000) and X(5001) are the antiorthocorrespondents of X(6); i.e. they share the same orthocorrespondent, X(6). See Bernard Gibert, Table 55: X(5000, X(5001) and related curves.
For a construction of X(5000) and X(5001) see AdGeom 5185.
If you have The Geometer's Sketchpad, you can view X(5000), or click here for a 3-dimensional representation of X(5000) as a gif.
The Walsmith point, as well as X(5001), lie on the Walsmith rectangular hyperbola, introduced in the preamble just before X(32110).
In the plane of a triangle ABC, let P be a point on the Euler line and P' = isogonal conjugate of P, and let DEF = medial triangle. Let ℓa = reflection of AP' in EF, and define ℓb and ℓc cyclically. The lines ℓa, ℓb, ℓc concur in a point, P", that lies on the Euler line. The mapping P → P'' is a projective transformation of the Euler line, and its fixed points are X(5000) and X(5001). If P=X(3) + t X(4), then P"= 4 t SA SB SC X(3) + (a^2 b^2 c^2 + 4 t SA SB SC) X(4). The fixed points are obtained for t = (2 SA SB SC ± S Sqrt[2 (a^2+b^2+c^2) SA SB SC])/(4 SA SB SC). (Angel Montesdeoca, April 19, 2021)
For a construction, see Antreas Hatzipolakis and Peter Moses, euclid 7200.
X(5000) lies on 8th Grozdev-Dekov-Parry circle, Dao-Moses-Telv circle, Moses radical circle, Stevanovic circle, Walsmith rectangular hyperbola; cubics K018, K270, K336, K337, K570, K608, K828, K829, K1091, K1092, K1129, K1133a, K1133b; curves Q019, Q021, Q024, Q026, Q037, Q049, Q054, Q098, Q115, Q116, Q117, Q118, Q144, Q146, Q147 and this line: {2,3}
X(5000) = isogonal conjugate of X(32618)
X(5000) = isotomic conjugate of X(42811)
X(5000) = complement of X(5002)
X(5000) = circumcircle-inverse of X(5001)
X(5000) = nine-point-circle-inverse of X(5001)
X(5000) = orthocentroidal-circle-inverse of X(5001)
X(5000) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5001)
X(5000) = polar-circle-inverse of X(5001)
X(5000) = tangential-circle-inverse of X(5001)
X(5000) = MacBeath-inconc-inverse of X(5001)
X(5000) = Yff-hypergbola-inverse of X(5001)
X(5000) = Walsmith-rectangular-hyperbola-inverse of X(5001)
X(5000) = polar conjugate of X(41194)
X(5000) = complementary conjugate of the complement of X(34136)
X(5000) = antigonal conjugate of X(42809)
X(5000) = orthoassociate of X(5001)
| PART 1: | Introduction and Centers X(1) - X(1000) | PART 2: | Centers X(1001) - X(3000) | PART 3: | Centers X(3001) - X(5000) |
| PART 4: | Centers X(5001) - X(7000) | PART 5: | Centers X(7001) - X(10000) | PART 6: | Centers X(10001) - X(12000) |
| PART 7: | Centers X(12001) - X(14000) | PART 8: | Centers X(14001) - X(16000) | PART 9: | Centers X(16001) - X(18000) |
| PART 10: | Centers X(18001) - X(20000) | PART 11: | Centers X(20001) - X(22000) | PART 12: | Centers X(22001) - X(24000) |
| PART 13: | Centers X(24001) - X(26000) | PART 14: | Centers X(26001) - X(28000) | PART 15: | Centers X(28001) - X(30000) |
| PART 16: | Centers X(30001) - X(32000) | PART 17: | Centers X(32001) - X(34000) | PART 18: | Centers X(34001) - X(36000) |
| PART 19: | Centers X(36001) - X(38000) | PART 20: | Centers X(38001) - X(40000) | PART 21: | Centers X(40001) - X(42000) |
| PART 22: | Centers X(42001) - X(44000) | PART 23: | Centers X(44001) - X(46000) | PART 24: | Centers X(46001) - X(48000) |
| PART 25: | Centers X(48001) - X(50000) | PART 26: | Centers X(50001) - X(52000) | PART 27: | Centers X(52001) - X(54000) |
| PART 28: | Centers X(54001) - X(56000) | PART 29: | Centers X(56001) - X(58000) | PART 30: | Centers X(58001) - X(60000) |
| PART 31: | Centers X(60001) - X(62000) | PART 32: | Centers X(62001) - X(64000) | PART 33: | Centers X(64001) - X(66000) |
| PART 34: | Centers X(66001) - X(68000) | PART 35: | Centers X(68001) - X(70000) | PART 36: | Centers X(70001) - X(72000) |
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