X₆₅₂

CROSSDIFFERENCE OF X₁ AND X₄

X₆₅₂ = crossdifference of every pair of points on line X₁X₄

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

X₆₅₂ (CROSSDIFFERENCE OF X₁ AND X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {44, 513}, {243, 522}

• Related Centers: X₆₅₃ (TRILINEAR POLE OF LINE X₁X₄), X₄₄ (-LINE CONJUGATE OF X₁), X₅₁₃ (ISOGONAL CONJUGATE OF X₁₀₀), X₂₄₃ (-LINE CONJUGATE OF X₇₃), X₅₂₂ (ISOGONAL CONJUGATE OF X₁₀₉)

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

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

Search Value (6, 9, 13): \(-14.802087816831\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations1 combination

• X₆₅₂ = \(2\) X₃₀₆₄ - \(3\) X₁₄₄₀₀, X₈₆₁₁ - \(3\) X₁₄₄₁₈, \(4\) X₁₄₃₃₁ - \(3\) X₁₄₄₀₀

▸ Incident Central Lines73 lines

{6, 2432}, {9, 3239}, {19, 42755}, {31, 21122}, {44, 513}, {55, 4105}, {57, 7658}, {63, 4025}, {73, 9240}, {101, 2149}, {109, 7115}, {112, 2762}, {142, 25604}, {228, 22388}, {243, 522}, {333, 35519}, {514, 23726}, {520, 647}, {521, 8611}, {653, 2639}, {663, 1951}, {905, 4091}, {926, 8641}, {1769, 6591}, {2170, 35065}, {2260, 42768}, {2504, 21107}, {2523, 14395}, {2636, 9394}, {2654, 2657}, {2655, 21173}, {3101, 20298}, {3215, 14825}, {3217, 27780}, {3219, 25259}, {3835, 28834}, {3928, 44551}, {4131, 24562}, {4382, 40166}, {4498, 21120}, {4524, 6139}, {4814, 42657}, {4885, 25924}, {4976, 6362}, {5075, 23655}, {5227, 9031}, {5271, 17894}, {6332, 15411}, {6608, 11934}, {7082, 23615}, {7117, 39006}, {7180, 20980}, {9029, 23865}, {10319, 20319}, {14330, 28589}, {16612, 21189}, {17094, 23727}, {17215, 25511}, {17975, 23146}, {20316, 26080}, {20760, 23093}, {22382, 23226}, {23090, 23189}, {25955, 31287}, {26049, 26652}, {26694, 27139}, {27527, 28960}, {32674, 36040}, {34591, 38344}, {35072, 38353}, {38983, 39687}, {38991, 43960}, {43060, 43924}

▸ Functional Relations28 relations

• X₆₅₂ = reflection of X_i in X_j for these {i,j}: {663, 21789}, {3064, 14331}, {23727, 17094}

• X₆₅₂ = isogonal conjugate of X₆₅₃

• X₆₅₂ = X_i-Ceva conjugate of X_j for these (i,j): (2, 38983), (101, 48), (109, 55)

• X₆₅₂ = crosspoint of X_i and X_j for these (i,j): (9, 101), (109, 222)

• X₆₅₂ = crosssum of X_i and X_j for these (i,j): (1, 652), (57, 514), (65, 650), (281, 522), (513, 1108)

• X₆₅₂ = crossdifference of every pair of points on line X₁X₄

• X₆₅₂ = trilinear pole of line {2637, 2638}

• X₆₅₂ = bicentric difference of PU(i) for i in (16, 76, 82, 126)

• X₆₅₂ = PU(16)-harmonic conjugate of X₇₃

• X₆₅₂ = PU(76)-harmonic conjugate of X₂₆₃₅

• X₆₅₂ = PU(82)-harmonic conjugate of X₂₆₅₄

• X₆₅₂ = PU(126)-harmonic conjugate of X₄

• X₆₅₂ = polar conjugate of isotomic conjugate of isogonal conjugate of X₃₆₁₂₇

• X₆₅₂ = trilinear product of PU(77)

• X₆₅₂ = trilinear pole of PU(77)

• X₆₅₂ = intersection of trilinear polars of X₁ and X₃

• X₆₅₂ = trilinear product of circumcircle intercepts of line X₂X₉₂

• X₆₅₂ = X₃₁-complementary conjugate of X₃₈₉₈₃

• X₆₅₂ = X_i-Ceva conjugate of X_j for these (i,j): {1, 2638}, {2, 38983}, {3, 3270}, {9, 34591}, {48, 38344}, {63, 7004}, {90, 2310}, {101, 48}, {109, 55}, {162, 14547}, {163, 2269}, {219, 7117}, {222, 1364}, {268, 35072}, {522, 663}, {651, 73}, {653, 1}, {658, 20277}, {823, 2654}, {905, 1459}, {906, 71}, {1020, 2646}, {1021, 650}, {1248, 42080}, {1331, 212}, {1332, 22072}, {1751, 11}, {1783, 22063}, {1813, 3}, {2762, 10535}, {3939, 7124}, {4558, 22361}, {4561, 20753}, {4563, 22421}, {4571, 2318}, {4574, 22074}, {4587, 219}, {7094, 38345}, {8059, 2188}, {8064, 1436}, {23067, 23207}, {23090, 36054}, {23189, 1946}, {36039, 672}, {36049, 6}, {36050, 42}, {36059, 40945}, {36093, 2356}, {37203, 35015}, {39943, 2170}, {39946, 3248}, {39947, 244}, {40117, 41087}, {40518, 40944}, {41206, 1936}

• X₆₅₂ = X_i-cross conjugate of X_j for these (i,j): {647, 650}, {822, 36054}, {1946, 1459}, {2638, 1}, {3270, 3}, {7117, 219}

• X₆₅₂ = cevapoint of X_i and X_j for these (i,j): {1, 2636}, {647, 822}

• X₆₅₂ = crosspoint of X_i and X_j for these (i,j): {1, 653}, {3, 1813}, {6, 32653}, {9, 101}, {21, 651}, {57, 8059}, {63, 1331}, {100, 40399}, {109, 222}, {190, 40436}, {219, 4587}, {268, 36049}, {333, 4636}, {521, 905}, {522, 6332}, {906, 2193}, {1021, 23090}, {1172, 40117}, {1812, 4571}

• X₆₅₂ = crosssum 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}

• X₆₅₂ = crossdifference of every pair of points on line {1, 4}

• X₆₅₂ = X_i-isoconjugate of X_j for these (i,j): {1, 653}, {2, 108}, {4, 651}, {6, 18026}, {7, 1783}, {8, 32714}, {9, 36118}, {19, 664}, {25, 4554}, {27, 4551}, {28, 4552}, {29, 1020}, {33, 658}, {34, 190}, {55, 13149}, {56, 6335}, {57, 1897}, {59, 17924}, {63, 36127}, {65, 648}, {73, 823}, {75, 32674}, {83, 46152}, {85, 8750}, {92, 109}, {99, 1880}, {100, 278}, {101, 273}, {102, 24035}, {107, 1214}, {110, 40149}, {112, 1441}, {158, 1813}, {162, 226}, {196, 13138}, {208, 44327}, {225, 662}, {264, 1415}, {270, 4605}, {281, 934}, {286, 4559}, {307, 24019}, {318, 1461}, {331, 692}, {342, 36049}, {347, 40117}, {349, 32676}, {388, 36099}, {393, 6516}, {513, 46102}, {514, 7012}, {521, 23984}, {522, 7128}, {607, 4569}, {608, 668}, {644, 1119}, {645, 1426}, {646, 1398}, {652, 24032}, {655, 1870}, {666, 1876}, {693, 7115}, {811, 1400}, {851, 41207}, {883, 8751}, {901, 37790}, {908, 36110}, {927, 5089}, {1016, 43923}, {1025, 36124}, {1041, 3732}, {1118, 1332}, {1172, 4566}, {1231, 32713}, {1262, 44426}, {1275, 18344}, {1290, 37799}, {1295, 2405}, {1309, 1465}, {1395, 1978}, {1396, 3952}, {1402, 6331}, {1409, 6528}, {1414, 1826}, {1427, 36797}, {1435, 3699}, {1476, 17906}, {1769, 39294}, {1785, 37136}, {1824, 4573}, {1829, 6648}, {1847, 3939}, {1848, 36098}, {1861, 36146}, {1874, 4584}, {1875, 13136}, {1877, 3257}, {1937, 1981}, {1952, 23353}, {1973, 4572}, {2006, 4242}, {2052, 36059}, {2149, 46107}, {2222, 17923}, {2333, 4625}, {2356, 34085}, {2406, 36121}, {2639, 9394}, {2766, 37798}, {2969, 31615}, {3064, 7045}, {3262, 32702}, {3669, 15742}, {4185, 32038}, {4564, 7649}, {4565, 41013}, {4617, 7046}, {4624, 5338}, {4626, 7079}, {4998, 6591}, {5236, 36086}, {5379, 7178}, {6198, 38340}, {6332, 24033}, {6336, 23703}, {6517, 6520}, {6614, 7101}, {7009, 37137}, {7071, 36838}, {7952, 37141}, {14733, 37805}, {14838, 34922}, {15352, 22341}, {16082, 23981}, {17080, 26704}, {21362, 40446}, {23706, 34234}, {23710, 37139}, {23985, 35518}, {23987, 36100}, {24027, 46110}, {24029, 36123}, {26706, 37800}, {32652, 40701}, {32667, 35516}, {32735, 46108}, {36126, 40152}, {41321, 43736}

• X₆₅₂ = barycentric product X_i*X_j for these {i,j}: {1, 521}, {3, 522}, {6, 6332}, {8, 1459}, {9, 905}, {10, 23189}, {11, 1331}, {21, 656}, {29, 520}, {31, 35518}, {33, 4131}, {41, 15413}, {48, 4391}, {55, 4025}, {60, 4064}, {63, 650}, {69, 663}, {71, 4560}, {72, 3737}, {73, 7253}, {75, 1946}, {77, 3900}, {78, 513}, {81, 8611}, {88, 14418}, {92, 36054}, {100, 7004}, {101, 26932}, {102, 39471}, {108, 24031}, {109, 2968}, {125, 4636}, {184, 35519}, {189, 10397}, {190, 7117}, {212, 693}, {219, 514}, {222, 3239}, {226, 23090}, {228, 18155}, {244, 4571}, {255, 44426}, {268, 14837}, {271, 6129}, {281, 4091}, {283, 523}, {284, 525}, {295, 3716}, {304, 3063}, {306, 7252}, {307, 21789}, {312, 22383}, {314, 810}, {318, 23224}, {326, 18344}, {332, 512}, {333, 647}, {345, 649}, {348, 657}, {394, 3064}, {517, 37628}, {518, 23696}, {577, 46110}, {603, 4397}, {604, 15416}, {607, 30805}, {643, 18210}, {644, 3942}, {651, 34591}, {653, 35072}, {661, 1812}, {664, 3270}, {667, 3718}, {692, 17880}, {822, 31623}, {885, 1818}, {895, 14432}, {906, 4858}, {926, 31637}, {1019, 3694}, {1021, 1214}, {1024, 25083}, {1036, 23874}, {1073, 14331}, {1086, 4587}, {1146, 1813}, {1156, 14414}, {1172, 24018}, {1259, 7649}, {1260, 3676}, {1265, 43924}, {1332, 2170}, {1334, 15419}, {1364, 1897}, {1400, 15411}, {1433, 8058}, {1437, 4086}, {1444, 4041}, {1565, 3939}, {1577, 2193}, {1639, 1797}, {1769, 1809}, {1790, 3700}, {1791, 17420}, {1792, 4017}, {1795, 2804}, {1796, 4976}, {1802, 24002}, {1807, 3738}, {1808, 4010}, {2188, 17896}, {2194, 14208}, {2269, 15420}, {2289, 17924}, {2299, 3265}, {2310, 6516}, {2311, 24459}, {2318, 7192}, {2319, 25098}, {2321, 7254}, {2327, 7178}, {2328, 17094}, {2338, 39470}, {2339, 2522}, {2349, 14395}, {2359, 3910}, {2501, 6514}, {2638, 18026}, {2785, 17973}, {3049, 28660}, {3271, 4561}, {3478, 9031}, {3504, 25128}, {3669, 3692}, {3699, 3937}, {3708, 4612}, {3709, 17206}, {3710, 3733}, {3719, 6591}, {3907, 7015}, {4105, 7056}, {4130, 7177}, {4147, 23086}, {4163, 7053}, {4451, 22093}, {4466, 5546}, {4467, 8606}, {4516, 4592}, {4518, 22384}, {4557, 17219}, {4558, 21044}, {4574, 17197}, {4768, 36058}, {4997, 22086}, {5440, 23838}, {5931, 42658}, {6056, 46107}, {6368, 35196}, {6510, 23893}, {6517, 42069}, {6608, 40443}, {7016, 8062}, {7100, 35057}, {7107, 17899}, {7108, 22382}, {7155, 22090}, {7182, 8641}, {7358, 8059}, {14298, 41081}, {14304, 36055}, {14400, 14919}, {16596, 36049}, {17418, 34259}, {17926, 40152}, {21186, 39167}, {22092, 36799}, {22350, 43728}, {23614, 24032}, {23978, 32660}, {23983, 32674}, {24026, 36059}, {32653, 40626}, {32656, 34387}, {34588, 36050}, {35014, 36037}, {39201, 44130}, {40399, 40628}, {42462, 44717}

• X₆₅₂ = barycentric quotient X_i/X_j for these {i,j}: {1, 18026}, {3, 664}, {6, 653}, {9, 6335}, {11, 46107}, {21, 811}, {25, 36127}, {29, 6528}, {31, 108}, {32, 32674}, {41, 1783}, {48, 651}, {55, 1897}, {56, 36118}, {57, 13149}, {63, 4554}, {69, 4572}, {71, 4552}, {73, 4566}, {77, 4569}, {78, 668}, {101, 46102}, {108, 24032}, {184, 109}, {212, 100}, {219, 190}, {222, 658}, {228, 4551}, {255, 6516}, {268, 44327}, {283, 99}, {284, 648}, {332, 670}, {333, 6331}, {345, 1978}, {512, 225}, {513, 273}, {514, 331}, {520, 307}, {521, 75}, {522, 264}, {525, 349}, {577, 1813}, {603, 934}, {604, 32714}, {647, 226}, {649, 278}, {650, 92}, {654, 17923}, {656, 1441}, {657, 281}, {661, 40149}, {663, 4}, {665, 5236}, {667, 34}, {692, 7012}, {798, 1880}, {810, 65}, {822, 1214}, {884, 36124}, {905, 85}, {906, 4564}, {926, 1861}, {1021, 31623}, {1092, 6517}, {1146, 46110}, {1172, 823}, {1259, 4561}, {1260, 3699}, {1331, 4998}, {1364, 4025}, {1409, 1020}, {1415, 7128}, {1437, 1414}, {1444, 4625}, {1459, 7}, {1635, 37790}, {1639, 46109}, {1790, 4573}, {1792, 7257}, {1802, 644}, {1807, 35174}, {1808, 4589}, {1812, 799}, {1813, 1275}, {1814, 34085}, {1818, 883}, {1919, 608}, {1946, 1}, {1951, 1981}, {1960, 1877}, {1964, 46152}, {1980, 1395}, {2170, 17924}, {2175, 8750}, {2182, 24035}, {2188, 13138}, {2193, 662}, {2194, 162}, {2197, 4605}, {2200, 4559}, {2204, 24019}, {2249, 41207}, {2289, 1332}, {2299, 107}, {2310, 44426}, {2318, 3952}, {2327, 645}, {2328, 36797}, {2342, 1309}, {2347, 17906}, {2359, 6648}, {2361, 4242}, {2605, 7282}, {2636, 39060}, {2638, 521}, {2968, 35519}, {3049, 1400}, {3063, 19}, {3064, 2052}, {3239, 7017}, {3248, 43923}, {3270, 522}, {3271, 7649}, {3669, 1847}, {3692, 646}, {3694, 4033}, {3709, 1826}, {3710, 27808}, {3716, 40717}, {3718, 6386}, {3737, 286}, {3900, 318}, {3937, 3676}, {3939, 15742}, {3942, 24002}, {3955, 6649}, {4025, 6063}, {4041, 41013}, {4055, 23067}, {4064, 34388}, {4079, 8736}, {4091, 348}, {4105, 7046}, {4130, 7101}, {4131, 7182}, {4391, 1969}, {4455, 1874}, {4516, 24006}, {4546, 44721}, {4558, 4620}, {4560, 44129}, {4571, 7035}, {4587, 1016}, {4612, 46254}, {4636, 18020}, {4895, 38462}, {5075, 17985}, {6056, 1331}, {6129, 342}, {6139, 23710}, {6332, 76}, {6514, 4563}, {6518, 15418}, {7004, 693}, {7053, 4626}, {7085, 14594}, {7099, 4617}, {7116, 37137}, {7117, 514}, {7118, 40117}, {7124, 3732}, {7177, 36838}, {7252, 27}, {7253, 44130}, {7254, 1434}, {8606, 6742}, {8611, 321}, {8638, 2356}, {8641, 33}, {8648, 1870}, {8676, 5125}, {8677, 22464}, {8748, 15352}, {9247, 1415}, {10397, 329}, {10581, 1855}, {14331, 15466}, {14395, 14206}, {14400, 46106}, {14413, 38461}, {14414, 30806}, {14418, 4358}, {14432, 44146}, {14578, 37136}, {14585, 32660}, {14837, 40701}, {14936, 3064}, {15411, 28660}, {15413, 20567}, {15416, 28659}, {17880, 40495}, {17973, 35154}, {18210, 4077}, {18344, 158}, {20752, 1025}, {20753, 3888}, {20775, 46153}, {21044, 14618}, {21132, 2973}, {21761, 1940}, {21789, 29}, {22053, 35312}, {22072, 21272}, {22074, 3882}, {22079, 35338}, {22086, 3911}, {22090, 3212}, {22091, 9312}, {22092, 43040}, {22093, 7176}, {22096, 43924}, {22361, 17136}, {22368, 46177}, {22379, 1443}, {22382, 1943}, {22383, 57}, {22384, 1447}, {23090, 333}, {23189, 86}, {23202, 23703}, {23220, 1457}, {23224, 77}, {23225, 1458}, {23226, 1442}, {23614, 24031}, {23615, 21666}, {23696, 2481}, {24018, 1231}, {24031, 35518}, {24562, 21609}, {25098, 30545}, {26932, 3261}, {31637, 46135}, {32320, 40152}, {32641, 39294}, {32656, 59}, {32658, 36146}, {32660, 1262}, {32674, 23984}, {32739, 7115}, {34591, 4391}, {34858, 36110}, {35014, 36038}, {35072, 6332}, {35196, 18831}, {35518, 561}, {35519, 18022}, {36054, 63}, {36057, 927}, {36059, 7045}, {37628, 18816}, {38344, 17496}, {39201, 73}, {39471, 35516}, {40628, 17862}, {42658, 5930}, {43924, 1119}, {46110, 18027}

• X₆₅₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {647, 22086, 22383}, {647, 22383, 1459}, {649, 657, 650}, {650, 654, 649}, {650, 4790, 14300}, {650, 9404, 657}, {650, 13401, 21127}, {653, 2639, 24032}, {654, 9404, 650}, {822, 2522, 22443}, {3064, 14331, 14400}, {4394, 40137, 650}, {4979, 21127, 13401}

Notes

X₆₅₂ is the perspector of triangle ABC and the tangential triangle of the conic that passes through the points A, B, C, X₁, and X₃. (Randy Hutson, 9/23/2011)