X₆₅₃

TRILINEAR POLE OF LINE X₁X_4

X₆₅₃ = polar conjugate of X₅₂₂

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

X₆₅₃ (TRILINEAR POLE OF LINE X₁X_4) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {29, 65}, {57, 92}

• Related Centers: X₆₅₂ (CROSSDIFFERENCE OF X₁ AND X₄), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER), X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE), X₅₇ (ISOGONAL CONJUGATE OF X₉), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT)

Trilinears: \(1/(\sec B - \sec C) : 1/(\sec C - \sec A) : 1/(\sec A - \sec B)\)

Barycentrics: \((\sin A)/(\sec B - \sec C) : (\sin B)/(\sec C - \sec A) : (\sin C)/(\sec A - \sec B)\)

Search Value (6, 9, 13): \(-0.177359197694\)

▸ Alternate Coordinates1 form

• Barycentrics: \(1/[(b - c)(b + c - a)(b^{2} + c^{2} - a^{2})] : :\)

▸ Linear Combinations1 combination

• X₆₅₃ = \(3\) X₂ - \(4\) X₄₀₅₃₅

X₆₅₃ lies on the curve CC₉.

▸ Incident Central Lines137 lines

{1, 2636}, {2, 196}, {3, 1148}, {4, 1156}, {7, 281}, {8, 44696}, {9, 342}, {19, 273}, {20, 3176}, {27, 18815}, {29, 65}, {34, 37129}, {40, 1895}, {46, 158}, {56, 23772}, {57, 92}, {63, 1767}, {77, 2331}, {78, 207}, {85, 2002}, {88, 278}, {100, 108}, {101, 4605}, {107, 109}, {112, 1305}, {190, 6335}, {208, 318}, {225, 897}, {226, 2349}, {241, 14571}, {242, 1876}, {243, 1155}, {297, 17950}, {307, 6330}, {329, 40837}, {331, 37130}, {347, 1249}, {349, 37220}, {425, 2651}, {450, 17975}, {468, 17985}, {484, 1784}, {514, 1461}, {607, 3212}, {608, 20332}, {646, 42384}, {648, 662}, {651, 1783}, {652, 2639}, {655, 17924}, {658, 13149}, {660, 43923}, {685, 36084}, {692, 23353}, {799, 4572}, {823, 15352}, {934, 2405}, {962, 40836}, {1013, 37541}, {1020, 2637}, {1025, 37206}, {1041, 40987}, {1047, 7138}, {1075, 20764}, {1118, 1788}, {1119, 8732}, {1214, 41083}, {1360, 4081}, {1400, 1821}, {1416, 21210}, {1430, 9364}, {1441, 2322}, {1447, 5089}, {1452, 37390}, {1585, 13437}, {1586, 13459}, {1708, 1748}, {1737, 7541}, {1826, 7282}, {1828, 40446}, {1844, 12432}, {1845, 12736}, {1846, 19636}, {1847, 7719}, {1857, 3474}, {1880, 7233}, {1957, 9316}, {2501, 9358}, {2656, 2658}, {3064, 37139}, {3162, 30918}, {3209, 9308}, {3257, 46102}, {3339, 39585}, {3559, 7098}, {3658, 35360}, {3729, 7101}, {3911, 17923}, {4296, 44698}, {4312, 39531}, {4554, 37215}, {4599, 42396}, {4848, 5174}, {4997, 37768}, {5236, 8756}, {5307, 43759}, {6332, 44327}, {6354, 18679}, {6360, 7011}, {7012, 7649}, {7013, 45738}, {7017, 32939}, {7049, 38284}, {7079, 8545}, {7115, 36087}, {7156, 34059}, {7551, 39542}, {8755, 22464}, {9056, 36067}, {9778, 44695}, {14257, 17555}, {14837, 23984}, {16813, 32660}, {17074, 18676}, {17077, 17913}, {18097, 37221}, {18588, 30841}, {21186, 24033}, {21617, 25993}, {23982, 23986}, {26003, 37787}, {30725, 46118}, {34393, 40626}, {35486, 38295}, {36090, 36110}, {37222, 37789}, {37797, 37799}

▸ Functional Relations25 relations

• X₆₅₃ = reflection of X₁₆₅₉₆ in X₄₀₅₃₅

• X₆₅₃ = isogonal conjugate of X₆₅₂

• X₆₅₃ = isotomic conjugate of X₆₃₃₂

• X₆₅₃ = anticomplement of X₁₆₅₉₆

• X₆₅₃ = polar conjugate of X₅₂₂

• X₆₅₃ = trilinear product of PU(76)

• X₆₅₃ = crossdifference of PU(77)

• X₆₅₃ = trilinear pole wrt tangential triangle of line X₁X₄

• X₆₅₃ = pole wrt polar circle of trilinear polar of X₅₂₂ (line X₁₁X₁₁₄₆)

• X₆₅₃ = X_i-anticomplementary conjugate of X_j for these (i,j): {7115, 6223}, {36049, 34188}, {40117, 33650}

• X₆₅₃ = X₃₁-complementary conjugate of X₃₉₀₅₃

• X₆₅₃ = X_i-Ceva conjugate of X_j for these (i,j): {1, 2639}, {2, 39053}, {648, 651}, {4564, 37279}, {4998, 5125}, {7045, 412}, {13149, 36118}, {18026, 1897}, {24032, 1}, {39294, 17923}, {46102, 278}

• X₆₅₃ = X_i-cross conjugate of X_j for these (i,j): {1, 24032}, {19, 7012}, {46, 7045}, {57, 7128}, {108, 36118}, {109, 664}, {196, 23984}, {278, 46102}, {514, 92}, {522, 7}, {579, 59}, {650, 29}, {652, 1}, {1020, 651}, {1021, 17097}, {1708, 4564}, {1721, 24011}, {1722, 7035}, {1723, 765}, {1734, 21453}, {1738, 39293}, {1781, 24000}, {1783, 1897}, {1786, 35049}, {1788, 4998}, {1826, 34922}, {2637, 23707}, {3064, 4}, {3144, 18020}, {3911, 39294}, {4142, 7249}, {6591, 40446}, {7649, 273}, {7661, 1440}, {8058, 342}, {8063, 329}, {10900, 5382}, {14331, 8}, {14400, 80}, {14837, 2}, {16612, 1220}, {17906, 6335}, {17924, 27}, {18679, 23582}, {21173, 40440}, {21180, 18815}, {21185, 1088}, {21186, 75}, {21189, 86}, {22443, 1037}, {32674, 36127}, {35349, 3699}, {42755, 22464}

• X₆₅₃ = cevapoint of X_i and X_j for these (i,j): {1, 652}, {4, 3064}, {6, 39199}, {9, 8058}, {19, 7649}, {57, 514}, {65, 650}, {108, 1783}, {109, 32674}, {196, 14837}, {281, 522}, {407, 2501}, {513, 1108}, {523, 1901}, {649, 3924}, {656, 2294}, {657, 4336}, {822, 2658}, {1828, 6591}, {1880, 43923}, {3676, 7195}, {6087, 23986}, {8063, 40837}, {8755, 42755}, {14331, 44696}, {17924, 40149}

• X₆₅₃ = crosspoint of X_i and X_j for these (i,j): {1, 9394}, {648, 823}, {13149, 18026}

• X₆₅₃ = crosssum of X_i and X_j for these (i,j): {1, 2636}, {647, 822}

• X₆₅₃ = trilinear pole of line {1, 4}

• X₆₅₃ = crossdifference of every pair of points on line {2637, 2638}

• X₆₅₃ = X_i-isoconjugate of X_j for these (i,j): {1, 652}, {2, 1946}, {3, 650}, {4, 36054}, {6, 521}, {8, 22383}, {9, 1459}, {11, 906}, {21, 647}, {29, 822}, {31, 6332}, {32, 35518}, {33, 4091}, {37, 23189}, {41, 4025}, {48, 522}, {55, 905}, {58, 8611}, {63, 663}, {65, 23090}, {69, 3063}, {71, 3737}, {72, 7252}, {73, 1021}, {74, 14395}, {77, 657}, {78, 649}, {84, 10397}, {100, 7117}, {101, 7004}, {106, 14418}, {108, 35072}, {109, 34591}, {184, 4391}, {210, 7254}, {212, 514}, {219, 513}, {222, 3900}, {228, 4560}, {244, 4587}, {255, 3064}, {268, 6129}, {281, 23224}, {283, 661}, {284, 656}, {295, 4435}, {314, 3049}, {332, 798}, {333, 810}, {345, 667}, {348, 8641}, {394, 18344}, {512, 1812}, {520, 1172}, {523, 2193}, {525, 2194}, {577, 44426}, {603, 3239}, {607, 4131}, {644, 3937}, {646, 22096}, {651, 3270}, {653, 2638}, {654, 1807}, {672, 23696}, {692, 26932}, {884, 25083}, {885, 20752}, {926, 1814}, {1015, 4571}, {1019, 2318}, {1024, 1818}, {1036, 2522}, {1146, 36059}, {1167, 40628}, {1214, 21789}, {1259, 6591}, {1260, 3669}, {1320, 22086}, {1331, 2170}, {1332, 3271}, {1364, 1783}, {1397, 15416}, {1402, 15411}, {1409, 7253}, {1415, 2968}, {1433, 14298}, {1437, 3700}, {1444, 3709}, {1639, 36058}, {1790, 4041}, {1792, 7180}, {1793, 21828}, {1797, 4895}, {1802, 3676}, {1803, 6608}, {1808, 21832}, {1809, 3310}, {1813, 2310}, {1896, 32320}, {1919, 3718}, {2053, 25098}, {2150, 4064}, {2175, 15413}, {2183, 37628}, {2188, 14837}, {2196, 3716}, {2200, 18155}, {2204, 3265}, {2212, 30805}, {2289, 7649}, {2291, 14414}, {2299, 24018}, {2319, 22090}, {2327, 4017}, {2359, 17420}, {2431, 6001}, {2637, 23707}, {2804, 14578}, {3287, 7015}, {3692, 43924}, {3694, 3733}, {3708, 4636}, {3907, 7116}, {3939, 3942}, {4105, 7177}, {4130, 7053}, {4147, 15373}, {4163, 7099}, {4516, 4558}, {4574, 18191}, {4575, 21044}, {4581, 22074}, {4612, 20975}, {4768, 32659}, {4858, 32656}, {4876, 22384}, {5546, 18210}, {6056, 17924}, {6510, 23351}, {6516, 14936}, {6588, 39167}, {7100, 9404}, {7105, 22382}, {7107, 8062}, {7110, 23226}, {8606, 14838}, {8851, 22092}, {9247, 35519}, {10581, 40443}, {14331, 19614}, {14400, 35200}, {14432, 36060}, {15420, 20967}, {16596, 32652}, {17880, 32739}, {17926, 22341}, {18026, 39687}, {22144, 42552}, {22356, 23838}, {22379, 36910}, {22381, 27527}, {23113, 40528}, {23146, 40505}, {23220, 36795}, {23225, 36796}, {23614, 23984}, {24026, 32660}, {24031, 32674}, {31623, 39201}, {32641, 35014}, {32653, 34588}, {32677, 39471}

• X₆₅₃ = barycentric product X_i*X_j for these {i,j}: {1, 18026}, {4, 664}, {7, 1897}, {8, 36118}, {9, 13149}, {19, 4554}, {25, 4572}, {27, 4552}, {29, 4566}, {33, 4569}, {34, 668}, {57, 6335}, {59, 46107}, {65, 811}, {69, 36127}, {73, 6528}, {75, 108}, {76, 32674}, {85, 1783}, {92, 651}, {99, 225}, {100, 273}, {101, 331}, {107, 307}, {109, 264}, {112, 349}, {158, 6516}, {162, 1441}, {190, 278}, {196, 44327}, {226, 648}, {281, 658}, {286, 4551}, {312, 32714}, {318, 934}, {342, 13138}, {514, 46102}, {521, 24032}, {608, 1978}, {644, 1847}, {646, 1435}, {655, 17923}, {662, 40149}, {666, 5236}, {693, 7012}, {799, 1880}, {823, 1214}, {883, 36124}, {927, 1861}, {1020, 31623}, {1093, 6517}, {1118, 4561}, {1119, 3699}, {1231, 24019}, {1262, 46110}, {1275, 3064}, {1305, 5125}, {1309, 22464}, {1395, 6386}, {1396, 4033}, {1400, 6331}, {1414, 41013}, {1415, 1969}, {1426, 7257}, {1461, 7017}, {1813, 2052}, {1824, 4625}, {1826, 4573}, {1848, 6648}, {1870, 35174}, {1874, 4589}, {1877, 4555}, {1952, 1981}, {2356, 46135}, {2501, 4620}, {3112, 46152}, {3257, 37790}, {3261, 7115}, {3262, 36110}, {3668, 36797}, {3676, 15742}, {4077, 5379}, {4242, 18815}, {4391, 7128}, {4467, 34922}, {4559, 44129}, {4564, 17924}, {4605, 46103}, {4610, 8736}, {4617, 7101}, {4626, 7046}, {4998, 7649}, {5089, 34085}, {5307, 32038}, {6063, 8750}, {6332, 23984}, {6742, 7282}, {7035, 43923}, {7045, 44426}, {7079, 36838}, {8680, 41207}, {9394, 39060}, {10015, 39294}, {15352, 40152}, {16082, 24029}, {17906, 40420}, {17985, 35154}, {18027, 32660}, {18097, 41676}, {18816, 23706}, {21272, 40446}, {23710, 35157}, {23987, 34393}, {24033, 35518}, {24035, 36100}, {26705, 33298}, {28017, 42384}, {35516, 36067}, {36049, 40701}, {36146, 46108}, {37139, 37805}, {40117, 40702}, {46104, 46153}

• X₆₅₃ = barycentric quotient X_i/X_j for these {i,j}: {1, 521}, {2, 6332}, {4, 522}, {6, 652}, {7, 4025}, {12, 4064}, {19, 650}, {25, 663}, {27, 4560}, {28, 3737}, {29, 7253}, {31, 1946}, {33, 3900}, {34, 513}, {37, 8611}, {44, 14418}, {48, 36054}, {56, 1459}, {57, 905}, {58, 23189}, {59, 1331}, {65, 656}, {73, 520}, {75, 35518}, {77, 4131}, {85, 15413}, {92, 4391}, {99, 332}, {100, 78}, {101, 219}, {104, 37628}, {105, 23696}, {107, 29}, {108, 1}, {109, 3}, {110, 283}, {112, 284}, {158, 44426}, {162, 21}, {163, 2193}, {190, 345}, {196, 14837}, {198, 10397}, {208, 6129}, {222, 4091}, {225, 523}, {226, 525}, {242, 3716}, {250, 4636}, {264, 35519}, {273, 693}, {278, 514}, {281, 3239}, {284, 23090}, {286, 18155}, {307, 3265}, {312, 15416}, {318, 4397}, {331, 3261}, {333, 15411}, {342, 17896}, {348, 30805}, {349, 3267}, {388, 23874}, {393, 3064}, {468, 14432}, {513, 7004}, {514, 26932}, {515, 39471}, {521, 24031}, {522, 2968}, {603, 23224}, {604, 22383}, {607, 657}, {608, 649}, {643, 1792}, {644, 3692}, {648, 333}, {649, 7117}, {650, 34591}, {651, 63}, {652, 35072}, {658, 348}, {662, 1812}, {663, 3270}, {664, 69}, {668, 3718}, {692, 212}, {693, 17880}, {765, 4571}, {811, 314}, {823, 31623}, {906, 2289}, {927, 31637}, {933, 35196}, {934, 77}, {1018, 3694}, {1020, 1214}, {1025, 25083}, {1096, 18344}, {1108, 40628}, {1118, 7649}, {1119, 3676}, {1155, 14414}, {1172, 1021}, {1214, 24018}, {1249, 14331}, {1252, 4587}, {1262, 1813}, {1331, 1259}, {1332, 3719}, {1365, 21134}, {1395, 667}, {1396, 1019}, {1398, 43924}, {1399, 23226}, {1400, 647}, {1402, 810}, {1403, 22090}, {1404, 22086}, {1409, 822}, {1412, 7254}, {1414, 1444}, {1415, 48}, {1423, 25098}, {1426, 4017}, {1428, 22384}, {1434, 15419}, {1435, 3669}, {1441, 14208}, {1445, 24562}, {1457, 8677}, {1459, 1364}, {1461, 222}, {1474, 7252}, {1633, 1040}, {1769, 35014}, {1783, 9}, {1785, 2804}, {1788, 20315}, {1813, 394}, {1824, 4041}, {1826, 3700}, {1827, 6608}, {1828, 6615}, {1829, 17420}, {1839, 4976}, {1840, 4140}, {1846, 23757}, {1847, 24002}, {1848, 3910}, {1870, 3738}, {1874, 4010}, {1875, 1769}, {1876, 2254}, {1877, 900}, {1880, 661}, {1893, 4804}, {1897, 8}, {1940, 8062}, {1946, 2638}, {1947, 17899}, {1950, 22382}, {1973, 3063}, {1981, 1944}, {1990, 14400}, {2052, 46110}, {2149, 906}, {2173, 14395}, {2201, 4435}, {2212, 8641}, {2222, 1807}, {2283, 1818}, {2285, 2522}, {2299, 21789}, {2331, 14298}, {2333, 3709}, {2356, 926}, {2501, 21044}, {2637, 33572}, {2638, 23614}, {2701, 17973}, {2720, 1795}, {2969, 21132}, {3064, 1146}, {3176, 14302}, {3186, 25128}, {3476, 9031}, {3668, 17094}, {3669, 3942}, {3676, 1565}, {3699, 1265}, {3732, 27509}, {3939, 1260}, {3952, 3710}, {4017, 18210}, {4064, 7068}, {4185, 17418}, {4186, 42312}, {4242, 4511}, {4551, 72}, {4552, 306}, {4554, 304}, {4557, 2318}, {4558, 6514}, {4559, 71}, {4561, 1264}, {4564, 1332}, {4565, 1790}, {4566, 307}, {4569, 7182}, {4572, 305}, {4573, 17206}, {4605, 26942}, {4617, 7177}, {4619, 44717}, {4620, 4563}, {4626, 7056}, {4998, 4561}, {5125, 20294}, {5236, 918}, {5307, 23880}, {5342, 4811}, {5379, 643}, {5546, 2327}, {5930, 8057}, {6198, 35057}, {6331, 28660}, {6332, 23983}, {6335, 312}, {6516, 326}, {6517, 3964}, {6528, 44130}, {6529, 8748}, {6558, 30681}, {6591, 2170}, {6614, 7053}, {7009, 3907}, {7012, 100}, {7045, 6516}, {7046, 4163}, {7071, 4105}, {7079, 4130}, {7115, 101}, {7119, 3287}, {7128, 651}, {7178, 4466}, {7192, 17219}, {7282, 4467}, {7649, 11}, {7952, 8058}, {8058, 7358}, {8059, 1433}, {8687, 2359}, {8735, 42462}, {8736, 4024}, {8748, 17926}, {8750, 55}, {8751, 1024}, {8756, 1639}, {9088, 3478}, {9316, 22091}, {13138, 271}, {13149, 85}, {13486, 1789}, {14257, 21186}, {14331, 40616}, {14776, 2342}, {14837, 16596}, {15439, 1794}, {15742, 3699}, {16609, 24459}, {17094, 17216}, {17906, 3452}, {17923, 3904}, {17924, 4858}, {17925, 17197}, {17985, 2785}, {18026, 75}, {18097, 4580}, {18344, 2310}, {20613, 2509}, {21186, 123}, {21189, 34588}, {21666, 23104}, {21859, 3949}, {23067, 3682}, {23353, 1936}, {23363, 22421}, {23703, 5440}, {23706, 517}, {23710, 6366}, {23845, 22072}, {23890, 6510}, {23979, 32660}, {23981, 22350}, {23985, 32674}, {23987, 515}, {24019, 1172}, {24027, 36059}, {24032, 18026}, {24033, 108}, {26700, 7100}, {26704, 10570}, {29055, 7015}, {32652, 2188}, {32656, 6056}, {32660, 577}, {32667, 32677}, {32669, 14578}, {32674, 6}, {32676, 2194}, {32691, 1036}, {32702, 909}, {32713, 2299}, {32714, 57}, {32727, 34078}, {32735, 36057}, {34922, 6742}, {35349, 42018}, {36037, 1809}, {36040, 36055}, {36044, 1295}, {36049, 268}, {36059, 255}, {36067, 102}, {36075, 22054}, {36076, 3422}, {36082, 1069}, {36093, 26703}, {36098, 1791}, {36099, 2339}, {36106, 45393}, {36110, 104}, {36118, 7}, {36123, 43728}, {36124, 885}, {36125, 23838}, {36126, 1896}, {36127, 4}, {36140, 32726}, {36146, 1814}, {36797, 1043}, {37141, 41081}, {37790, 3762}, {38462, 4768}, {39199, 38983}, {39294, 13136}, {39534, 35015}, {40116, 2338}, {40117, 282}, {40149, 1577}, {40663, 14429}, {40983, 2488}, {40987, 17115}, {41013, 4086}, {41207, 35145}, {41321, 40869}, {42069, 23615}, {42070, 4543}, {42755, 10017}, {43035, 39470}, {43290, 44722}, {43923, 244}, {43924, 3937}, {44327, 44189}, {44426, 24026}, {44696, 21172}, {46102, 190}, {46107, 34387}, {46110, 23978}, {46152, 38}, {46153, 3917}, {46254, 4631}

• X₆₅₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 2636, 2638}, {46, 158, 412}, {65, 1940, 29}, {652, 24032, 2639}, {1118, 1788, 5125}, {1148, 8762, 3}, {1783, 32714, 651}, {2636, 2638, 23707}, {4566, 14543, 651}, {14837, 23984, 39053}, {16596, 40535, 2}

• X₆₅₃ = cevapoint of X_i and X_j for these (i,j): (1, 652), (57, 514), (65, 650), (281, 522), (513, 1108)

• X₆₅₃ = X₆-isoconjugate of X₅₂₁

• X₆₅₃ = perspector of conic {A, B, C,PU(76)}}