X₁₀₁₉

X₉₉-CEVA CONJUGATE OF X₁

X₁₀₁₉ = isogonal conjugate of X₁₀₁₈

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

X₁₀₁₉ (X₉₉-CEVA CONJUGATE OF X₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {36, 238}, {1, 512}

• Related Centers: X₉₉ (STEINER POINT), X₁ (INCENTER), X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER), X₂₃₈ (-LINE CONJUGATE OF X₃₇), X₅₁₂ (ISOGONAL CONJUGATE OF X₉₉)

Trilinears: \((b - c)/(b + c) : (c - a)/(c + a) : (a - b)/(a + b)\)

Barycentrics: \(a(b - c)/(b + c) : b(c - a)/(c + a) : c(a - b)/(a + b)\)

Search Value (6, 9, 13): \(22.939901199774\)

▸ Linear Combinations1 combination

• X₁₀₁₉ = X₁ + \(2\) X₄₇₈₄, \(3\) X₁ - \(2\) X₄₈₇₉, \(3\) X₆₄₉ - X₄₄₉₈, \(4\) X₆₄₉ - X₂₁₃₈₅, \(5\) X₁₆₉₈ - \(4\) X₂₁₀₅₁, \(7\) X₃₆₂₄ - \(4\) X₄₈₀₆, \(3\) X₃₇₃₃ - X₄₈₃₃, \(3\) X₃₇₃₇ - \(2\) X₄₈₃₃, X₃₇₃₇ + \(2\) X₄₈₄₀, \(2\) X₃₉₆₀ + X₄₉₇₉, \(3\) X₄₀₆₃ - \(2\) X₄₄₉₈, \(3\) X₄₃₆₇ - X₄₈₇₉, \(3\) X₄₃₇₉ - \(2\) X₄₈₂₃, \(4\) X₄₄₉₈ - \(3\) X₂₁₃₈₅, \(2\) X₄₅₆₀ + X₄₉₆₀, \(3\) X₄₇₅₀ - X₂₁₁₂₄, \(3\) X₄₇₅₀ - \(2\) X₂₁₁₉₂, \(3\) X₄₇₈₄ + X₄₈₇₉, \(2\) X₄₇₉₄ - \(3\) X₈₆₄₃, X₄₈₃₃ + \(3\) X₄₈₄₀, X₄₉₈₃ - \(3\) X₁₄₄₁₉, X₂₃₇₂₉ - \(3\) X₃₀₇₂₄

X₁₀₁₉ lies on thje cubics K035 and K10174.

▸ Incident Central Lines134 lines

{1, 512}, {2, 4129}, {3, 39577}, {8, 4807}, {21, 35355}, {36, 238}, {39, 14991}, {40, 28473}, {57, 7178}, {58, 1027}, {81, 1022}, {83, 4444}, {86, 23892}, {99, 813}, {110, 1308}, {163, 1414}, {239, 514}, {522, 5214}, {523, 20369}, {525, 4897}, {659, 6372}, {661, 1931}, {662, 3257}, {663, 6005}, {693, 29013}, {741, 14665}, {759, 840}, {798, 2669}, {799, 4607}, {812, 4509}, {824, 21389}, {830, 2254}, {918, 2483}, {1020, 1262}, {1025, 4237}, {1424, 7031}, {1429, 3669}, {1434, 23599}, {1577, 4369}, {1634, 35338}, {1698, 21051}, {1734, 8678}, {1924, 20981}, {2084, 3572}, {2484, 15411}, {2533, 2787}, {2786, 7265}, {2832, 23738}, {3249, 4375}, {3293, 22320}, {3309, 7659}, {3333, 34958}, {3337, 29136}, {3624, 4806}, {3667, 7253}, {3676, 17925}, {3708, 7266}, {3751, 9040}, {3762, 29402}, {3800, 39545}, {3801, 29029}, {3835, 25510}, {3907, 4761}, {3960, 4979}, {4010, 29150}, {4033, 37205}, {4041, 4160}, {4049, 27003}, {4083, 4378}, {4122, 29090}, {4151, 17166}, {4379, 4823}, {4380, 4801}, {4382, 29270}, {4391, 24601}, {4401, 4724}, {4449, 29350}, {4458, 29118}, {4467, 23879}, {4705, 9508}, {4762, 18196}, {4774, 29268}, {4782, 29198}, {4785, 17179}, {4794, 8643}, {4813, 27646}, {4817, 16737}, {4874, 29170}, {4922, 29298}, {5307, 14618}, {5539, 9424}, {6003, 6909}, {6006, 28591}, {6626, 27929}, {7153, 9336}, {7255, 16549}, {8027, 18169}, {8034, 29821}, {8690, 28520}, {8712, 18199}, {9013, 33844}, {10455, 21211}, {10566, 20950}, {16744, 23355}, {16755, 28859}, {16874, 16877}, {16887, 26843}, {17096, 30723}, {17174, 20295}, {17205, 31647}, {17210, 25381}, {17761, 26845}, {18827, 35172}, {20517, 29132}, {21146, 29070}, {21188, 23788}, {21392, 23875}, {22108, 28902}, {23729, 30724}, {23815, 24719}, {24019, 32668}, {24237, 26856}, {24624, 37222}, {26702, 28838}, {26775, 27013}, {26821, 26853}, {26823, 26825}, {27168, 27169}, {27527, 31286}, {29458, 29459}, {29770, 29772}, {32678, 35049}, {32944, 35353}, {39568, 39600}

▸ Functional Relations19 relations

• X₁₀₁₉ = midpoint of X_i and X_j for these {i,j}: {3669, 4790}, {3733, 4840}, {4367, 4784}, {4378, 4834}, {4380, 4801}, {4560, 7192}

• X₁₀₁₉ = reflection of X_i in X_j for these {i,j}: {1, 4367}, {8, 4807}, {661, 14838}, {1577, 4369}, {3737, 3733}, {4040, 667}, {4063, 649}, {4705, 9508}, {4724, 4401}, {4960, 7192}, {7265, 8045}, {14349, 905}, {21124, 21192}, {21385, 4063}, {24719, 23815}

• X₁₀₁₉ = isogonal conjugate of X₁₀₁₈

• X₁₀₁₉ = isotomic conjugate of X₄₀₃₃

• X₁₀₁₉ = anticomplement of X₄₁₂₉

• X₁₀₁₉ = anticomplement of the isotomic conjugate of X₃₇₂₀₅

• X₁₀₁₉ = isogonal conjugate of the anticomplement of X₁₇₇₆₁

• X₁₀₁₉ = isogonal conjugate of the isotomic conjugate of X₇₁₉₉

• X₁₀₁₉ = X_i-anticomplementary conjugate of X_j for these (i,j): {110, 18133}, {163, 24068}, {596, 21294}, {8050, 21287}, {34594, 69}, {37205, 6327}

• X₁₀₁₉ = X_i-Ceva conjugate of X_j for these (i,j): {27, 17197}, {28, 18184}, {81, 16726}, {99, 1}, {100, 18166}, {110, 18164}, {190, 8025}, {662, 81}, {757, 244}, {799, 86}, {873, 3248}, {932, 18171}, {1014, 18191}, {1414, 58}, {1434, 17205}, {3733, 18197}, {4565, 57}, {4584, 18206}, {4589, 18792}, {4598, 16738}, {4603, 17207}, {4614, 4658}, {4623, 18169}, {4637, 1014}, {7192, 3737}, {8690, 18186}, {17096, 7203}, {34594, 16696}, {37205, 2}

• X₁₀₁₉ = X_i-cross conjugate of X_j for these (i,j): {244, 757}, {513, 7192}, {649, 3733}, {1015, 1}, {1086, 57}, {1977, 87}, {3123, 32010}, {3248, 873}, {3669, 17925}, {3733, 7203}, {3937, 269}, {3960, 1022}, {4979, 513}, {7252, 3737}, {8042, 16726}, {8632, 1027}, {16726, 81}, {16742, 274}, {17217, 18197}, {18191, 1014}, {23470, 36598}, {38346, 6}

• X₁₀₁₉ = X_i-isoconjugate of X_j for these (i,j): {1, 1018}, {2, 4557}, {4, 4574}, {6, 3952}, {8, 4559}, {9, 4551}, {10, 101}, {12, 5546}, {21, 21859}, {31, 4033}, {32, 27808}, {37, 100}, {42, 190}, {55, 4552}, {56, 30730}, {57, 4069}, {58, 4103}, {59, 3700}, {65, 644}, {71, 1897}, {72, 1783}, {82, 35309}, {99, 1500}, {106, 4169}, {108, 3694}, {109, 2321}, {110, 594}, {112, 3695}, {162, 3949}, {163, 1089}, {181, 645}, {200, 1020}, {210, 651}, {213, 668}, {220, 4566}, {225, 4587}, {226, 3939}, {228, 6335}, {281, 23067}, {306, 8750}, {313, 32739}, {321, 692}, {512, 1016}, {523, 1252}, {643, 2171}, {646, 1402}, {647, 15742}, {648, 3690}, {653, 2318}, {660, 2238}, {661, 765}, {662, 756}, {664, 1334}, {666, 20683}, {669, 31625}, {670, 7109}, {677, 17747}, {740, 813}, {787, 28593}, {789, 3774}, {798, 7035}, {799, 872}, {825, 3773}, {831, 28594}, {850, 23990}, {901, 3943}, {919, 3932}, {932, 20691}, {934, 4515}, {983, 7239}, {1023, 4674}, {1026, 18785}, {1042, 6558}, {1110, 1577}, {1126, 4115}, {1211, 32736}, {1213, 8701}, {1254, 7259}, {1275, 4524}, {1292, 3991}, {1293, 3950}, {1331, 1826}, {1332, 1824}, {1400, 3699}, {1415, 3701}, {1427, 4578}, {1461, 4082}, {1576, 28654}, {1834, 29163}, {1880, 4571}, {1918, 1978}, {1962, 37212}, {1983, 15065}, {2092, 8707}, {2149, 4086}, {2197, 36797}, {2205, 6386}, {2276, 4613}, {2284, 13576}, {2292, 36147}, {2295, 3903}, {2328, 4605}, {2333, 4561}, {2346, 35310}, {2427, 38955}, {2702, 6541}, {2748, 16611}, {2901, 29014}, {3122, 6632}, {3257, 21805}, {3610, 32691}, {3696, 8693}, {3704, 8687}, {3709, 4998}, {3710, 32674}, {3714, 32693}, {3724, 36804}, {3747, 4562}, {3778, 4621}, {3842, 28841}, {3930, 36086}, {3948, 34067}, {3971, 34071}, {3986, 28226}, {3992, 32665}, {3994, 34075}, {4024, 4570}, {4029, 6014}, {4041, 4564}, {4058, 28162}, {4072, 8699}, {4076, 7180}, {4078, 28847}, {4079, 4600}, {4080, 23344}, {4085, 28883}, {4095, 29055}, {4120, 9268}, {4125, 34073}, {4136, 8685}, {4158, 6529}, {4171, 7045}, {4516, 31615}, {4553, 18098}, {4556, 6535}, {4558, 7140}, {4565, 6057}, {4567, 4705}, {4573, 7064}, {4585, 34857}, {4595, 23493}, {4596, 21816}, {4603, 21803}, {4606, 37593}, {4618, 21821}, {4628, 15523}, {4629, 8013}, {4646, 6574}, {4729, 5382}, {4730, 5376}, {4756, 28625}, {4767, 28658}, {4770, 5385}, {4849, 27834}, {4878, 37206}, {5257, 8694}, {5295, 36080}, {5377, 24290}, {5378, 21832}, {5380, 21839}, {5381, 14404}, {6065, 7178}, {6539, 35327}, {6540, 20970}, {6543, 17943}, {6577, 21070}, {6606, 21795}, {6725, 6733}, {7012, 8611}, {7141, 32661}, {8684, 18904}, {8706, 21796}, {8708, 16589}, {8709, 21830}, {9271, 21885}, {13138, 21871}, {17757, 32641}, {20491, 20640}, {20501, 20696}, {20680, 39272}, {20693, 37135}, {20964, 27805}, {21033, 36098}, {21045, 31616}, {21075, 36049}, {21078, 36050}, {21759, 36863}, {21801, 36037}, {21868, 29227}, {29127, 37715}

• X₁₀₁₉ = cevapoint of X_i and X_j for these (i,j): {6, 23404}, {512, 14991}, {513, 649}, {661, 4132}, {3733, 7252}, {3768, 38349}, {7192, 17217}, {8042, 16726}

• X₁₀₁₉ = crosspoint of X_i and X_j for these (i,j): {81, 662}, {86, 799}, {99, 1509}, {190, 1255}, {593, 4565}, {823, 837}, {1014, 4637}, {1414, 1434}, {7192, 17096}

• X₁₀₁₉ = crosssum of X_i and X_j for these (i,j): {10, 4151}, {37, 661}, {42, 798}, {71, 8611}, {210, 4171}, {512, 1500}, {513, 3720}, {523, 3925}, {594, 3700}, {649, 1100}, {650, 2269}, {656, 17441}, {822, 836}, {1334, 4041}, {4024, 21675}, {4120, 21942}, {4705, 21816}, {4730, 21821}, {4770, 21822}, {4931, 21943}, {8061, 17456}, {20680, 24290}

• X₁₀₁₉ = trilinear pole of line {244, 659}

• X₁₀₁₉ = crossdifference of every pair of points on line {37, 42}

• X₁₀₁₉ = barycentric product X_i*X_j for these {i,j}: {1, 7192}, {6, 7199}, {7, 3737}, {8, 7203}, {9, 17096}, {11, 1414}, {19, 15419}, {21, 3676}, {27, 905}, {28, 4025}, {56, 18155}, {57, 4560}, {58, 693}, {60, 4077}, {63, 17925}, {75, 3733}, {81, 514}, {85, 7252}, {86, 513}, {87, 17217}, {92, 7254}, {99, 244}, {100, 17205}, {101, 16727}, {104, 23788}, {105, 23829}, {108, 17219}, {110, 1111}, {141, 39179}, {162, 1565}, {163, 23989}, {190, 16726}, {256, 17212}, {257, 18200}, {261, 4017}, {269, 7253}, {270, 17094}, {272, 23800}, {273, 23189}, {274, 649}, {279, 1021}, {284, 24002}, {286, 1459}, {310, 667}, {330, 18197}, {333, 3669}, {512, 873}, {522, 1014}, {523, 757}, {552, 4041}, {593, 1577}, {643, 1358}, {648, 3942}, {650, 1434}, {651, 17197}, {659, 18827}, {661, 1509}, {662, 1086}, {664, 18191}, {670, 3248}, {741, 3766}, {759, 4453}, {763, 4024}, {764, 4600}, {799, 1015}, {811, 3937}, {812, 37128}, {849, 850}, {876, 33295}, {893, 16737}, {932, 23824}, {982, 7255}, {1016, 8042}, {1020, 26856}, {1022, 16704}, {1027, 30941}, {1088, 21789}, {1146, 4637}, {1169, 4509}, {1171, 4978}, {1178, 4374}, {1333, 3261}, {1357, 7257}, {1396, 6332}, {1408, 35519}, {1412, 4391}, {1435, 15411}, {1444, 7649}, {1474, 15413}, {1647, 4622}, {1790, 17924}, {1847, 23090}, {1919, 6385}, {1977, 4602}, {2087, 4615}, {2160, 16755}, {2170, 4573}, {2185, 7178}, {2215, 15417}, {2310, 4616}, {2363, 3004}, {2533, 7303}, {2969, 4592}, {2973, 4575}, {3122, 4623}, {3125, 4610}, {3271, 4625}, {3337, 7372}, {3572, 30940}, {3801, 7305}, {3960, 24624}, {4049, 30576}, {4086, 7341}, {4131, 8747}, {4228, 26721}, {4367, 32010}, {4481, 14621}, {4556, 16732}, {4565, 4858}, {4567, 6545}, {4584, 27918}, {4589, 27846}, {4598, 16742}, {4601, 21143}, {4603, 7200}, {4635, 14936}, {4705, 6628}, {4840, 30598}, {4960, 25417}, {4979, 32014}, {5317, 30805}, {6384, 16695}, {6591, 17206}, {7058, 7216}, {7153, 27527}, {7177, 17926}, {7215, 36126}, {8034, 24037}, {9309, 17218}, {9311, 18199}, {10566, 16696}, {13478, 16754}, {14377, 16751}, {16887, 18108}, {20028, 21173}, {21208, 34594}, {23345, 30939}, {24018, 36419}, {30581, 31010}

• X₁₀₁₉ = barycentric quotient X_i/X_j for these {i,j}: {1, 3952}, {2, 4033}, {6, 1018}, {9, 30730}, {11, 4086}, {21, 3699}, {27, 6335}, {28, 1897}, {31, 4557}, {37, 4103}, {39, 35309}, {44, 4169}, {48, 4574}, {55, 4069}, {56, 4551}, {57, 4552}, {58, 100}, {60, 643}, {75, 27808}, {81, 190}, {86, 668}, {99, 7035}, {110, 765}, {162, 15742}, {163, 1252}, {244, 523}, {261, 7257}, {269, 4566}, {270, 36797}, {274, 1978}, {283, 4571}, {284, 644}, {310, 6386}, {333, 646}, {512, 756}, {513, 10}, {514, 321}, {521, 3710}, {522, 3701}, {523, 1089}, {552, 4625}, {593, 662}, {603, 23067}, {604, 4559}, {643, 4076}, {647, 3949}, {649, 37}, {650, 2321}, {652, 3694}, {656, 3695}, {657, 4515}, {659, 740}, {661, 594}, {662, 1016}, {663, 210}, {665, 3930}, {667, 42}, {669, 872}, {693, 313}, {741, 660}, {757, 99}, {763, 4610}, {764, 3120}, {798, 1500}, {799, 31625}, {810, 3690}, {812, 3948}, {849, 110}, {873, 670}, {891, 3994}, {900, 3992}, {905, 306}, {985, 4613}, {1014, 664}, {1015, 661}, {1021, 346}, {1022, 4080}, {1027, 13576}, {1086, 1577}, {1098, 7256}, {1100, 4115}, {1111, 850}, {1169, 36147}, {1171, 37212}, {1178, 3903}, {1333, 101}, {1357, 4017}, {1358, 4077}, {1396, 653}, {1400, 21859}, {1407, 1020}, {1408, 109}, {1412, 651}, {1414, 4998}, {1427, 4605}, {1434, 4554}, {1437, 1331}, {1444, 4561}, {1459, 72}, {1474, 1783}, {1475, 35310}, {1491, 3773}, {1509, 799}, {1565, 14208}, {1576, 1110}, {1577, 28654}, {1635, 3943}, {1769, 17757}, {1790, 1332}, {1919, 213}, {1924, 7109}, {1946, 2318}, {1960, 21805}, {1977, 798}, {1980, 1918}, {2087, 4120}, {2150, 5546}, {2170, 3700}, {2185, 645}, {2193, 4587}, {2194, 3939}, {2203, 8750}, {2206, 692}, {2254, 3932}, {2275, 7239}, {2287, 6558}, {2328, 4578}, {2363, 8707}, {2423, 2250}, {2483, 28594}, {2488, 21039}, {2516, 4072}, {2522, 3610}, {2530, 15523}, {2605, 3678}, {2832, 4442}, {2969, 24006}, {3004, 18697}, {3063, 1334}, {3120, 4036}, {3121, 4079}, {3122, 4705}, {3123, 21051}, {3125, 4024}, {3248, 512}, {3249, 3121}, {3261, 27801}, {3271, 4041}, {3285, 1023}, {3286, 1026}, {3287, 4095}, {3310, 21801}, {3337, 6758}, {3669, 226}, {3675, 4088}, {3676, 1441}, {3733, 1}, {3736, 3799}, {3737, 8}, {3756, 4404}, {3766, 35544}, {3776, 20234}, {3777, 2887}, {3900, 4082}, {3937, 656}, {3942, 525}, {3960, 3936}, {4014, 21052}, {4017, 12}, {4025, 20336}, {4040, 4651}, {4041, 6057}, {4057, 3293}, {4063, 3995}, {4077, 34388}, {4079, 762}, {4083, 3971}, {4091, 3998}, {4132, 4075}, {4164, 4039}, {4273, 4752}, {4367, 1215}, {4369, 3963}, {4374, 1237}, {4379, 4377}, {4391, 30713}, {4394, 3950}, {4401, 3896}, {4435, 3985}, {4448, 4783}, {4449, 3967}, {4453, 35550}, {4475, 4122}, {4481, 3661}, {4491, 31855}, {4498, 3175}, {4509, 1228}, {4556, 4567}, {4560, 312}, {4565, 4564}, {4567, 6632}, {4591, 5376}, {4610, 4601}, {4637, 1275}, {4653, 4767}, {4658, 4756}, {4705, 6535}, {4724, 3696}, {4762, 4044}, {4777, 4125}, {4782, 3993}, {4784, 3842}, {4790, 5257}, {4802, 4066}, {4833, 3679}, {4840, 1698}, {4874, 4710}, {4879, 4096}, {4960, 28605}, {4977, 4647}, {4978, 1230}, {4979, 1213}, {4983, 8013}, {5009, 3573}, {5029, 20693}, {5323, 14594}, {6085, 4695}, {6129, 21075}, {6363, 4642}, {6371, 2292}, {6372, 21020}, {6377, 21834}, {6545, 16732}, {6586, 4006}, {6588, 21074}, {6589, 21078}, {6591, 1826}, {6615, 21031}, {6628, 4623}, {6729, 6725}, {7054, 7259}, {7058, 7258}, {7117, 8611}, {7178, 6358}, {7180, 2171}, {7192, 75}, {7199, 76}, {7202, 7265}, {7203, 7}, {7216, 6354}, {7234, 21803}, {7250, 1254}, {7252, 9}, {7253, 341}, {7254, 63}, {7255, 7033}, {7303, 4594}, {7304, 36860}, {7341, 1414}, {8027, 3122}, {8034, 2643}, {8042, 1086}, {8054, 4132}, {8578, 22321}, {8632, 2238}, {8642, 4878}, {8643, 4849}, {8656, 21870}, {8712, 4656}, {9002, 4424}, {9508, 6541}, {14419, 4062}, {14438, 4144}, {14838, 3969}, {14936, 4171}, {15419, 304}, {16695, 43}, {16696, 4568}, {16704, 24004}, {16726, 514}, {16727, 3261}, {16737, 1920}, {16742, 3835}, {16751, 17233}, {16754, 4417}, {16755, 33939}, {16757, 4150}, {16947, 1415}, {17096, 85}, {17187, 4553}, {17197, 4391}, {17205, 693}, {17212, 1909}, {17214, 4106}, {17217, 6376}, {17219, 35518}, {17302, 21604}, {17418, 3714}, {17420, 3704}, {17477, 4139}, {17494, 4043}, {17925, 92}, {17926, 7101}, {18108, 18082}, {18155, 3596}, {18163, 25268}, {18184, 25259}, {18186, 25272}, {18191, 522}, {18196, 25264}, {18197, 192}, {18199, 3729}, {18200, 894}, {18210, 4064}, {18211, 3667}, {18268, 813}, {18600, 21580}, {18792, 23354}, {18827, 4583}, {19945, 14431}, {20979, 20691}, {20981, 2295}, {21007, 3294}, {21123, 3954}, {21143, 3125}, {21173, 17751}, {21191, 22028}, {21196, 27569}, {21758, 2245}, {21789, 200}, {21828, 4053}, {21832, 4037}, {22096, 810}, {22383, 71}, {23090, 3692}, {23092, 22370}, {23189, 78}, {23224, 3682}, {23345, 4674}, {23355, 18793}, {23572, 21877}, {23788, 3262}, {23824, 20906}, {23829, 3263}, {23989, 20948}, {24002, 349}, {24006, 7141}, {24624, 36804}, {26822, 18040}, {27527, 4110}, {27644, 4595}, {27846, 4010}, {28209, 4714}, {29226, 4135}, {29545, 26772}, {29821, 21295}, {30940, 27853}, {30966, 4505}, {31947, 21081}, {33295, 874}, {33296, 36863}, {36419, 823}, {36420, 24019}, {37128, 4562}, {38238, 21100}, {39179, 83}

Notes

Evans conjectured that X₁, X₄₈₄, X₁₂₇₆, X₁₂₇₇ are concyclic, and reported that Paul Yiu confirmed this conjecture and noted that the center of this circle is X₁₀₁₉. (Lawrence Evans, 2/24/2003)

The circle is now known as the Evans circle. (September 2020)