X₆₇₃

TRILINEAR POLE OF LINE X₁X_514

X₆₇₃ = BSS(a^2→a) of X₉₈

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

X₆₇₃ (TRILINEAR POLE OF LINE X₁X_514) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 11}, {6, 7}

• Related Centers: X₂ (CENTROID), X₁₁ (FEUERBACH POINT), X₆ (SYMMEDIAN POINT), X₇ (GERGONNE POINT)

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

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

Search Value (6, 9, 13): \(2.666912049087\)

▸ Linear Combinations1 combination

• X₆₇₃ = \(3\) X₂ + X₄₁₈₄₅, \(4\) X₁₄₂ - \(5\) X₂₇₁₉₁, \(4\) X₄₄₂₂ - \(5\) X₁₈₂₃₀, \(3\) X₅₈₁₇ - \(2\) X₂₄₈₂₈, \(2\) X₁₆₅₉₃ + X₄₁₈₄₅, \(3\) X₂₁₁₆₈ - X₂₄₈₁₇, \(4\) X₂₅₃₅₁ - \(3\) X₃₈₀₅₂, \(3\) X₂₇₄₈₄ - X₃₃₈₈₈

X₆₇₃ lies on the conic {A, B, C,X₂,X₇}}, the cubics K251, K323, K385, K506, K623, K980, K983, the curves CC₉ and Q012.

▸ Incident Central Lines261 lines

{1, 17682}, {2, 11}, {4, 41785}, {6, 7}, {8, 4437}, {9, 75}, {10, 17681}, {19, 273}, {27, 162}, {56, 4209}, {57, 658}, {58, 29775}, {63, 37206}, {65, 27000}, {81, 39734}, {85, 2082}, {86, 142}, {87, 27498}, {88, 1638}, {101, 17761}, {103, 43672}, {144, 4373}, {150, 4904}, {169, 3673}, {218, 17753}, {226, 1174}, {234, 43192}, {238, 516}, {239, 335}, {241, 14189}, {272, 24606}, {277, 17170}, {310, 333}, {329, 42361}, {480, 24820}, {514, 30188}, {527, 666}, {537, 5223}, {545, 6172}, {655, 2161}, {664, 2170}, {668, 29479}, {675, 919}, {759, 4237}, {812, 1024}, {823, 8748}, {871, 37133}, {885, 900}, {893, 3752}, {897, 21180}, {909, 30379}, {910, 1447}, {918, 2400}, {927, 2291}, {954, 37502}, {958, 17691}, {971, 24813}, {1010, 16818}, {1027, 37129}, {1043, 29960}, {1111, 3732}, {1121, 4530}, {1125, 42335}, {1150, 37209}, {1155, 14197}, {1220, 17686}, {1268, 6666}, {1334, 32008}, {1416, 12573}, {1429, 9454}, {1434, 1475}, {1436, 1440}, {1465, 1945}, {1479, 17671}, {1492, 5135}, {1659, 31566}, {1714, 29461}, {1724, 29743}, {1733, 24846}, {1751, 15467}, {1760, 39943}, {1837, 26531}, {1861, 26001}, {1911, 27846}, {1981, 36123}, {2140, 4251}, {2160, 38340}, {2164, 7318}, {2246, 9318}, {2258, 2999}, {2259, 21617}, {2264, 41246}, {2280, 14828}, {2319, 4598}, {2329, 20257}, {2339, 19804}, {2345, 4422}, {2348, 10025}, {2364, 4604}, {2402, 6084}, {2432, 36100}, {2635, 23694}, {2669, 24378}, {2898, 5435}, {3216, 29434}, {3218, 37143}, {3243, 16834}, {3254, 5377}, {3286, 14953}, {3303, 27253}, {3451, 7175}, {3512, 16609}, {3570, 30997}, {3616, 17683}, {3684, 20335}, {3739, 24358}, {3871, 28742}, {3880, 40872}, {3885, 28961}, {3912, 5853}, {4008, 15299}, {4089, 34578}, {4253, 14377}, {4254, 25521}, {4312, 16468}, {4343, 45223}, {4361, 9055}, {4383, 39741}, {4393, 42871}, {4432, 16832}, {4534, 39351}, {4561, 18061}, {4599, 18087}, {4607, 24602}, {4649, 5542}, {4657, 25357}, {4859, 16779}, {5030, 17729}, {5086, 26526}, {5088, 43065}, {5220, 16816}, {5248, 17687}, {5253, 26964}, {5292, 29566}, {5299, 24790}, {5308, 8236}, {5698, 14267}, {5728, 36019}, {5759, 29243}, {5762, 24833}, {5817, 24828}, {6008, 24601}, {6650, 17768}, {6651, 15254}, {7008, 37279}, {7176, 40133}, {7384, 42356}, {7397, 35514}, {7402, 38149}, {7676, 8053}, {7677, 11349}, {7678, 44412}, {7770, 27299}, {8049, 32911}, {8257, 33794}, {8616, 31200}, {8751, 36099}, {9311, 9312}, {9328, 9329}, {9503, 43035}, {9669, 17675}, {10030, 20459}, {10099, 37142}, {11038, 17014}, {11375, 27183}, {11495, 20992}, {11683, 24633}, {12701, 27129}, {13390, 31565}, {13740, 30107}, {14923, 26653}, {14936, 24499}, {15481, 31349}, {15485, 31183}, {15570, 29584}, {15668, 28626}, {16060, 17030}, {16477, 30424}, {16484, 29571}, {16786, 17067}, {16826, 42819}, {16831, 38316}, {17050, 41239}, {17181, 20269}, {17284, 32941}, {17300, 20180}, {17308, 38200}, {17316, 20162}, {17370, 20195}, {17371, 28650}, {17379, 30712}, {17380, 20159}, {17680, 26561}, {17877, 21602}, {17963, 35466}, {20028, 27644}, {20059, 36606}, {20131, 26626}, {20132, 25557}, {20137, 29586}, {20147, 26806}, {20148, 39745}, {20153, 29592}, {20158, 39720}, {20332, 43929}, {20356, 25800}, {20367, 20605}, {20880, 33950}, {21044, 31640}, {21151, 37474}, {21168, 24817}, {21241, 30837}, {21264, 26244}, {23692, 36084}, {23696, 23707}, {24512, 40761}, {24584, 39732}, {24589, 37210}, {24610, 39746}, {24617, 24625}, {24619, 24624}, {24631, 40038}, {24632, 29756}, {24827, 31671}, {24880, 29564}, {25351, 29598}, {25532, 35342}, {26273, 27918}, {26801, 33826}, {26959, 33828}, {28916, 41228}, {29433, 29467}, {29437, 29483}, {29456, 29459}, {29473, 29742}, {29541, 40018}, {30332, 42318}, {30821, 32943}, {30946, 37658}, {32735, 36094}, {33150, 39723}, {36630, 40027}, {36662, 38037}, {36698, 43161}, {36799, 36803}, {36942, 38055}, {37207, 37686}, {37272, 42884}, {37650, 41325}, {38048, 39716}, {38941, 43057}

▸ Functional Relations22 relations

• X₆₇₃ = midpoint of X_i and X_j for these {i,j}: {144, 4440}, {20533, 41845}

• X₆₇₃ = reflection of X_i in X_j for these {i,j}: {7, 1086}, {190, 9}, {20533, 16593}, {31671, 24827}

• X₆₇₃ = isogonal conjugate of X₆₇₂

• X₆₇₃ = isotomic conjugate of X₃₉₁₂

• X₆₇₃ = complement of X₂₀₅₃₃

• X₆₇₃ = anticomplement of X₁₆₅₉₃

• X₆₇₃ = BSS(a^2→a) of X₉₈

• X₆₇₃ = antipode of X₇ in hyperbola {A, B, C,X₂,X₇}}

• X₆₇₃ = polar conjugate of X₁₈₆₁

• X₆₇₃ = X_i-anticomplementary conjugate of X_j for these (i,j): {1280, 20552}, {39272, 20295}

• X₆₇₃ = X_i-complementary conjugate of X_j for these (i,j): {2115, 3452}, {9499, 141}, {9500, 10}

• X₆₇₃ = X_i-Ceva conjugate of X_j for these (i,j): {927, 885}, {2481, 14942}, {6185, 6654}, {18031, 31637}, {36802, 2402}, {39293, 36086}

• X₆₇₃ = X_i-cross conjugate of X_j for these (i,j): {57, 9503}, {238, 86}, {239, 6654}, {291, 3226}, {294, 14942}, {516, 7}, {672, 1}, {812, 190}, {885, 927}, {1024, 36086}, {1438, 36124}, {1738, 75}, {2254, 664}, {3008, 2}, {4444, 35148}, {9436, 9311}, {9441, 21453}, {13576, 2481}, {18785, 105}, {20367, 81}, {20459, 6}, {20520, 43190}, {20605, 82}, {24618, 34234}, {24712, 1121}, {24715, 903}, {33854, 1220}, {36057, 31637}

• X₆₇₃ = cevapoint of X_i and X_j for these (i,j): {1, 672}, {2, 239}, {6, 20470}, {7, 14189}, {9, 5853}, {57, 43035}, {105, 294}, {649, 27846}, {1086, 6084}, {1438, 36057}, {2170, 2254}, {3120, 21832}, {9312, 9436}, {13576, 18785}

• X₆₇₃ = crosspoint of X_i and X_j for these (i,j): {57, 2111}, {291, 9442}, {2481, 34018}, {34085, 39293}

• X₆₇₃ = crosssum of X_i and X_j for these (i,j): {1, 39341}, {6, 20672}, {9, 24578}, {238, 9441}, {17435, 24290}, {20683, 39258}

• X₆₇₃ = trilinear pole of line {1, 514}

• X₆₇₃ = crossdifference of every pair of points on line {665, 926}

• X₆₇₃ = X_i-isoconjugate of X_j for these (i,j): {1, 672}, {2, 2223}, {3, 5089}, {4, 20752}, {6, 518}, {9, 1458}, {19, 1818}, {25, 25083}, {31, 3912}, {32, 3263}, {37, 3286}, {41, 9436}, {42, 18206}, {44, 34230}, {48, 1861}, {55, 241}, {56, 3693}, {57, 2340}, {58, 3930}, {59, 17435}, {63, 2356}, {75, 9454}, {76, 9455}, {81, 20683}, {86, 39258}, {100, 665}, {101, 2254}, {103, 9502}, {105, 6184}, {106, 14439}, {110, 24290}, {163, 4088}, {184, 46108}, {212, 5236}, {213, 30941}, {219, 1876}, {220, 34855}, {228, 15149}, {238, 3252}, {239, 40730}, {292, 8299}, {294, 1362}, {513, 2284}, {604, 3717}, {647, 4238}, {649, 1026}, {650, 2283}, {651, 926}, {657, 41353}, {663, 1025}, {667, 42720}, {673, 42079}, {692, 918}, {840, 1642}, {883, 3063}, {893, 4447}, {919, 3126}, {1214, 37908}, {1252, 3675}, {1280, 20662}, {1333, 3932}, {1438, 4712}, {1449, 14626}, {1814, 42071}, {1911, 17755}, {1914, 22116}, {1918, 18157}, {2175, 40704}, {2183, 36819}, {2210, 40217}, {2291, 35293}, {2414, 8642}, {2428, 3309}, {2481, 39686}, {2991, 20455}, {3290, 34159}, {4554, 8638}, {4899, 38266}, {4925, 34080}, {4966, 28615}, {5377, 35505}, {6168, 9439}, {6185, 23612}, {6335, 23225}, {7077, 34253}, {15344, 20728}, {17798, 40781}, {23102, 41934}, {32641, 42758}, {32658, 34337}

• X₆₇₃ = barycentric product X_i*X_j for these {i,j}: {1, 2481}, {4, 31637}, {6, 18031}, {7, 14942}, {9, 34018}, {11, 39293}, {57, 36796}, {69, 36124}, {75, 105}, {76, 1438}, {85, 294}, {86, 13576}, {92, 1814}, {264, 36057}, {274, 18785}, {277, 31638}, {279, 6559}, {304, 8751}, {312, 1462}, {335, 6654}, {514, 666}, {522, 927}, {649, 36803}, {650, 34085}, {658, 28132}, {663, 46135}, {664, 885}, {668, 1027}, {693, 36086}, {811, 10099}, {884, 4572}, {919, 3261}, {1024, 4554}, {1088, 28071}, {1111, 5377}, {1416, 3596}, {1447, 33676}, {1969, 32658}, {1978, 43929}, {2195, 6063}, {2402, 37206}, {3112, 46149}, {3227, 36816}, {3676, 36802}, {3699, 43930}, {3912, 6185}, {4391, 36146}, {4449, 14727}, {7035, 43921}, {7261, 40724}, {9503, 30807}, {18026, 23696}, {29956, 37133}, {32666, 40495}, {32735, 35519}, {40754, 40845}

• X₆₇₃ = barycentric quotient X_i/X_j for these {i,j}: {1, 518}, {2, 3912}, {3, 1818}, {4, 1861}, {6, 672}, {7, 9436}, {8, 3717}, {9, 3693}, {10, 3932}, {19, 5089}, {25, 2356}, {27, 15149}, {31, 2223}, {32, 9454}, {34, 1876}, {37, 3930}, {42, 20683}, {44, 14439}, {48, 20752}, {55, 2340}, {56, 1458}, {57, 241}, {58, 3286}, {63, 25083}, {75, 3263}, {81, 18206}, {85, 40704}, {86, 30941}, {92, 46108}, {100, 1026}, {101, 2284}, {104, 36819}, {105, 1}, {106, 34230}, {109, 2283}, {145, 4899}, {162, 4238}, {171, 4447}, {190, 42720}, {213, 39258}, {238, 8299}, {239, 17755}, {244, 3675}, {269, 34855}, {274, 18157}, {278, 5236}, {291, 22116}, {292, 3252}, {294, 9}, {335, 40217}, {513, 2254}, {514, 918}, {518, 4712}, {523, 4088}, {560, 9455}, {649, 665}, {651, 1025}, {661, 24290}, {663, 926}, {664, 883}, {666, 190}, {672, 6184}, {884, 663}, {885, 522}, {910, 9502}, {919, 101}, {927, 664}, {934, 41353}, {1024, 650}, {1027, 513}, {1125, 4966}, {1155, 35293}, {1416, 56}, {1429, 34253}, {1438, 6}, {1447, 39775}, {1458, 1362}, {1462, 57}, {1738, 120}, {1769, 42758}, {1814, 63}, {1861, 34337}, {1911, 40730}, {2170, 17435}, {2195, 55}, {2223, 42079}, {2246, 1642}, {2254, 3126}, {2299, 37908}, {2334, 14626}, {2356, 42071}, {2402, 4468}, {2481, 75}, {3008, 16593}, {3290, 17464}, {3512, 40781}, {3616, 4684}, {3667, 4925}, {3676, 43042}, {3729, 40883}, {3912, 4437}, {4366, 27919}, {4449, 42341}, {4712, 23102}, {5377, 765}, {5853, 40609}, {6180, 6168}, {6559, 346}, {6654, 239}, {7192, 23829}, {7193, 20778}, {8751, 19}, {9453, 9451}, {9454, 39686}, {9503, 36101}, {10099, 656}, {13576, 10}, {14189, 36905}, {14197, 28850}, {14267, 1738}, {14625, 5257}, {14942, 8}, {18031, 76}, {18206, 16728}, {18785, 37}, {20470, 39046}, {20780, 20749}, {21051, 21959}, {21138, 23773}, {21956, 20482}, {23694, 2808}, {23696, 521}, {23770, 20504}, {27818, 10029}, {27846, 38989}, {28071, 200}, {28132, 3239}, {29956, 3250}, {31637, 69}, {31638, 344}, {32658, 48}, {32666, 692}, {32735, 109}, {33674, 17794}, {33676, 4518}, {34018, 85}, {34063, 25302}, {34085, 4554}, {34906, 40865}, {35333, 4553}, {36041, 1292}, {36057, 3}, {36086, 100}, {36111, 26706}, {36124, 4}, {36138, 8693}, {36146, 651}, {36796, 312}, {36802, 3699}, {36803, 1978}, {36816, 536}, {37206, 2414}, {39293, 4998}, {40724, 4645}, {40754, 3509}, {40761, 39252}, {41934, 1438}, {42079, 23612}, {42754, 42770}, {43035, 39063}, {43921, 244}, {43929, 649}, {43930, 3676}, {46135, 4572}, {46149, 38}, {46163, 46148}

• X₆₇₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 39341, 42079}, {2, 20172, 5263}, {2, 20533, 16593}, {2, 31058, 30857}, {2, 41845, 20533}, {11, 26007, 2}, {75, 20445, 20431}, {101, 17761, 24203}, {105, 13576, 14942}, {142, 16503, 86}, {239, 335, 32029}, {294, 1462, 1814}, {335, 32029, 24841}, {335, 32096, 239}, {379, 5222, 41245}, {1001, 2550, 5263}, {1111, 5540, 3732}, {2170, 9317, 664}, {2280, 30949, 14828}, {2481, 31638, 6559}, {4000, 5819, 7}, {4366, 26582, 5263}, {4384, 17738, 17755}, {10707, 30857, 31058}, {10707, 31226, 30857}, {14936, 43063, 24499}, {16706, 20179, 86}, {17023, 24588, 16054}, {17026, 24586, 14829}, {17259, 20181, 2345}, {17686, 26965, 1220}, {17691, 27304, 958}, {17738, 17755, 190}, {30857, 31226, 2}, {39341, 42079, 37138}, {40565, 40566, 14942}