X₃₄₆

ISOTOMIC CONJUGATE OF X₂₇₉

The cevian triangle of X₃₄₆ is perspective to the Ayme triangle; see X₃₆₁₀.

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

X₃₄₆ (ISOTOMIC CONJUGATE OF X₂₇₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {8, 9}, {2, 37}

• Related Centers: X₂₇₉ (ISOGONAL CONJUGATE OF X₂₂₀), X₈ (NAGEL POINT), X₉ (MITTENPUNKT), X₂ (CENTROID), X₃₇ (CROSSPOINT OF INCENTER AND CENTROID)

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

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

Search Value (6, 9, 13): \(5.609320090643\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\cos(A/2) \csc^{3}(A/2) : :\)

• Barycentrics: \((1 + \cos A)(1 - \cos A) : :\)

▸ Linear Combinations1 combination

• X₃₄₆ = \(2\) X₂ + \(19\) X₃₄₅ = X₁₇₄₃ + \(2\) X₄₀₇₂ = \(17\) X₄₆₈₆ - \(2\) X₄₆₈₈ = \(19\) X₄₆₈₈ - \(4\) X₄₇₈₈ = \(5\) X₃₇ - \(14\) X₄₈₂₁

X₃₄₆ lies on the cubics K605 and K697.

▸ Incident Central Lines136 lines

{1, 1219}, {2, 37}, {4, 3695}, {6, 145}, {7, 3729}, {8, 9}, {10, 3731}, {19, 3610}, {20, 1766}, {44, 3621}, {45, 594}, {55, 3974}, {69, 144}, {78, 280}, {100, 198}, {101, 2370}, {141, 4419}, {142, 4659}, {193, 6542}, {200, 4082}, {210, 11997}, {219, 644}, {220, 1043}, {253, 306}, {279, 304}, {281, 318}, {341, 3965}, {347, 4552}, {387, 2901}, {480, 4012}, {497, 3703}, {519, 1743}, {527, 4488}, {545, 7232}, {573, 1018}, {646, 3596}, {672, 10453}, {726, 4310}, {894, 3945}, {941, 1500}, {962, 10445}, {1023, 6790}, {1086, 4373}, {1089, 3085}, {1100, 3623}, {1125, 4098}, {1146, 6558}, {1212, 4673}, {1249, 1897}, {1260, 2322}, {1376, 1696}, {1400, 3501}, {1441, 8232}, {1449, 3241}, {1698, 3986}, {1761, 4427}, {1901, 3936}, {2178, 4188}, {2256, 5782}, {2264, 3189}, {2268, 2329}, {2285, 3600}, {2298, 4195}, {2303, 11115}, {2310, 4073}, {2550, 3932}, {2551, 3704}, {2899, 8165}, {2951, 9950}, {3008, 4402}, {3039, 8168}, {3062, 10324}, {3146, 7270}, {3217, 3684}, {3219, 3719}, {3244, 4898}, {3247, 3616}, {3294, 9534}, {3416, 5698}, {3452, 8055}, {3618, 4360}, {3619, 4389}, {3620, 6646}, {3632, 3973}, {3633, 4856}, {3661, 5232}, {3662, 4346}, {3679, 4058}, {3705, 5274}, {3712, 5218}, {3715, 4046}, {3730, 10449}, {3767, 7230}, {3771, 4135}, {3836, 7613}, {3871, 4254}, {3875, 5222}, {3923, 4307}, {3949, 3952}, {3969, 5739}, {3970, 11036}, {3975, 4110}, {3977, 5744}, {3985, 4451}, {3991, 4385}, {3996, 6605}, {4030, 10385}, {4066, 10198}, {4123, 9539}, {4130, 4397}, {4171, 4529}, {4222, 5687}, {4293, 5525}, {4294, 7206}, {4361, 4422}, {4363, 4648}, {4366, 4473}, {4384, 4431}, {4416, 6172}, {4470, 7227}, {4644, 4851}, {4675, 7222}, {4872, 10513}, {4876, 7155}, {4936, 6737}, {5257, 9780}, {5273, 11679}, {5281, 7081}, {5308, 7229}, {5430, 7027}, {5552, 7110}, {6556, 6736}, {7991, 10443}, {8557, 12649}, {8609, 10529}, {9025, 9309}

▸ Functional Relations26 relations

• X₃₄₆ = reflection of X_i in X_j for these {i,j}: {8, 4901}, {4452, 4000}, {5838, 9}

• X₃₄₆ = isogonal conjugate of X₁₄₀₇

• X₃₄₆ = isotomic conjugate of X₂₇₉

• X₃₄₆ = X_i-Ceva conjugate of X_j for these (i,j): {2, 6552}, {8, 6555}, {304, 10327}, {312, 8}, {341, 5423}, {345, 7080}, {646, 4397}, {1016, 3699}, {1043, 200}, {3699, 4163}, {4076, 4578}, {6557, 6556}, {7258, 7253}

• X₃₄₆ = X_i-cross conjugate of X_j for these (i,j): {200, 8}, {220, 7046}, {728, 5423}, {1260, 1265}, {2310, 7253}, {2321, 7101}, {3239, 6558}, {3694, 3692}, {3965, 2287}, {4081, 4397}, {4082, 341}, {4130, 4578}, {4163, 3699}, {4515, 200}, {4529, 7258}, {4907, 7}, {4953, 522}, {5423, 6556}, {6554, 2}

• X₃₄₆ = cevapoint of X_i and X_j for these (i,j): {1, 10860}, {6, 1604}, {9, 2324}, {37, 8804}, {200, 728}, {220, 1260}, {650, 4534}, {1146, 3239}, {2310, 4171}, {2321, 3694}, {4081, 4130}, {4082, 4515}

• X₃₄₆ = crosspoint of X_i and X_j for these (i,j): {2, 6553}, {8, 6557}, {312, 341}, {646, 4076}, {1016, 3699}

• X₃₄₆ = crosssum of X_i and X_j for these (i,j): {6, 1616}, {604, 1106}, {649, 3937}

• X₃₄₆ = crossdifference of every pair of points on line {667, 6363, 7250}

• X₃₄₆ = complement of X₄₄₅₂

• X₃₄₆ = anticomplement of X₄₀₀₀

• X₃₄₆ = pole wrt polar circle of trilinear polar of X₁₁₁₉

• X₃₄₆ = X₄₈-isoconjugate (polar conjugate) of X₁₁₁₉

• X₃₄₆ = centroid of the set consisting of the interiors (with or without boundaries) of the 3 Soddy circles

• X₃₄₆ = polar conjugate of X₁₁₁₉

• X₃₄₆ = X_i-beth conjugate of X_j for these (i,j): {190, 347}, {346, 9}, {644, 573}, {645, 144}, {646, 346}, {1043, 390}, {6558, 346}, {7258, 3596}, {7259, 219}

• X₃₄₆ = X_i-gimel conjugate of X_j for these (i,j): {4462, 346}, {8712, 346}

• X₃₄₆ = X_i-zayin conjugate of X_j for these (i,j): {1, 1407}, {905, 649}

• X₃₄₆ = X_i-anticomplementary conjugate of X_j for these (i,j): {1037, 7}, {7084, 2}, {7123, 8}, {7131, 3434}

• X₃₄₆ = X_i-complementary conjugate of X_j for these (i,j): {31, 6552}, {2137, 2886}, {6553, 2887}

• X₃₄₆ = X_i-Hirst inverse of X_j for these (i,j): {8, 3717}, {3975, 4110}

• X₃₄₆ = trilinear pole of line {3239, 3900, 4148, 4163, 4524, 4528, 4546, 4990}

• X₃₄₆ = barycentric square of X₈

• X₃₄₆ = isoconjugate of X_j and X_j for these (i,j): {1, 1407}, {2, 1106}, {3, 1435}, {4, 7099}, {6, 269}, {7, 604}, {8, 7366}, {9, 7023}, {19, 7053}, {25, 7177}, {27, 1410}, {31, 279}, {32, 1088}, {34, 222}, {40, 6612}, {41, 479}, {48, 1119}, {55, 738}, {56, 57}, {58, 1427}, {60, 7147}, {63, 1398}, {65, 1412}, {73, 1396}, {77, 608}, {81, 1042}, {84, 6611}, {85, 1397}, {109, 3669}, {110, 7216}, {184, 1847}, {221, 1422}, {223, 1413}, {226, 1408}, {241, 1416}, {244, 1262}, {266, 7370}, {278, 603}, {348, 1395}, {512, 4637}, {513, 1461}, {593, 1254}, {649, 934}, {650, 6614}, {658, 667}, {662, 7250}, {663, 4617}, {669, 4635}, {764, 4619}, {798, 4616}, {849, 6354}, {1014, 1400}, {1015, 7045}, {1020, 3733}, {1118, 7125}, {1122, 3451}, {1275, 3248}, {1333, 3668}, {1357, 4564}, {1358, 2149}, {1402, 1434}, {1403, 7153}, {1414, 7180}, {1415, 3676}, {1417, 3911}, {1426, 1790}, {1431, 7175}, {1439, 1474}, {1440, 2199}, {1446, 2206}, {1458, 1462}, {1472, 7365}, {1919, 4569}, {1973, 7056}, {2150, 6046}, {2170, 7339}, {2171, 7341}, {2185, 7143}, {2221, 4320}, {3063, 4626}, {3937, 7128}, {4017, 4565}, {4559, 7203}, {6358, 7342}, {7132, 7248}, {7183, 7337}

• X₃₄₆ = barycentric product X_i*X_j for these {i,j}: {1, 341}, {4, 1265}, {7, 5423}, {8, 8}, {9, 312}, {10, 1043}, {11, 4076}, {21, 3701}, {29, 3710}, {33, 3718}, {55, 3596}, {63, 7101}, {69, 7046}, {75, 200}, {76, 220}, {78, 318}, {85, 728}, {86, 4082}, {92, 3692}, {100, 4397}, {145, 6556}, {179, 400}, {188, 7027}, {190, 3239}, {210, 314}, {219, 7017}, {261, 6057}, {264, 1260}, {274, 4515}, {280, 7080}, {281, 345}, {304, 7079}, {305, 7071}, {306, 2322}, {313, 2328}, {321, 2287}, {333, 2321}, {480, 6063}, {514, 6558}, {522, 3699}, {523, 7256}, {556, 6731}, {561, 1253}, {594, 7058}, {643, 4086}, {644, 4391}, {645, 3700}, {646, 650}, {657, 1978}, {661, 7258}, {664, 4163}, {668, 3900}, {670, 4524}, {693, 4578}, {799, 4171}, {1016, 1146}, {1021, 4033}, {1089, 1098}, {1222, 6736}, {1229, 6605}, {1264, 1857}, {1320, 4723}, {1577, 7259}, {1639, 4582}, {1802, 1969}, {2310, 7035}, {2319, 4110}, {2325, 4997}, {3161, 6557}, {3685, 4518}, {3686, 4102}, {3912, 6559}, {3952, 7253}, {3975, 4876}, {4012, 8817}, {4041, 7257}, {4073, 7033}, {4081, 4998}, {4087, 7077}, {4092, 6064}, {4105, 4572}, {4130, 4554}, {4148, 4562}, {4373, 6555}, {4451, 7081}, {4528, 4555}, {4673, 4866}, {4990, 6540}, {6386, 8641}, {6552, 6553}

• X₃₄₆ = barycentric quotient X_i/X_j for these {i,j}: {1, 269}, {2, 279}, {3, 7053}, {4, 1119}, {6, 1407}, {7, 479}, {8, 7}, {9, 57}, {10, 3668}, {11, 1358}, {12, 6046}, {19, 1435}, {21, 1014}, {25, 1398}, {31, 1106}, {33, 34}, {37, 1427}, {41, 604}, {42, 1042}, {48, 7099}, {55, 56}, {56, 7023}, {57, 738}, {59, 7339}, {60, 7341}, {63, 7177}, {69, 7056}, {72, 1439}, {75, 1088}, {78, 77}, {92, 1847}, {99, 4616}, {100, 934}, {101, 1461}, {109, 6614}, {144, 9533}, {181, 7143}, {188, 7371}, {190, 658}, {198, 6611}, {200, 1}, {210, 65}, {212, 603}, {219, 222}, {220, 6}, {228, 1410}, {259, 7370}, {261, 552}, {280, 1440}, {281, 278}, {282, 1422}, {284, 1412}, {294, 1462}, {312, 85}, {318, 273}, {321, 1446}, {333, 1434}, {341, 75}, {345, 348}, {388, 7197}, {390, 3598}, {480, 55}, {497, 7195}, {512, 7250}, {522, 3676}, {556, 555}, {594, 6354}, {604, 7366}, {607, 608}, {612, 4320}, {643, 1414}, {644, 651}, {645, 4573}, {646, 4554}, {650, 3669}, {651, 4617}, {657, 649}, {661, 7216}, {662, 4637}, {664, 4626}, {668, 4569}, {728, 9}, {756, 1254}, {765, 7045}, {799, 4635}, {984, 7204}, {1016, 1275}, {1018, 1020}, {1021, 1019}, {1043, 86}, {1098, 757}, {1146, 1086}, {1172, 1396}, {1212, 1418}, {1252, 1262}, {1253, 31}, {1259, 1804}, {1260, 3}, {1261, 1476}, {1264, 7055}, {1265, 69}, {1334, 1400}, {1436, 6612}, {1604, 6609}, {1792, 1444}, {1802, 48}, {1824, 1426}, {1857, 1118}, {1863, 1851}, {2171, 7147}, {2175, 1397}, {2192, 1413}, {2194, 1408}, {2195, 1416}, {2212, 1395}, {2287, 81}, {2289, 7125}, {2310, 244}, {2318, 73}, {2319, 7153}, {2321, 226}, {2322, 27}, {2324, 223}, {2325, 3911}, {2327, 1790}, {2328, 58}, {2329, 7175}, {2332, 1474}, {2340, 1458}, {2345, 7365}, {2968, 1565}, {3022, 3271}, {3056, 7248}, {3057, 1122}, {3059, 354}, {3119, 2170}, {3158, 1420}, {3161, 5435}, {3190, 4306}, {3208, 1423}, {3239, 514}, {3270, 3937}, {3271, 1357}, {3596, 6063}, {3684, 1429}, {3685, 1447}, {3686, 553}, {3687, 3674}, {3688, 1401}, {3689, 1319}, {3690, 1425}, {3692, 63}, {3693, 241}, {3694, 1214}, {3695, 6356}, {3699, 664}, {3700, 7178}, {3701, 1441}, {3703, 3665}, {3705, 7185}, {3706, 4059}, {3707, 4031}, {3709, 7180}, {3710, 307}, {3711, 2099}, {3712, 7181}, {3713, 940}, {3715, 5221}, {3717, 9436}, {3718, 7182}, {3719, 7183}, {3737, 7203}, {3790, 7179}, {3811, 4341}, {3870, 4350}, {3900, 513}, {3939, 109}, {3952, 4566}, {3965, 3666}, {3974, 388}, {3975, 10030}, {4007, 4654}, {4012, 497}, {4030, 7198}, {4041, 4017}, {4046, 3649}, {4060, 3982}, {4061, 3671}, {4069, 4551}, {4073, 982}, {4076, 4998}, {4081, 11}, {4082, 10}, {4086, 4077}, {4092, 1365}, {4095, 4032}, {4103, 4605}, {4105, 663}, {4113, 4955}, {4123, 7210}, {4130, 650}, {4148, 812}, {4152, 1317}, {4157, 7214}, {4163, 522}, {4171, 661}, {4178, 7217}, {4183, 28}, {4319, 614}, {4397, 693}, {4420, 1442}, {4433, 1284}, {4451, 7249}, {4477, 4367}, {4511, 1443}, {4512, 3361}, {4513, 6180}, {4515, 37}, {4517, 1469}, {4518, 7233}, {4524, 512}, {4528, 900}, {4529, 4369}, {4546, 3667}, {4548, 7251}, {4571, 6516}, {4578, 100}, {4587, 1813}, {4827, 4790}, {4847, 10481}, {4853, 7271}, {4873, 5219}, {4882, 4328}, {4907, 5573}, {4936, 1743}, {4953, 3756}, {4990, 4977}, {5423, 8}, {5532, 7336}, {5546, 4565}, {5547, 7316}, {6056, 7335}, {6057, 12}, {6058, 7314}, {6059, 7337}, {6060, 7338}, {6061, 60}, {6062, 1354}, {6064, 7340}, {6065, 59}, {6068, 3321}, {6335, 13149}, {6552, 4452}, {6554, 4000}, {6555, 145}, {6556, 4373}, {6558, 190}, {6559, 673}, {6600, 1617}, {6602, 41}, {6603, 6610}, {6605, 1170}, {6607, 2488}, {6726, 266}, {6731, 174}, {6736, 3663}, {6737, 3664}, {6745, 1323}, {7017, 331}, {7027, 4146}, {7046, 4}, {7054, 593}, {7058, 1509}, {7062, 1355}, {7063, 1356}, {7064, 181}, {7065, 1363}, {7067, 1366}, {7068, 1367}, {7069, 1393}, {7070, 1394}, {7071, 25}, {7074, 221}, {7075, 1424}, {7079, 19}, {7080, 347}, {7081, 7176}, {7101, 92}, {7102, 7103}, {7156, 3213}, {7172, 3600}, {7253, 7192}, {7256, 99}, {7257, 4625}, {7258, 799}, {7259, 662}, {7283, 6359}, {7322, 7273}, {7359, 6357}, {7360, 5088}, {7367, 1436}, {7368, 198}, {8012, 1475}, {8641, 667}, {8706, 6613}, {8805, 8810}, {10382, 1467}

Notes

Let A' be the point where BC cuts the radical axis of the circumcircle of ABC and its A-excircle, and define B' and C' cyclically. Then A', B', C' are collinear on the tripolar of X₃₄₆. (César Lozada, October 8, 2019)