X₆₆₆

TRILINEAR POLE OF LINE X₂X_11

X₆₆₆ = trilinear pole of line {2, 11}

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

X₆₆₆ (TRILINEAR POLE OF LINE X₂X_11) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {239, 294}, {101, 514}

• Related Centers: X₂₃₉ (-LINE CONJUGATE OF X₄₂), X₂₉₄ (ISOGONAL CONJUGATE OF X₂₄₁), X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₅₁₄ (ISOGONAL CONJUGATE OF X₁₀₁)

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

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

Search Value (6, 9, 13): \(-2.629117015619\)

▸ Linear Combinations1 combination

• X₆₆₆ = \(3\) X₂ - \(4\) X₄₀₅₄₀, \(3\) X₁₈₈₂₁ - \(4\) X₃₅₀₉₄, X₁₈₈₂₁ - \(4\) X₃₅₁₁₃, \(3\) X₁₈₈₂₁ - \(2\) X₃₉₃₅₃, X₁₈₈₂₁ + \(2\) X₃₉₃₆₃, \(3\) X₁₈₈₂₁ - \(8\) X₄₀₅₄₀, \(5\) X₂₉₅₉₀ - X₄₀₈₆₈, X₃₅₀₉₄ - \(3\) X₃₅₁₁₃, \(2\) X₃₅₀₉₄ + \(3\) X₃₉₃₆₃, \(6\) X₃₅₁₁₃ - X₃₉₃₅₃, \(2\) X₃₅₁₁₃ + X₃₉₃₆₃, \(3\) X₃₅₁₁₃ - \(2\) X₄₀₅₄₀, X₃₉₃₅₃ + \(3\) X₃₉₃₆₃, X₃₉₃₅₃ - \(4\) X₄₀₅₄₀, \(3\) X₃₉₃₆₃ + \(4\) X₄₀₅₄₀

X₆₆₆ lies on the Steiner circumellipse.

▸ Incident Central Lines85 lines

{1, 35167}, {2, 18821}, {6, 35119}, {8, 35158}, {10, 35163}, {37, 35152}, {99, 919}, {100, 31150}, {101, 514}, {105, 898}, {144, 30228}, {190, 522}, {220, 35120}, {238, 33674}, {239, 294}, {290, 15628}, {519, 1121}, {527, 673}, {644, 668}, {645, 670}, {648, 5379}, {650, 4998}, {651, 1275}, {659, 34067}, {671, 5080}, {693, 5375}, {742, 43099}, {883, 41075}, {885, 2398}, {889, 5381}, {894, 40754}, {993, 31169}, {1026, 1027}, {1252, 17494}, {1438, 3226}, {1462, 40400}, {1494, 15627}, {1757, 33676}, {1783, 17924}, {1814, 1944}, {2311, 3509}, {2338, 3912}, {2341, 14616}, {2397, 2402}, {2401, 32641}, {2414, 32644}, {3008, 9436}, {3241, 35168}, {3758, 35175}, {3872, 35957}, {4416, 35150}, {4453, 37143}, {4552, 6606}, {4555, 5376}, {4573, 35326}, {4581, 6648}, {4582, 6635}, {4585, 35171}, {4586, 5384}, {4597, 5385}, {4762, 40865}, {5081, 36124}, {5382, 6631}, {5383, 18830}, {5386, 43097}, {5387, 24004}, {5388, 46132}, {5389, 43098}, {5452, 6063}, {6542, 35141}, {6654, 16834}, {7760, 35172}, {8751, 46108}, {14827, 21218}, {15629, 34393}, {16588, 40419}, {17350, 34361}, {18031, 43093}, {26777, 43986}, {28132, 35157}, {29574, 35153}, {29590, 40868}, {31633, 40166}, {39272, 42720}, {40863, 46137}

▸ Functional Relations18 relations

• X₆₆₆ = midpoint of X_i and X_j for these {i,j}: {2, 39363}, {239, 10025}

• X₆₆₆ = reflection of X_i in X_j for these {i,j}: {2, 35113}, {927, 34906}, {3912, 40869}, {9436, 3008}, {18821, 2}, {35094, 40540}, {39353, 35094}

• X₆₆₆ = isogonal conjugate of X₆₆₅

• X₆₆₆ = isotomic conjugate of X₉₁₈

• X₆₆₆ = complement of X₃₉₃₅₃

• X₆₆₆ = anticomplement of X₃₅₀₉₄

• X₆₆₆ = X₉₂-isoconjugate of X₂₃₂₂₅

• X₆₆₆ = Steiner-circumellipse-X₁-antipode of X₃₅₁₆₇

• X₆₆₆ = X_i-anticomplementary conjugate of X_j for these (i,j): {5377, 20552}, {32666, 39353}

• X₆₆₆ = X_i-Ceva conjugate of X_j for these (i,j): {34085, 927}, {39293, 14942}

• X₆₆₆ = X_i-cross conjugate of X_j for these (i,j): {239, 1016}, {294, 5377}, {518, 4998}, {644, 39272}, {660, 8709}, {883, 30610}, {885, 2481}, {918, 2}, {926, 40419}, {2284, 100}, {2348, 4076}, {2398, 664}, {2402, 34018}, {3573, 99}, {3766, 83}, {4435, 1}, {10025, 1275}, {14942, 39293}, {28132, 14942}, {30807, 46102}, {32922, 7035}, {36086, 927}, {36236, 4555}, {43929, 105}, {46163, 919}

• X₆₆₆ = cevapoint of X_i and X_j for these (i,j): {2, 918}, {6, 659}, {100, 2284}, {105, 43929}, {190, 3570}, {294, 885}, {385, 3287}, {513, 3290}, {514, 3008}, {518, 650}, {522, 40869}, {523, 2238}, {812, 27942}, {926, 16588}, {1027, 18785}, {14942, 28132}

• X₆₆₆ = trilinear pole of line {2, 11}

• X₆₆₆ = crossdifference of every pair of points on line {15615, 35505}

• X₆₆₆ = X_i-isoconjugate of X_j for these (i,j): {1, 665}, {6, 2254}, {31, 918}, {41, 43042}, {57, 926}, {58, 24290}, {85, 8638}, {92, 23225}, {101, 3675}, {109, 17435}, {213, 23829}, {241, 663}, {244, 2284}, {512, 18206}, {513, 672}, {514, 2223}, {518, 649}, {650, 1458}, {652, 1876}, {657, 34855}, {659, 3252}, {661, 3286}, {667, 3912}, {669, 18157}, {693, 9454}, {798, 30941}, {810, 15149}, {812, 40730}, {813, 38989}, {875, 17755}, {905, 2356}, {909, 42758}, {1015, 1026}, {1019, 20683}, {1024, 1362}, {1025, 3271}, {1027, 6184}, {1333, 4088}, {1438, 3126}, {1459, 5089}, {1635, 34230}, {1818, 6591}, {1861, 22383}, {1919, 3263}, {1946, 5236}, {2170, 2283}, {2340, 3669}, {2424, 9502}, {3063, 9436}, {3248, 42720}, {3261, 9455}, {3310, 36819}, {3572, 8299}, {3693, 43924}, {3733, 3930}, {4712, 43929}, {4790, 14626}, {4925, 38266}, {5091, 34905}, {7192, 39258}, {7649, 20752}, {8632, 22116}, {9315, 42341}, {14439, 23345}, {14936, 41353}, {15615, 34085}, {20662, 35355}, {32666, 35094}, {35505, 36086}, {37136, 42771}

• X₆₆₆ = barycentric product X_i*X_j for these {i,j}: {6, 36803}, {7, 36802}, {8, 927}, {9, 34085}, {55, 46135}, {75, 36086}, {76, 919}, {99, 13576}, {100, 2481}, {101, 18031}, {105, 668}, {190, 673}, {294, 4554}, {308, 46163}, {312, 36146}, {390, 41075}, {522, 39293}, {561, 32666}, {644, 34018}, {646, 1462}, {651, 36796}, {658, 6559}, {664, 14942}, {693, 5377}, {799, 18785}, {885, 4998}, {1027, 7035}, {1275, 28132}, {1376, 14727}, {1438, 1978}, {1814, 6335}, {1897, 31637}, {2195, 4572}, {3112, 35333}, {3596, 32735}, {4076, 43930}, {4561, 36124}, {4562, 6654}, {4569, 28071}, {4607, 36816}, {4633, 14625}, {5388, 29956}, {6185, 42720}, {9503, 42719}, {14267, 35574}, {21615, 36138}, {31625, 43929}, {31638, 37206}

• X₆₆₆ = barycentric quotient X_i/X_j for these {i,j}: {1, 2254}, {2, 918}, {6, 665}, {7, 43042}, {10, 4088}, {37, 24290}, {55, 926}, {59, 2283}, {86, 23829}, {99, 30941}, {100, 518}, {101, 672}, {105, 513}, {108, 1876}, {109, 1458}, {110, 3286}, {145, 4925}, {184, 23225}, {190, 3912}, {294, 650}, {513, 3675}, {517, 42758}, {518, 3126}, {644, 3693}, {648, 15149}, {650, 17435}, {651, 241}, {653, 5236}, {659, 38989}, {660, 22116}, {662, 18206}, {664, 9436}, {665, 35505}, {668, 3263}, {673, 514}, {692, 2223}, {765, 1026}, {799, 18157}, {813, 3252}, {884, 3271}, {885, 11}, {901, 34230}, {906, 20752}, {918, 35094}, {919, 6}, {927, 7}, {934, 34855}, {1016, 42720}, {1018, 3930}, {1023, 14439}, {1024, 2170}, {1026, 4712}, {1027, 244}, {1252, 2284}, {1331, 1818}, {1332, 25083}, {1376, 42341}, {1416, 43924}, {1438, 649}, {1462, 3669}, {1738, 20504}, {1783, 5089}, {1814, 905}, {1897, 1861}, {2175, 8638}, {2195, 663}, {2283, 1362}, {2284, 6184}, {2402, 4904}, {2481, 693}, {3570, 17755}, {3573, 8299}, {3699, 3717}, {3835, 23773}, {3939, 2340}, {3952, 3932}, {3971, 21959}, {4427, 4966}, {4554, 40704}, {4557, 20683}, {4562, 40217}, {4564, 1025}, {4579, 4447}, {4998, 883}, {5377, 100}, {5379, 4238}, {6335, 46108}, {6559, 3239}, {6654, 812}, {7045, 41353}, {8638, 15615}, {8694, 14626}, {8750, 2356}, {8751, 6591}, {9319, 34905}, {10015, 42770}, {10099, 18210}, {13576, 523}, {14267, 23770}, {14625, 4841}, {14727, 32023}, {14942, 522}, {18031, 3261}, {18785, 661}, {21450, 14347}, {23696, 7004}, {28071, 3900}, {28132, 1146}, {31637, 4025}, {31638, 4468}, {32658, 22383}, {32666, 31}, {32735, 56}, {32739, 9454}, {34018, 24002}, {34067, 40730}, {34085, 85}, {34906, 9318}, {35185, 3433}, {35313, 3035}, {35333, 38}, {36037, 36819}, {36041, 2191}, {36057, 1459}, {36086, 1}, {36124, 7649}, {36138, 2279}, {36146, 57}, {36796, 4391}, {36802, 8}, {36803, 76}, {36816, 4728}, {39272, 1280}, {39293, 664}, {40724, 4458}, {41934, 43929}, {42720, 4437}, {43042, 3323}, {43290, 4899}, {43921, 764}, {43929, 1015}, {43930, 1358}, {46135, 6063}, {46149, 2530}, {46163, 39}

• X₆₆₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 39353, 35094}, {885, 35313, 5377}, {35094, 35113, 40540}, {35094, 39353, 18821}, {35094, 40540, 2}, {35113, 39363, 18821}