X₆₇₂

CROSSDIFFERENCE OF X₁ AND X₅₁₄

X₆₇₂ = X₂₃₂-of-excentral-triangle

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

X₆₇₂ (CROSSDIFFERENCE OF X₁ AND X₅₁₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 7}, {6, 31}

• Related Centers: X₂ (CENTROID), X₇ (GERGONNE POINT), X₆ (SYMMEDIAN POINT), X₃₁ (2nd POWER POINT)

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

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

Search Value (6, 9, 13): \(-5.422115840045\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a(as - S_{\omega }) : :\)

• Trilinears: \((\cot A + \cot B + \cot C) \csc A - (\csc A + \csc B + \csc C) \cot A : :\)

▸ Linear Combinations1 combination

• X₆₇₂ = X₆₇₂ = \(2\) X₃₆₉₃ - \(3\) X₁₄₄₃₉, X₃₉₃₀ - \(3\) X₁₄₄₃₉

X₆₇₂ lies on the cubics K135, K155, K225, K382, K385, K506, K981, K982, K1243 K1244, the curve Q090.

▸ Incident Central Lines424 lines

{1, 1002}, {2, 7}, {3, 41}, {6, 31}, {8, 3501}, {10, 3691}, {11, 17747}, {19, 1851}, {21, 41239}, {34, 41320}, {35, 4251}, {36, 101}, {37, 38}, {39, 213}, {40, 2082}, {43, 165}, {44, 513}, {45, 4860}, {46, 169}, {48, 2911}, {56, 220}, {58, 4476}, {59, 1252}, {65, 1212}, {72, 1009}, {75, 20632}, {100, 3684}, {103, 919}, {105, 238}, {106, 6017}, {109, 43079}, {141, 24690}, {145, 3208}, {163, 5127}, {171, 5276}, {172, 22065}, {187, 2251}, {190, 350}, {191, 29633}, {192, 17027}, {193, 22370}, {194, 17033}, {198, 1473}, {210, 44798}, {219, 604}, {237, 20777}, {239, 2111}, {241, 9502}, {244, 3290}, {284, 1174}, {292, 3009}, {294, 9441}, {325, 4766}, {344, 30962}, {346, 10453}, {377, 26036}, {484, 5011}, {497, 41325}, {516, 13576}, {517, 2170}, {518, 3693}, {519, 1018}, {570, 22058}, {572, 3449}, {594, 33162}, {595, 5299}, {602, 22131}, {607, 1593}, {609, 4257}, {610, 37262}, {612, 16517}, {614, 16970}, {660, 36906}, {663, 9320}, {665, 14411}, {726, 17031}, {728, 6762}, {750, 5275}, {758, 24036}, {799, 41535}, {800, 22071}, {857, 26012}, {901, 2291}, {942, 16601}, {956, 4390}, {960, 39244}, {966, 25631}, {970, 23637}, {971, 44424}, {982, 26242}, {992, 28242}, {993, 16788}, {1015, 1149}, {1023, 24496}, {1100, 3748}, {1107, 2295}, {1108, 17452}, {1125, 3294}, {1146, 40663}, {1190, 37499}, {1201, 2176}, {1208, 7124}, {1213, 32781}, {1269, 41681}, {1282, 1757}, {1319, 6603}, {1362, 1458}, {1376, 37658}, {1405, 15288}, {1434, 32008}, {1438, 8647}, {1449, 10389}, {1457, 4559}, {1468, 5021}, {1477, 6078}, {1479, 17732}, {1500, 20963}, {1509, 46194}, {1621, 16503}, {1642, 3675}, {1655, 41240}, {1713, 8804}, {1723, 1766}, {1724, 4456}, {1731, 16548}, {1732, 8557}, {1736, 11028}, {1737, 5179}, {1738, 20605}, {1739, 16611}, {1761, 29850}, {1765, 10445}, {1778, 4269}, {1783, 2202}, {1788, 6554}, {1802, 37579}, {1818, 3286}, {1876, 5089}, {1901, 3136}, {1903, 41509}, {1911, 38874}, {1931, 2311}, {1953, 21853}, {1967, 2107}, {2054, 9506}, {2092, 21753}, {2099, 34522}, {2106, 37128}, {2110, 2117}, {2112, 2115}, {2171, 5173}, {2174, 5124}, {2178, 38269}, {2195, 41934}, {2201, 35993}, {2212, 45786}, {2223, 2340}, {2262, 21866}, {2264, 7964}, {2271, 31448}, {2273, 5019}, {2278, 34879}, {2284, 34230}, {2287, 13588}, {2292, 40978}, {2297, 15479}, {2300, 7109}, {2301, 7688}, {2317, 4268}, {2318, 2352}, {2321, 17135}, {2325, 29824}, {2329, 2975}, {2333, 4185}, {2345, 31330}, {2356, 37908}, {2382, 6016}, {2384, 6014}, {2670, 21838}, {2702, 28471}, {2957, 5536}, {3002, 8776}, {3003, 22059}, {3006, 4071}, {3008, 18785}, {3010, 46177}, {3057, 21872}, {3061, 3869}, {3097, 16468}, {3124, 20461}, {3126, 38379}, {3160, 34497}, {3169, 20012}, {3177, 3212}, {3195, 45141}, {3207, 5204}, {3240, 4266}, {3247, 29814}, {3252, 23612}, {3263, 17755}, {3271, 20863}, {3419, 11355}, {3446, 7113}, {3474, 5819}, {3496, 33950}, {3508, 17794}, {3510, 9359}, {3555, 3991}, {3583, 5134}, {3588, 21877}, {3589, 25349}, {3621, 4050}, {3634, 46196}, {3660, 43947}, {3666, 21840}, {3670, 16600}, {3678, 17746}, {3682, 19762}, {3685, 24727}, {3686, 4651}, {3690, 40956}, {3709, 5098}, {3729, 4441}, {3731, 10980}, {3741, 17355}, {3747, 16782}, {3758, 37632}, {3760, 29455}, {3778, 24513}, {3780, 20691}, {3781, 37507}, {3789, 5220}, {3792, 24484}, {3812, 21921}, {3831, 27040}, {3868, 25082}, {3874, 3970}, {3912, 18206}, {3915, 14974}, {3924, 16968}, {3934, 4721}, {3938, 16973}, {3950, 42057}, {3973, 16569}, {3985, 4358}, {3986, 25501}, {4032, 25001}, {4051, 14923}, {4059, 6706}, {4095, 4696}, {4153, 30171}, {4192, 5755}, {4210, 7293}, {4258, 5217}, {4262, 5010}, {4271, 16669}, {4284, 16470}, {4300, 10822}, {4368, 20372}, {4370, 34590}, {4414, 36404}, {4426, 13733}, {4438, 30751}, {4447, 19593}, {4465, 20530}, {4473, 30967}, {4513, 12513}, {4517, 21010}, {4641, 22097}, {4642, 41015}, {4672, 40718}, {4700, 19998}, {4735, 4749}, {4759, 24491}, {4805, 7761}, {4848, 41006}, {4857, 41326}, {4859, 31200}, {4872, 24712}, {4875, 5836}, {4878, 15624}, {4919, 38460}, {5036, 22354}, {5044, 25068}, {5060, 5546}, {5080, 26074}, {5088, 9317}, {5096, 20780}, {5164, 20982}, {5222, 37555}, {5227, 33171}, {5230, 5286}, {5247, 23640}, {5248, 16783}, {5254, 21935}, {5283, 17750}, {5338, 44103}, {5537, 18771}, {5709, 36670}, {5782, 16405}, {5783, 11358}, {5838, 9778}, {6205, 21373}, {6244, 18766}, {6656, 24995}, {6734, 21029}, {6735, 21013}, {6763, 17744}, {6817, 15656}, {7075, 37683}, {7076, 37386}, {7131, 7183}, {7146, 24635}, {7175, 37659}, {7181, 17044}, {7193, 19554}, {7201, 24554}, {7741, 24045}, {7778, 30816}, {8609, 18839}, {8616, 16779}, {8731, 11018}, {9263, 10027}, {9334, 16676}, {9343, 16671}, {9367, 28271}, {9605, 16466}, {10030, 18031}, {10916, 21073}, {11227, 25075}, {11509, 32561}, {12782, 16476}, {12915, 21809}, {14096, 22449}, {14964, 23531}, {15254, 28600}, {15629, 36040}, {16567, 24892}, {16583, 24443}, {16604, 28352}, {16609, 30807}, {16667, 42042}, {16673, 30350}, {16784, 40091}, {16850, 31445}, {16885, 37673}, {16972, 17017}, {16980, 23630}, {17032, 17379}, {17034, 25264}, {17046, 33839}, {17053, 28360}, {17081, 26658}, {17090, 20089}, {17120, 40721}, {17122, 37675}, {17137, 27109}, {17152, 30038}, {17165, 21101}, {17264, 41851}, {17275, 33074}, {17278, 31199}, {17279, 24691}, {17280, 31027}, {17281, 31136}, {17289, 30966}, {17303, 25611}, {17336, 30963}, {17339, 31028}, {17349, 19591}, {17351, 21264}, {17369, 30970}, {17443, 21863}, {17444, 21864}, {17499, 27020}, {17603, 30944}, {17751, 26770}, {17752, 21226}, {19346, 37504}, {20228, 22199}, {20244, 20257}, {20247, 25237}, {20457, 20669}, {20593, 20718}, {20594, 20719}, {20666, 20668}, {20769, 21495}, {20778, 20860}, {20785, 46148}, {20888, 29433}, {21078, 25078}, {21139, 43037}, {21232, 30806}, {21281, 30036}, {21369, 24259}, {21477, 23151}, {21760, 23579}, {21956, 33136}, {22079, 42447}, {23569, 23656}, {23622, 23988}, {24170, 33792}, {24214, 26978}, {24266, 28916}, {24549, 33819}, {24631, 26234}, {25427, 29578}, {26006, 43054}, {26043, 27091}, {26107, 27158}, {26244, 32918}, {27622, 28246}, {27919, 42720}, {28244, 28269}, {28535, 28875}, {30965, 33157}, {30969, 33115}, {31036, 41233}, {31508, 42043}, {33105, 37661}, {33888, 33889}, {37195, 40957}, {37548, 39247}, {39798, 40147}

▸ Functional Relations26 relations

• X₆₇₂ = midpoint of X₁₀₁₈ and X₄₅₇₅₁

• X₆₇₂ = reflection of X_i in X_j for these {i,j}: {2170, 43065}, {3930, 3693}, {20347, 20335}, {21139, 43037}, {30806, 21232}

• X₆₇₂ = isogonal conjugate of X₆₇₃

• X₆₇₂ = isotomic conjugate of X₁₈₀₃₁

• X₆₇₂ = complement of X₂₀₃₄₇

• X₆₇₂ = anticomplement of X₂₀₃₃₅

• X₆₇₂ = X_i-Hirst inverse of X_j for these (i,j): (6, 55), (1362, 1458)

• X₆₇₂ = X₂₃₂-of-excentral-triangle

• X₆₇₂ = trilinear pole of line X₆₆₅X₉₂₆

• X₆₇₂ = crossdifference of PU(i) for i in (47, 51)

• X₆₇₂ = bicentric sum of PU(49)

• X₆₇₂ = PU(49)-harmonic conjugate of X₆₆₃

• X₆₇₂ = perspector of conic {A, B, C,X₁,X₁₀₁,PU(93)}

• X₆₇₂ = perspector of 4th mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle

• X₆₇₂ = X₃₁-complementary conjugate of X₃₉₀₄₆

• X₆₇₂ = X_i-Ceva conjugate of X_j for these (i,j): {1, 42079}, {2, 39046}, {9, 9502}, {19, 17464}, {103, 55}, {238, 3009}, {241, 1458}, {291, 42}, {292, 3252}, {518, 2340}, {673, 1}, {813, 649}, {1024, 17439}, {1025, 2254}, {1861, 2356}, {2195, 9310}, {2283, 926}, {2284, 665}, {3286, 2223}, {3912, 1818}, {7096, 20589}, {9442, 2293}, {18206, 518}, {34085, 4449}, {36039, 652}, {36086, 663}, {39293, 46177}, {39979, 1193}, {43672, 14547}

• X₆₇₂ = X_i-cross conjugate of X_j for these (i,j): {665, 2284}, {926, 2283}, {2223, 1458}, {6184, 3252}, {9454, 2356}, {20662, 6}, {20683, 518}, {39258, 2223}, {42079, 1}

• X₆₇₂ = cevapoint of X_i and X_j for these (i,j): {1, 39341}, {6, 20672}, {9, 24578}, {238, 9441}, {17435, 24290}, {20683, 39258}

• X₆₇₂ = crosspoint of X_i and X_j for these (i,j): {1, 673}, {6, 292}, {9, 2338}, {57, 1477}, {190, 5378}, {241, 518}, {1252, 6078}, {1861, 3912}, {2195, 9439}, {3286, 18206}, {4564, 36086}, {4570, 4584}

• X₆₇₂ = crosssum 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}

• X₆₇₂ = trilinear pole of line {665, 926}

• X₆₇₂ = crossdifference of every pair of points on line {1, 514}

• X₆₇₂ = X_i-isoconjugate of X_j for these (i,j): {1, 673}, {2, 105}, {4, 1814}, {6, 2481}, {7, 294}, {8, 1462}, {19, 31637}, {31, 18031}, {55, 34018}, {56, 36796}, {57, 14942}, {63, 36124}, {69, 8751}, {75, 1438}, {81, 13576}, {83, 46149}, {85, 2195}, {86, 18785}, {92, 36057}, {190, 1027}, {264, 32658}, {269, 6559}, {279, 28071}, {291, 6654}, {312, 1416}, {513, 666}, {514, 36086}, {516, 9503}, {518, 6185}, {522, 36146}, {644, 43930}, {648, 10099}, {650, 927}, {651, 885}, {653, 23696}, {663, 34085}, {664, 1024}, {667, 36803}, {668, 43929}, {693, 919}, {789, 29956}, {884, 4554}, {934, 28132}, {1016, 43921}, {1086, 5377}, {1292, 2402}, {1429, 33676}, {2111, 33674}, {2170, 39293}, {2191, 31638}, {2721, 38895}, {2991, 14267}, {3063, 46135}, {3261, 32666}, {3263, 41934}, {3512, 40724}, {3669, 36802}, {4391, 32735}, {4468, 36041}, {6084, 39272}, {6169, 9312}, {7261, 40754}, {10566, 35333}, {14727, 20980}, {26546, 35185}, {36816, 37129}

• X₆₇₂ = barycentric product X_i*X_j for these {i,j}: {1, 518}, {3, 1861}, {4, 1818}, {6, 3912}, {7, 2340}, {8, 1458}, {9, 241}, {10, 3286}, {19, 25083}, {31, 3263}, {37, 18206}, {41, 40704}, {42, 30941}, {48, 46108}, {55, 9436}, {56, 3717}, {57, 3693}, {58, 3932}, {63, 5089}, {69, 2356}, {71, 15149}, {75, 2223}, {76, 9454}, {78, 1876}, {81, 3930}, {86, 20683}, {88, 14439}, {92, 20752}, {100, 2254}, {101, 918}, {105, 4712}, {110, 4088}, {190, 665}, {200, 34855}, {213, 18157}, {219, 5236}, {238, 22116}, {239, 3252}, {256, 4447}, {274, 39258}, {291, 8299}, {292, 17755}, {307, 37908}, {350, 40730}, {513, 1026}, {514, 2284}, {517, 36819}, {519, 34230}, {522, 2283}, {561, 9455}, {649, 42720}, {650, 1025}, {656, 4238}, {662, 24290}, {663, 883}, {664, 926}, {673, 6184}, {765, 3675}, {1126, 4966}, {1156, 35293}, {1293, 4925}, {1362, 14942}, {1438, 4437}, {1477, 40609}, {1642, 37131}, {1738, 34159}, {1914, 40217}, {2334, 4684}, {2338, 39063}, {2428, 4468}, {2481, 42079}, {2991, 17464}, {3126, 36086}, {3445, 4899}, {72) = 3509, 40781}, {3616, 14626}, {3900, 41353}, {3939, 43042}, {4557, 23829}, {4564, 17435}, {4572, 8638}, {4876, 34253}, {5378, 38989}, {7077, 39775}, {9315, 40883}, {9502, 36101}, {16728, 18785}, {18031, 39686}, {20662, 36807}, {31637, 42071}, {34337, 36057}, {34905, 40865}, {36037, 42758}, {39252, 40788}

• X₆₇₂ = barycentric quotient X_i/X_j for these {i,j}: {1, 2481}, {2, 18031}, {3, 31637}, {6, 673}, {9, 36796}, {25, 36124}, {31, 105}, {32, 1438}, {41, 294}, {42, 13576}, {48, 1814}, {55, 14942}, {57, 34018}, {59, 39293}, {101, 666}, {109, 927}, {184, 36057}, {190, 36803}, {213, 18785}, {218, 31638}, {220, 6559}, {241, 85}, {518, 75}, {604, 1462}, {651, 34085}, {657, 28132}, {663, 885}, {664, 46135}, {665, 514}, {667, 1027}, {692, 36086}, {810, 10099}, {883, 4572}, {911, 9503}, {918, 3261}, {926, 522}, {1025, 4554}, {1026, 668}, {1110, 5377}, {1253, 28071}, {1362, 9436}, {1397, 1416}, {1415, 36146}, {1438, 6185}, {1458, 7}, {1818, 69}, {1861, 264}, {1876, 273}, {1914, 6654}, {1919, 43929}, {1946, 23696}, {1964, 46149}, {1973, 8751}, {2110, 33674}, {2175, 2195}, {2223, 1}, {2254, 693}, {2283, 664}, {2284, 190}, {2340, 8}, {2356, 4}, {2428, 37206}, {3063, 1024}, {3230, 36816}, {3248, 43921}, {3252, 335}, {3263, 561}, {3286, 86}, {3675, 1111}, {3693, 312}, {3717, 3596}, {3912, 76}, {3930, 321}, {3932, 313}, {3939, 36802}, {4088, 850}, {4238, 811}, {4447, 1909}, {4712, 3263}, {4966, 1269}, {5089, 92}, {5236, 331}, {6184, 3912}, {7077, 33676}, {8299, 350}, {8638, 663}, {9247, 32658}, {9436, 6063}, {9454, 6}, {9455, 31}, {9502, 30807}, {14439, 4358}, {14626, 5936}, {15149, 44129}, {16728, 18157}, {17435, 4858}, {17755, 1921}, {17798, 40724}, {18157, 6385}, {18206, 274}, {19554, 40754}, {20455, 1738}, {20662, 3008}, {20683, 10}, {20752, 63}, {20776, 1818}, {22116, 334}, {23225, 1459}, {23612, 4712}, {24290, 1577}, {25083, 304}, {25302, 34086}, {30941, 310}, {32739, 919}, {34230, 903}, {34253, 10030}, {34855, 1088}, {35293, 30806}, {36819, 18816}, {37908, 29}, {39046, 20347}, {39258, 37}, {39341, 33675}, {39775, 18033}, {40217, 18895}, {40704, 20567}, {40730, 291}, {40781, 40845}, {41353, 4569}, {42071, 1861}, {42079, 518}, {42341, 20907}, {42720, 1978}, {42758, 36038}, {42771, 35015}, {43924, 43930}, {46108, 1969}

• X₆₇₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 1475, 17474}, {1, 3730, 1334}, {1, 4253, 1475}, {2, 7, 30949}, {2, 144, 30946}, {2, 329, 30961}, {2, 5905, 30985}, {2, 6646, 31004}, {2, 17350, 24514}, {2, 20347, 20335}, {3, 218, 41}, {6, 31, 21764}, {6, 55, 2280}, {6, 71, 2269}, {6, 2276, 42}, {6, 17735, 1914}, {6, 21793, 5332}, {6, 42316, 55}, {7, 27626, 28351}, {9, 57, 40131}, {9, 63, 5282}, {9, 579, 1400}, {9, 17754, 2}, {10, 16552, 3691}, {35, 17745, 4251}, {36, 101, 1055}, {36, 5526, 101}, {37, 583, 2260}, {37, 24512, 3720}, {39, 213, 1193}, {40, 16572, 2082}, {42, 35270, 55}, {43, 1743, 37657}, {44, 910, 2348}, {44, 1155, 2246}, {44, 1575, 2238}, {44, 2245, 2183}, {44, 20331, 899}, {55, 42316, 41423}, {56, 220, 9310}, {57, 1423, 3598}, {63, 36276, 9318}, {65, 1212, 17451}, {72, 25066, 33299}, {75, 20646, 20632}, {101, 5030, 36}, {144, 27624, 1423}, {190, 37686, 350}, {219, 5120, 604}, {220, 5022, 56}, {292, 16514, 3009}, {329, 27659, 28387}, {484, 5540, 5011}, {573, 1743, 2347}, {583, 24512, 2350}, {910, 2348, 2246}, {942, 16601, 21808}, {1015, 3230, 1149}, {1025, 9436, 6168}, {1107, 2295, 10459}, {1108, 21871, 17452}, {1155, 2348, 910}, {1201, 23649, 2275}, {1202, 2269, 2280}, {1319, 6603, 17439}, {1334, 1475, 1}, {1334, 4253, 17474}, {1400, 8012, 40131}, {1447, 10025, 9318}, {1575, 2238, 899}, {1723, 1766, 40968}, {1737, 5179, 21044}, {1757, 2108, 3783}, {1783, 37305, 2202}, {1914, 17735, 902}, {2066, 5414, 1253}, {2174, 5124, 22054}, {2176, 2275, 1201}, {2183, 2272, 910}, {2223, 20683, 2340}, {2238, 20331, 1575}, {2280, 41423, 55}, {2323, 5053, 1404}, {2350, 40586, 3720}, {2590, 2591, 2246}, {2911, 36743, 48}, {3056, 36635, 20978}, {3501, 21384, 8}, {3729, 17026, 4441}, {3730, 4253, 1}, {3779, 20992, 2293}, {3869, 26690, 3061}, {3930, 14439, 3693}, {4251, 24047, 35}, {4438, 30953, 30751}, {4559, 43039, 1457}, {5030, 5526, 1055}, {5905, 27661, 28388}, {6203, 6204, 1445}, {6646, 27678, 28402}, {7113, 17796, 22356}, {14974, 16502, 3915}, {16549, 16552, 10}, {17137, 27109, 29960}, {17279, 24691, 30945}, {17279, 30945, 30821}, {17351, 21264, 24330}, {17756, 37657, 43}, {19037, 19038, 38293}, {21477, 23151, 25940}, {21872, 40133, 3057}, {24578, 39252, 239}, {27626, 44421, 7}