[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point.
For P = X(355) [ = the midpoint of the orthocenter and the Nagel Point = the Fuhrmann circle center ] the triangles ABC, IOP share the same centroid G. (*)
[Angel Montesdeoca]¨
*** The locus of P such that the centroid of IOP lies on the Euler line of ABC is the line through X(40) parallel to Euler line.
The centroid of IOP is the projection of P from X(1385) on a Euler Line.
Pairs {P=X(i), Q=X(j)}, for {i, j}: {30,30}, {40,3}, {355,2}, {3652,21}, {3654,3524}, {3679,5054}, {5690,549}, {5691,381}, {7330,16418}, {12438,26451}, {16139,21161}.
Other pairs {P=X(i), Qi}:
Q191 = a (3 a^6-3 a^5 (b+c)+a^4 (-6 b^2+b c-6 c^2)-2 b c (b^2-c^2)^2+6 a^3 (b^3+c^3)+a^2 (3 b^4+b^3 c+6 b^2 c^2+b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :
= (6 r+7 R) X[3] + 2 R X[4]
= lies on these lines: {1,22937}, {2,3}, {36,4870}, {79,5204}, {191,1385}, {399,16164}, {517,5426}, {551,22765}, {758,4930}, {993,3655}, {999,5427}, {1125,16159}, {1482,4428}, {2771,3576}, {3579,3922}, {3584,5172}, {3612,17637}, {3647,13465}, {3648,5303}, {3654,11849}, {3656,5248}, {3679,12331}, {3683,13624}, {5096,10168}, {5217,5441}, {5251,18524}, {5453,16948}, {7701,7987}, {10543,10573}, {11263,16150}, {12645,21677}, {15178,16126}, {16118,17605}, {16143,26202}, {18253,18526}, {25055,26286}
= the midpoint of X(21) and X(21161)
= reflection of X(i) in X(j), for these {i, j}: {3,21161}, {5055,15671}, {21161,5428}
(6 - 9 - 13) - search numbers of Q191: (4.37889887160875, 3.49782730292207, -0.801938514704167).
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Q1709 = a (-3 a^6+3 a^5 (b+c)+4 b c (b^2-c^2)^2+a^4 (6 b^2-8 b c+6 c^2)-6 a^3 (b^3+c^3)+a^2 (-3 b^4+4 b^3 c-6 b^2 c^2+4 b c^3-3 c^4)+3 a (b^5-b^4 c-b c^4+c^5)) : :
= (3 r+R) X[3] + 2 R X[4]
= lies on these liens: {2,3}, {35,18518}, {55,18519}, {84,24299}, {355,4421}, {498,18542}, {519,10679}, {551,5450}, {958,3654}, {999,11551}, {1001,3653}, {1319,7284}, {1385,1709}, {1470,3582}, {1482,11260}, {1727,2099}, {1836,18493}, {2077,19875}, {3058,10949}, {3085,18545}, {3189,12645}, {3241,12000}, {3295,22759}, {3652,12635}, {3655,4428}, {3656,10680}, {3679,11248}, {3829,11928}, {3873,10247}, {3928,24474}, {4302,18499}, {4640,12702}, {4653,18451}, {4861,8148}, {4870,22766}, {5010,18491}, {5204,9955}, {5217,18480}, {5432,18516}, {6001,10179}, {6284,18544}, {6767,12735}, {7171,13151}, {8069,11237}, {8071,11238}, {10039,18525}, {10056,10058}, {10269,25055}, {12686,24927}, {15171,18543}, {15338,18517}, {15446,26437}, {24467,24473}
= midpoint of X(1012) and X(16370)
= reflection of X(i) in X(j), for these {i, j}: {3,16370}, {16370,6914}
(6 - 9 - 13) - search numbers of Q1709: (0.0180894385147809, -0.851478204778399, 4.22180042128568).
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Q1710 = a (-3 a^9+9 a^5 b (b-c)^2 c+3 a^8 (b+c)+2 b (b-c)^4 c (b+c)^3+a^7 (6 b^2-7 b c+6 c^2)+a^6 (-9 b^3+b^2 c+b c^2-9 c^3)+a (b^2-c^2)^2 (3 b^4-5 b^3 c+6 b^2 c^2-5 b c^3+3 c^4)-a^2 (b-c)^2 (3 b^5+3 b^4 c+10 b^3 c^2+10 b^2 c^3+3 b c^4+3 c^5)+a^4 (9 b^5-9 b^4 c+8 b^3 c^2+8 b^2 c^3-9 b c^4+9 c^5)+a^3 (-6 b^6+3 b^5 c+12 b^4 c^2-14 b^3 c^3+12 b^2 c^4+3 b c^5-6 c^6)) : :
= (9 r^2+20 r R+14 R^2-3 s^2) X[3] + 4 R (r+R) X[4]
= lies on these lines: {2,3}, {1385,1710}, {2217,3655}
(6 - 9 - 13) - search numbers of Q1710: (2.16278810494452, 1.28756269241637,1 .75106503102942)
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Q1762 = a (3 a^9-3 a^8 (b+c)-2 b (b-c)^4 c (b+c)^3+a^7 (-6 b^2+b c-6 c^2)+a^5 b c (3 b^2+16 b c+3 c^2)+a^6 (9 b^3+5 b^2 c+5 b c^2+9 c^3)-a (b^2-c^2)^2 (3 b^4-5 b^3 c+2 b^2 c^2-5 b c^3+3 c^4)+a^3 (b+c)^2 (6 b^4-21 b^3 c+22 b^2 c^2-21 b c^3+6 c^4)+a^2 (b-c)^2 (3 b^5+9 b^4 c+16 b^3 c^2+16 b^2 c^3+9 b c^4+3 c^5)-a^4 (9 b^5+3 b^4 c+8 b^3 c^2+8 b^2 c^3+3 b c^4+9 c^5)) : :
= (9 r^2+32 r R+28 R^2-3 s^2) X[3] + 4 R (r+2 R) X[4]
= lies on these lines: {2, 3}, {1385, 1762}, {11903, 12132}
(6 - 9 - 13) - search numbers of Q1762: (3.51501502392203, 2.63622240263833, 0.193272807501520).
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Q2100 = a (4 a^4 b c-2 b c (b^2-c^2)^2-2 a^2 b c (b^2+c^2)-3 a^3 Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]+3 a (b^2+c^2) Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]) : :
= (2R-3 OH) X[3] - 2 R X[4]
= lies on these lines: {2,3}, {182,15162}, {1385,2100}, {2575,15041}, {14500,20127}, {14810,15163}
(6 - 9 - 13) - search numbers of Q2100: (2.52150451259463, 1.64533279712562, 1.33781661654604).
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Q2101 = a (4 a^4 b c-2 b c (b^2-c^2)^2-2 a^2 b c (b^2+c^2)+3 a^3 Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]-3 a (b^2+c^2) Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]) : :
= (3 OH+2R) X[3] - 2R X[4]
= lies on these lines: {2,3}, {182,15163}, {1385,2101}, {2574,15041}, {14499,20127}, {14810,15162}
(6 - 9 - 13) - search numbers of Q2101: (11.0432212757981, 10.1445690563372, -8.47937006900201).
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Q2941 = a (-3 a^6-13 a^4 b c-3 a^5 (b+c)+2 b c (b^2-c^2)^2+3 a (b+c) (b^2+c^2)^2+a^2 (3 b^4+11 b^3 c+6 b^2 c^2+11 b c^3+3 c^4)) : :
= (3 r^2+r R-3 s^2) X[3] + 2 r R X[4]
= lies on these lines: {2,3}, {1385,2941}
(6 - 9 - 13) - search numbers of Q2941: (7.35887107283956, 6.46993826224272, -4.23492557941730).
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Q2960 = a (-3 a^9+7 a^6 b c (b+c)+2 b (b-c)^4 c (b+c)^3+12 a^5 b c (b^2-b c+c^2)+a^7 (6 b^2-7 b c+6 c^2)+a^2 b (b-c)^2 c (3 b^3-b^2 c-b c^2+3 c^3)-2 a^4 b c (6 b^3-b^2 c-b c^2+6 c^3)+a (b^2-c^2)^2 (3 b^4-2 b^3 c+6 b^2 c^2-2 b c^3+3 c^4)-a^3 (6 b^6+3 b^5 c-6 b^4 c^2+14 b^3 c^3-6 b^2 c^4+3 b c^5+6 c^6)) : :
= (3 r^2+7 r R+7 R^2-3 s^2) X[3] + 2 R (r+R) X[4]
= lies on these lines: {2,3}, {1385,2960}
(6 - 9 - 13) - search numbers of Q2960: (-52.4799208109804, -53.2109973776780, 64.7005491945986).
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Q3929 = a (-9 a^6+9 a^5 (b+c)+8 b c (b^2-c^2)^2+2 a^4 (9 b^2+b c+9 c^2)-18 a^3 (b^3+c^3)-a^2 (9 b^4+10 b^3 c+18 b^2 c^2+10 b c^3+9 c^4)+9 a (b^5-b^4 c-b c^4+c^5)) : :
= (9 r+14 R) X[3] + 4 R X[4]
= lies on these lines: {2,3}, {527,3653}, {1385,3929}, {3655,5325}
(6 - 9 - 13) - search numbers of Q3929: (4.29693931414626, 3.41608395684161, -0.707519479358086).
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Q6253 = -2 a^7+2 a^6 (b+c)-2 a^2 b (b-c)^2 c (b+c)+(b-c)^4 (b+c)^3-a (b-c)^2 (b+c)^4+4 a^3 b c (b^2-b c+c^2)+a^5 (3 b^2-2 b c+3 c^2)+a^4 (-3 b^3+b^2 c+b c^2-3 c^3) : :
= (R-r) X[3] - (r+2 R) X[4]
= lies on these lines: {2,3}, {11,18407}, {36,18406}, {56,18517}, {57,80}, {142,13151}, {355,529}, {386,13408}, {388,18518}, {495,18524}, {497,18499}, {515,5883}, {517,5891}, {519,24474}, {528,3656}, {542,4260}, {551,24299}, {553,18389}, {942,5434}, {952,3873}, {997,12699}, {1385,6253}, {1389,3241}, {1478,11502}, {1482,3189}, {1699,5840}, {1737,7354}, {2771,11246}, {3058,15950}, {3086,18544}, {3296,18526}, {3476,15934}, {3601,5443}, {3679,5709}, {3878,28194}, {4293,18519}, {4299,18761}, {4511,22791}, {5138,5476}, {5229,18542}, {5442,10483}, {5587,5841}, {5708,18391}, {5755,17330}, {5842,5886}, {6284,9955}, {6796,10197}, {7956,10738}, {7965,24466}, {9940,28208}, {9956,11827}, {10056,11501}, {10072,26475}, {10532,11239}, {10954,11237}, {11227,28160}, {11235,22753}, {11499,26332}, {11826,22793}, {12943,18516}, {14986,18543}, {15171,18493}
= midpoint of X(4) and X(17579)
= the reflection of X(i) in X(j), for these {i, j}: {7491,11113}, {11113,5}
(6 - 9 - 13) - search numbers of Q6253: (-2.47017667143147, -3.33318020589872, 7.08833231895956).
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Q7701 = a (3 a^6-3 a^5 (b+c)+a^4 (-6 b^2+5 b c-6 c^2)-4 b c (b^2-c^2)^2+6 a^3 (b^3+c^3)+a^2 (3 b^4-b^3 c+6 b^2 c^2-b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :
= (6 r+5 R) X[3] + 4 R X[4]
= lies on these lines: {1,13465}, {2,3}, {55,9897}, {191,1482}, {758,10247}, {993,3656}, {1001,18515}, {1385,7701}, {1621,12773}, {1749,2099}, {2771,5426}, {3647,8148}, {3653,5450}, {3655,5248}, {3679,11849}, {4265,11178}, {4421,5790}, {4428,22758}, {4995,10058}, {5204,16118}, {5901,14450}, {6690,10742}, {10543,12647}, {10902,28208}, {12409,25055}, {12702,22937}, {13089,26287}, {13624,16143}, {16132,26202}, {16159,18493}, {19875,26285}
= the reflection of X(i) in X(j), for these {i, j}: {5054,15672}, {10246,5426}
(6 - 9 - 13) - search numbers of Q7701: ( 1.97543484902115, 1.10070367911273, 1.96689969681965).
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Q8141 = a (-3 a^9+2 a^6 b c (b+c)-2 a^2 b^2 (b-c)^2 c^2 (b+c)+b (b-c)^4 c (b+c)^3+a^5 b c (3 b^2-8 b c+3 c^2)+a^7 (6 b^2-2 b c+6 c^2)+a^4 b c (-3 b^3+b^2 c+b c^2-3 c^3)+a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+a^3 (-6 b^6+4 b^4 c^2-4 b^3 c^3+4 b^2 c^4-6 c^6)) : :
= (3 r^2+11 r R+10 R^2-3 s^2) X[3] + R (r+2 R) X[4]
= lies on these lines: {2,3}, {517,11202}, {1385,8141}, {3654,5285}
(6 - 9 - 13) - search numbers of Q8141: (0.173505684107798, -0.696471951165465, 4.04275782466456).
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Q9572 = a (-6 a^9-22 a^5 b^2 c^2+a^6 b c (b+c)+2 a^4 b^2 c^2 (b+c)+2 b (b-c)^4 c (b+c)^3+a^7 (12 b^2-b c+12 c^2)-a^2 b (b-c)^2 c (3 b^3+7 b^2 c+7 b c^2+3 c^3)+2 a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)-a^3 (b+c)^2 (12 b^4-27 b^3 c+28 b^2 c^2-27 b c^3+12 c^4)) : :
= (6 r^2+25 r R+26 R^2-6 s^2) X[3] + 2 R (r+2 R) X[4]
= lies on these lines: {2,3}, {1385,9572}.
(6 - 9 - 13) - search numbers of Q9572: (3.10192984221503, 2.22422695007094, 0.669155127759102).
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Q9573 = a (6 a^9-7 a^6 b c (b+c)-2 b (b-c)^4 c (b+c)^3+a^7 (-12 b^2+7 b c-12 c^2)-2 a^5 b c (6 b^2-5 b c+6 c^2)-a^2 b (b-c)^2 c (3 b^3-b^2 c-b c^2+3 c^3)+2 a^4 b c (6 b^3-b^2 c-b c^2+6 c^3)-2 a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+a^3 (b+c)^2 (12 b^4-21 b^3 c+28 b^2 c^2-21 b c^3+12 c^4)) : :
= (6 r^2+19 r R+14 R^2-6 s^2) X[3] + 2 R (r+2 R) X[4]
= lies on these lines: {2,3}, {1385,9573}, {10251,12645}
(6 - 9 - 13) - search numbers of Q9573 : (-25.5622765242673, -26.3643625162548, 33.6908892351303).
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Q10251 = -4 a^10+3 a^7 b c (b+c)-(b^2-c^2)^4 (b^2+c^2)+a^8 (9 b^2-3 b c+9 c^2)-6 a^5 b c (b^3+c^3)-2 a^6 (b^4-3 b^3 c+5 b^2 c^2-3 b c^3+c^4)+2 a^2 (b^2-c^2)^2 (3 b^4+4 b^2 c^2+3 c^4)-a^4 (b+c)^2 (8 b^4-13 b^3 c+16 b^2 c^2-13 b c^3+8 c^4)+3 a^3 b c (b^5-b^4 c-b c^4+c^5) : :
= (5 r^2+17 r R+14 R^2-5 s^2) X[3] + ((r+2 R)^2-s^2) X[4]
= lies on these lines: {2,3}, {1385,10251}, {3656,18453}, {8251,25055}
(6 - 9 - 13) - search numbers of Q10251: (-3.38029401127337, -4.24089663321795, 8.13680554087682
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Q11826 = -2 a^7+2 a^6 (b+c)-2 a^2 b (b-c)^2 c (b+c)-a (b-c)^4 (b+c)^2+(b-c)^4 (b+c)^3+4 a^3 b c (2 b^2-b c+2 c^2)+a^5 (3 b^2-10 b c+3 c^2)+a^4 (-3 b^3+b^2 c+b c^2-3 c^3) : :
= (3 R-r) X[3] + r X[4]
= lies on these lines: {2,3}, {165,5841}, {517,5434}, {519,5884}, {528,3655}, {529,3654}, {553,24474}, {1385,3058}, {1768,3359}, {2077,3584}, {2550,18519}, {2829,26446}, {3474,12702}, {3576,5840}, {3579,7354}, {3820,10742}, {4299,22759}, {4304,13151}, {4316,7688}, {4413,18516}, {4430,5844}, {4995,26285}, {5298,15908}, {5790,14647}, {6284,7743}, {7080,18545}, {10056,11248}, {10072,10269}, {10106,16004}, {10246,15170}, {10247,11038}, {10270,19875}, {10310,11237}, {10385,16202}, {10525,11238}, {10950,13145}, {20292,22791}, {26200,28198}
= midpoint of X(i) and X(j), for these {i, j}: {376,17579}, {3058,11826}
= reflection of X(i) in X(j), for these {i, j}: {3058,1385}, {11113,549}, {24474,553}
(6 - 9 - 13) - search numbers of Q11826: (5.83581153198378, 4.95089659091247, -2.48033078810138).
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Q11827 = -2 a^7+2 a^6 (b+c)-2 a^2 b (b-c)^2 c (b+c)-a (b-c)^4 (b+c)^2+(b-c)^4 (b+c)^3-4 a^3 b c (b^2+b c+c^2)+a^5 (3 b^2+2 b c+3 c^2)+a^4 (-3 b^3+b^2 c+b c^2-3 c^3) : :
= (r+3 R) X[3] - r X[4]
= lies on these lines: {2,3}, {165,5840}, {226,13151}, {515,10176}, {517,3058}, {528,3654}, {529,3655}, {553,10202}, {580,3017}, {582,1834}, {912,17781}, {952,3681}, {997,18481}, {1385,5434}, {1482,15170}, {1708,5722}, {1737,3579}, {2550,18499}, {2551,18518}, {3336,16113}, {3428,11238}, {3475,10246}, {3576,5841}, {3582,11012}, {3583,7688}, {3584,10902}, {3586,3587}, {3820,18524}, {3925,18407}, {4302,11502}, {4654,18443}, {5178,5690}, {5298,26286}, {5506,5691}, {5584,10525}, {5758,15933}, {5842,26446}, {6253,9956}, {7354,13624}, {7742,10953}, {8148,15172}, {10056,10267}, {10072,11249}, {10157,28160}, {10268,19875}, {10385,10679}, {10526,11237}, {12702,15171}, {18544,19843}
= midpoint of X(i) and X(j), for these {i, j}: {376,11114}, {5434,11827}
= reflection of X(i) in X(j), for these {i, j}: {1482,15170}, {5434,1385}, {11112,549}
(6 - 9 - 13) - search numbers of Q11827: (7.72891425640891, 6.83900526255034, -4.66122266435458).
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Q16113 = 5 a^7-5 a^6 (b+c)+a (b-c)^4 (b+c)^2-(b-c)^4 (b+c)^3+a^5 (-9 b^2+b c-9 c^2)-a^2 (b-c)^2 (3 b^3+b^2 c+b c^2+3 c^3)+a^4 (9 b^3-b^2 c-b c^2+9 c^3)+a^3 (3 b^4+b^3 c+10 b^2 c^2+b c^3+3 c^4) : :
= (8 r+9 R) X[3] - 2 r X[4]
= lies on these lines: {2,3}, {79,4870}, {519,16139}, {758,3655}, {1385,16113}, {2193,3163}, {3579,5441}, {3612,18977}, {3647,18481}, {3652,4297}, {3653,16159}, {3683,26202}, {5427,10072}, {5655,16164}, {8148,10385}, {10543,12702}, {16140,21578}, {18253,18525}, {22937,28204}
= the midpoint of X(i) and X(j), for these {i, j}: {376,15677}, {3534,13743}, {3651,15678}
= reflection of X(i) in X(j), for these {i, j}: {2,5428}, {381,15670}, {3651,8703}, {3830,6841}, {3845,10021}, {5499,12100}, {5655,16164}, {6175,549}, {6841,15673}, {13743,17525}, {15679,5499}
(6 - 9 - 13) - search numbers of Q16113: (7.33016096202258, 6.44130388942299, -4.20185096247273).
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Q16138 = a (-3 a^6+3 a^5 (b+c)+5 b c (b^2-c^2)^2+a^4 (6 b^2-7 b c+6 c^2)-6 a^3 (b^3+c^3)+a^2 (-3 b^4+2 b^3 c-6 b^2 c^2+2 b c^3-3 c^4)+3 a (b^5-b^4 c-b c^4+c^5)),b (-3 a^4 b (b+c)-3 b (b-c)^3 (b+c)^2+a^5 (3 b+5 c)+6 a^2 (b^4-b^2 c^2)-2 a^3 (3 b^3-b^2 c+5 c^3)+a (3 b^5-7 b^4 c+2 b^2 c^3-3 b c^4+5 c^5)) : :
= 2 (3 r+2 R) X[3] + 5 R X[4]
= lies on these lines: {2,3}, {104,551}, {191,11531}, {758,16200}, {944,4428}, {999,16133}, {1385,16138}, {1482,19919}, {1621,3655}, {2077,3828}, {2975,3656}, {3241,22758}, {3584,10058}, {3822,10728}, {4861,11278}, {5303,9955}, {5441,10039}, {5450,25055}, {5603,11194}, {10308,16132}, {11281,16116}
= midpoint of X(21161) and X(21669)
= reflection of X(i) in X(j), for these {i, j}: {3651,21161}, {21161,21}
(6 - 9 - 13) - search numbers of Q16138: (0.773702837727355, -0.0978581327919375, 3.35131880258156).
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Q16309 = -10 a^10+4 a^9 (b+c)-(b-c)^4 (b+c)^6-2 a (b-c)^4 (b+c)^3 (b^2-b c+c^2)+a^8 (21 b^2-8 b c+21 c^2)-2 a^7 (5 b^3-b^2 c-b c^2+5 c^3)-2 a^6 (b^4-4 b^3 c+22 b^2 c^2-4 b c^3+c^4)+4 a^2 (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+2 a^3 (b-c)^2 (b^5+3 b^4 c-2 b^3 c^2-2 b^2 c^3+3 b c^4+c^5)+2 a^5 (3 b^5-6 b^4 c+5 b^3 c^2+5 b^2 c^3-6 b c^4+3 c^5)+a^4 (-20 b^6+6 b^5 c+28 b^4 c^2-20 b^3 c^3+28 b^2 c^4+6 b c^5-20 c^6) : :
= (6 r^2+21 r R+21 R^2-5 s^2) X[3] + (3 r R+6 R^2-s^2) X[4]
= lies on these lines: {2,3}, {1385,16309}
(6 - 9 - 13) - search numbers of Q16309: (3.37240655223218, 2.49399013569761, 0.357560594625028).
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Q19919 = a (-6 a^6+6 a^5 (b+c)+5 b c (b^2-c^2)^2+4 a^4 (3 b^2-b c+3 c^2)-12 a^3 (b^3+c^3)-a^2 (6 b^4+b^3 c+12 b^2 c^2+b c^3+6 c^4)+6 a (b^5-b^4 c-b c^4+c^5)) : :
= (12 r+13 R) X[3] + 5 R X[4]
= lies on these lines: {2,3}, {1385,19919}, {1397,10222}, {5426,16200}, {7987,16138}, {11278,22937}, {11531,16139}
(6 - 9 - 13) - search numbers of Q19919: (3.77803286596185, 2.89854639696973, -0.109728961823213).
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Q21375 = a (-3 a^6+2 b c (b^2-c^2)^2+2 a^2 b c (b^2+c^2)+a^4 (3 b^2-4 b c+3 c^2)-3 a^3 (b^3+c^3)+3 a (b^5+b^3 c^2+b^2 c^3+c^5)) : :
= (9 r^2+14 r R-3 s^2) X[3] + 4 r R X[4]
= lies on these lines: {2,3}, {1385,21375}, {3052,12702}
(6 - 9 - 13) - search numbers of Q21375: (8.63812473977162, 7.74581722570067, -5.70865116962607).
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Q21677 = -4 a^7+4 a^6 (b+c)-(b-c)^4 (b+c)^3+a (b-c)^2 (b+c)^4+a^5 (9 b^2+2 b c+9 c^2)+2 a^2 (b-c)^2 (3 b^3+4 b^2 c+4 b c^2+3 c^3)-a^4 (9 b^3+b^2 c+b c^2+9 c^3)-2 a^3 (3 b^4+2 b^3 c+4 b^2 c^2+2 b c^3+3 c^4) : :
= (5 r+7 R) X[3] + (r+2 R) X[4]
= lies on these lines: {2,3}, {57,5444}, {119,17009}, {214,5745}, {519,24299}, {551,24474}, {758,10165}, {942,5298}, {997,18253}, {1385,21677}, {1737,10543}, {2771,11227}, {3601,5445}, {3649,22937}, {3655,5791}, {4260,10168}, {4995,24929}, {5427,5432}, {5708,16137}, {5709,25055}, {6174,12619}, {6699,16164}, {10246,24477}, {11231,21155}, {11281,16139}, {15174,18391}
= midpoint of X(i) and X(j), for these {i, j}: {2,21161}, {3524,15671}
(6 - 9 - 13) - search numbers of Q21677: (4.10499983769563, 3.22465082157627, -0.486401396581795).
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Q26921 = a (3 a^6-3 a^5 (b+c)-2 b c (b^2-c^2)^2-2 a^4 (3 b^2+b c+3 c^2)+6 a^3 (b^3+c^3)+a^2 (3 b^4+4 b^3 c+6 b^2 c^2+4 b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :
= (3 r+5 R) X[3] + R X[4]
= lies on these lines: {2,3}, {36,4654}, {55,3654}, {56,3653}, {63,13151}, {517,4428}, {519,10267}, {527,10269}, {551,11249}, {582,19765}, {846,7986}, {912,3576}, {958,28204}, {971,17502}, {993,5325}, {997,13624}, {1385,11194}, {1480,8616}, {1708,24929}, {1737,5217}, {3189,5690}, {3241,16202}, {3428,3656}, {3601,10399}, {3679,10902}, {3715,12738}, {3828,6796}, {3873,10246}, {3927,4511}, {3928,18443}, {4995,8069}, {5010,11502}, {5248,28194}, {5260,18518}, {5298,8071}, {5434,7742}, {7330,7987}, {10072,26357}, {10202,21165}, {11012,25055}, {11496,28198}, {11499,19875}, {14831,22076}, {15931,22758}
= midpoint of X(3) and X(16418)
= reflection of X(3560) in X(16418)
(6 - 9 - 13) - search numbers of Q26921: (4.91829520915878, 4.03580069931406, -1.42333379107556).
Angel Montesdeoca
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