X(45186) Complete Combos [by Peter Moses]

X(45186) = X(3)X(51)∩X(4)X(69)

Barycentrics    a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-6 a^4 b^2 c^2+a^2 b^4 c^2+4 b^6 c^2-3 a^4 c^4+a^2 b^2 c^4-6 b^4 c^4+3 a^2 c^6+4 b^2 c^6-c^8) : :
Barycentrics    (SB+SC) (S^2-4 R^2 SA+SB SC) : :
X(45186) = 3*X(2)-4*X(10110),2*X(3)-3*X(51),3*X(3)-4*X(5462),5*X(3)-6*X(5892),4*X(3)-3*X(36987),3*X(4)-2*X(5907),3*X(4)-X(11412),5*X(4)-3*X(11459),4*X(4)-3*X(15030),7*X(4)-5*X(15058),4*X(5)-3*X(3917),3*X(5)-2*X(10627),5*X(5)-4*X(32142),X(20)-3*X(3060),3*X(20)-5*X(10574),3*X(51)-4*X(5446),9*X(51)-8*X(5462),5*X(51)-4*X(5892),3*X(52)-2*X(6102),3*X(52)-X(10575),5*X(52)-2*X(13491),3*X(52)-4*X(14449),4*X(52)-3*X(14831),2*X(52)-3*X(21969),X(64)-3*X(34751),8*X(140)-9*X(373),4*X(143)-3*X(9730),3*X(185)-4*X(6102),X(185)-4*X(10263),3*X(185)-2*X(10575),5*X(185)-4*X(13491),3*X(185)-8*X(14449),2*X(185)-3*X(14831),X(185)-3*X(21969),3*X(376)-5*X(3567),3*X(376)-4*X(9729),2*X(376)-3*X(16226),3*X(381)-2*X(1216),3*X(381)-X(37484),2*X(382)+X(14531),3*X(382)-X(18439),2*X(389)-3*X(3060),6*X(389)-5*X(10574),4*X(546)-3*X(5891),2*X(548)-3*X(5946),3*X(549)-4*X(10095),2*X(550)-3*X(9730),3*X(568)-X(1657),3*X(568)-2*X(40647),4*X(576)-3*X(40673),5*X(631)-6*X(5943),5*X(631)-7*X(9781),5*X(631)-4*X(13348),5*X(632)-6*X(13364),4*X(1112)-3*X(16223),3*X(1351)-2*X(32284),5*X(1656)-4*X(5447),10*X(1656)-9*X(5650),5*X(1656)-8*X(12002),5*X(1656)-3*X(13340),5*X(1843)-4*X(41714),3*X(2979)-5*X(3091),3*X(2979)-4*X(11793),9*X(3060)-5*X(10574),7*X(3090)-6*X(3819),5*X(3091)-4*X(11793),5*X(3522)-9*X(11002),5*X(3522)-7*X(15043),5*X(3522)-6*X(16836),7*X(3523)-9*X(5640),7*X(3523)-8*X(11695),7*X(3528)-9*X(15045),7*X(3528)-8*X(17704),X(3529)-3*X(5890),X(3529)-4*X(16625),2*X(3530)-3*X(13451),4*X(3530)-5*X(15026),3*X(3534)-5*X(37481),3*X(3543)-X(12111),3*X(3543)-2*X(13474),9*X(3545)-7*X(7999),5*X(3567)-4*X(9729),10*X(3567)-9*X(16226),5*X(3567)-6*X(21849),3*X(3574)-2*X(12363),4*X(3627)-3*X(32062),3*X(3627)-2*X(32137),8*X(3628)-9*X(14845),3*X(3830)-X(18436),7*X(3832)-5*X(11444),7*X(3832)-4*X(15606),9*X(3839)-7*X(15056),5*X(3843)-3*X(23039),3*X(3845)-2*X(11591),4*X(3850)-3*X(15067),7*X(3851)-6*X(10170),4*X(3853)-3*X(16194),5*X(3858)-4*X(14128),4*X(3861)-3*X(15060),3*X(3917)-2*X(10625),9*X(3917)-8*X(10627),X(5059)-4*X(13382),X(5059)-3*X(15072),X(5059)-9*X(16981),5*X(5076)-3*X(18435),3*X(5102)-2*X(32366),3*X(5446)-2*X(5462),5*X(5446)-3*X(5892),8*X(5446)-3*X(36987),8*X(5447)-9*X(5650),4*X(5447)-3*X(13340),10*X(5462)-9*X(5892),3*X(5562)-4*X(5907),3*X(5562)-2*X(11412),5*X(5562)-6*X(11459),X(5562)-4*X(13598),2*X(5562)-3*X(15030),7*X(5562)-10*X(15058),9*X(5640)-8*X(11695),3*X(5650)-2*X(13340),3*X(5657)-4*X(23841),2*X(5876)-3*X(16194),3*X(5890)-4*X(16625),3*X(5891)-2*X(6101),8*X(5892)-5*X(36987),10*X(5907)-9*X(11459),X(5907)-3*X(13598),8*X(5907)-9*X(15030),6*X(5943)-7*X(9781),3*X(5943)-2*X(13348),X(6102)-3*X(10263),5*X(6102)-3*X(13491),8*X(6102)-9*X(14831),4*X(6102)-9*X(21969),2*X(6243)+X(11381),3*X(6243)+X(18439),3*X(6467)-4*X(32284),4*X(6759)-3*X(34750),3*X(8703)-4*X(12006),8*X(9729)-9*X(16226),2*X(9729)-3*X(21849),7*X(9781)-4*X(13348),4*X(9822)-3*X(10519),6*X(10263)-X(10575),5*X(10263)-X(13491),3*X(10263)-2*X(14449),8*X(10263)-3*X(14831),4*X(10263)-3*X(21969),5*X(10575)-6*X(13491),X(10575)-4*X(14449),4*X(10575)-9*X(14831),2*X(10575)-9*X(21969),3*X(10625)-4*X(10627),5*X(10625)-8*X(32142),5*X(10627)-6*X(32142),9*X(11002)-7*X(15043),3*X(11002)-2*X(16836),8*X(11017)-9*X(23046),3*X(11381)-2*X(18439),5*X(11412)-9*X(11459),X(11412)-6*X(13598),4*X(11412)-9*X(15030),5*X(11444)-4*X(15606),3*X(11459)-10*X(13598),4*X(11459)-5*X(15030),9*X(11539)-8*X(11592),2*X(11574)-3*X(14853),4*X(11746)-3*X(38727),8*X(12002)-3*X(13340),6*X(12099)-5*X(38729),2*X(12162)-3*X(32062),3*X(12162)-4*X(32137),X(12290)-3*X(15682),2*X(12362)-3*X(16657),4*X(12811)-3*X(44324),9*X(13321)-5*X(15696),6*X(13363)-5*X(15712),4*X(13382)-3*X(15072),4*X(13382)-9*X(16981),2*X(13419)-3*X(34603),4*X(13446)-3*X(37943),6*X(13451)-5*X(15026),3*X(13491)-10*X(14449),8*X(13598)-3*X(15030),4*X(13630)-3*X(14855),8*X(14449)-9*X(21969),X(14516)-3*X(34603),3*X(14531)+2*X(18439),2*X(14689)-3*X(16225),3*X(14855)-2*X(15704),3*X(14855)-8*X(16982),7*X(14869)-8*X(32205),8*X(15012)-5*X(17538),3*X(15035)-4*X(41671),7*X(15043)-6*X(16836),9*X(15045)-8*X(17704),X(15072)-3*X(16981),3*X(15305)-5*X(17578),X(15704)-4*X(16982),3*X(15800)-X(22815),2*X(16163)-3*X(16223),3*X(16222)-2*X(38726),3*X(16226)-4*X(21849),X(16659)-3*X(34613),2*X(17712)-3*X(43573),3*X(20423)-2*X(44479),2*X(22815)-3*X(43581),4*X(30531)-3*X(44325),9*X(32062)-8*X(32137),2*X(34782)-3*X(41580),3*X(36518)-2*X(41673)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2736.

X(45186) lies on these lines: {2,10110}, {3,51}, {4,69}, {5,3917}, {6,10984}, {20,389}, {22,578}, {23,10282}, {24,13346}, {25,1092}, {26,13352}, {30,52}, {49,5899}, {54,12088}, {64,34751}, {68,11550}, {74,11800}, {110,11807}, {125,23335}, {140,373}, {143,550}, {155,18534}, {181,601}, {182,10323}, {184,7387}, {186,15010}, {193,34781}, {195,37924}, {235,1568}, {343,1595}, {376,3567}, {381,1216}, {382,6243}, {394,1598}, {500,7420}, {517,16980}, {524,16621}, {542,16659}, {546,5891}, {548,5946}, {549,10095}, {567,13564}, {568,1657}, {569,22352}, {576,7592}, {602,3271}, {631,5943}, {632,13364}, {674,14872}, {970,6906}, {1012,5752}, {1064,10475}, {1112,16163}, {1147,1495}, {1154,3627}, {1181,1351}, {1199,5097}, {1204,12085}, {1350,7395}, {1370,39571}, {1498,2393}, {1519,35631}, {1532,37536}, {1533,2883}, {1593,17834}, {1614,34986}, {1656,5447}, {1899,34938}, {1907,11576}, {1941,8884}, {1986,16879}, {1993,6759}, {1994,12087}, {2070,12038}, {2777,21649}, {2781,41362}, {2794,39817}, {2807,41869}, {2937,18475}, {2979,3091}, {3090,3819}, {3098,7509}, {3133,23217}, {3146,5889}, {3149,37482}, {3199,3289}, {3292,7530}, {3313,5480}, {3448,14864}, {3515,37497}, {3517,35602}, {3518,43574}, {3522,11002}, {3523,5640}, {3524,15024}, {3525,6688}, {3528,15045}, {3529,5890}, {3530,13451}, {3534,37481}, {3538,18928}, {3543,12111}, {3545,7999}, {3560,22076}, {3564,16655}, {3574,12363}, {3580,20299}, {3628,14845}, {3796,11426}, {3830,18436}, {3832,11444}, {3839,15056}, {3843,23039}, {3845,11591}, {3850,15067}, {3851,10170}, {3853,5876}, {3855,13570}, {3858,14128}, {3861,15060}, {3937,37532}, {4297,31757}, {5012,37505}, {5056,7998}, {5059,13382}, {5068,33884}, {5073,14915}, {5076,18435}, {5092,43651}, {5102,32366}, {5188,27375}, {5198,17814}, {5422,37515}, {5448,11799}, {5651,7529}, {5657,23841}, {5663,13421}, {5706,37516}, {5709,26892}, {5751,9122}, {5925,31978}, {5965,16658}, {6153,12307}, {6193,31383}, {6241,33703}, {6515,14216}, {6636,13434}, {6642,34417}, {6823,9967}, {6834,37521}, {6907,18180}, {6950,15489}, {6979,33852}, {7330,26893}, {7383,14561}, {7391,18381}, {7400,11574}, {7404,43653}, {7409,33523}, {7488,11430}, {7500,9833}, {7506,44106}, {7512,15033}, {7525,37513}, {7526,37478}, {7527,7691}, {7555,10610}, {8276,9686}, {8541,44492}, {8703,12006}, {8718,15032}, {9019,12233}, {9306,10594}, {9545,26881}, {9715,11425}, {9777,37198}, {9786,21312}, {9818,37486}, {9820,37971}, {9822,10519}, {9909,19357}, {9914,34777}, {9927,11572}, {9973,15811}, {10018,32223}, {10112,29012}, {10303,11451}, {10531,35645}, {10540,41597}, {10564,37814}, {10619,43595}, {10628,10733}, {11017,23046}, {11064,21841}, {11250,32110}, {11387,14826}, {11413,11438}, {11456,37517}, {11479,33878}, {11539,11592}, {11557,12121}, {11692,18859}, {11746,38727}, {11806,20127}, {11826,22300}, {12022,29317}, {12058,18390}, {12083,13366}, {12084,21663}, {12099,38729}, {12102,31834}, {12103,16881}, {12107,43394}, {12163,12235}, {12164,34382}, {12225,13403}, {12236,16111}, {12237,12256}, {12238,12257}, {12239,42258}, {12240,42259}, {12241,19161}, {12280,13433}, {12290,15682}, {12295,21650}, {12359,41586}, {12362,16657}, {12811,44324}, {12897,18563}, {13321,15696}, {13358,14677}, {13363,15712}, {13417,17702}, {13419,14516}, {13446,37943}, {13621,37496}, {13630,14855}, {13851,18569}, {14110,42450}, {14520,36012}, {14627,44111}, {14641,17800}, {14689,16225}, {14788,19130}, {14869,32205}, {14984,15063}, {15004,35243}, {15012,17538}, {15028,15717}, {15035,41671}, {15038,37471}, {15073,34621}, {15305,17578}, {15472,22109}, {15559,21243}, {15705,40284}, {15800,22815}, {15801,43605}, {16197,37649}, {16222,38726}, {16386,32411}, {16654,34380}, {16661,34545}, {17080,34956}, {17712,43573}, {17928,37480}, {18281,32225}, {18378,22115}, {18483,31738}, {18925,34608}, {19124,37488}, {19467,31305}, {19925,31737}, {20423,44479}, {22416,33843}, {23154,24474}, {23698,39846}, {25739,43896}, {26882,32237}, {28150,31728}, {28164,31732}, {30531,44325}, {31074,32767}, {31304,34785}, {31730,31760}, {31810,44544}, {31817,31871}, {32171,37936}, {34565,36753}, {34782,41580}, {36518,41673}, {36742,40952}, {36978,42147}, {36979,42432}, {36980,42148}, {36981,42431}, {37200,41365}, {37406,39271}, {37437,41723}, {37474,40954}, {38281,44436}, {38738,39835}, {38749,39806}, {44076,44407}, {44102,44469}

X(45186) = midpoint of X(i) and X(j) for these {i,j}: {382,6243}, {3146,5889}, {5073,34783}, {6241,33703}, {11381,14531}
X(45186) = reflection of X(i) in X(j) for these (i,j): (3,5446), (4,13598), (20,389), (52,10263), (74,11800), (110,11807), (185,52), (376,21849), (550,143), (1350,9969), (1657,40647), (3313,5480), (4297,31757), (5188,27375), (5447,12002), (5562,4), (5876,3853), (5925,31978), (6101,546), (6102,14449), (6146,13142), (6467,1351), (7691,11808), (9967,21850), (10575,6102), (10625,5), (11381,382), (11412,5907), (11750,12370), (11826,22300), (12058,18390), (12103,16881), (12111,13474), (12121,11557), (12162,3627), (12163,12235), (12225,13403), (12256,12237), (12257,12238), (12280,13433), (12294,31670), (12307,6153), (13630,16982), (14110,42450), (14516,13419), (14531,6243), (14677,13358), (14831,21969), (15063,16105), (15644,10110), (15704,13630), (16111,12236), (16163,1112), (16386,32411), (17800,14641), (18563,12897), (18859,11692), (20127,11806), (21650,12295), (23154,24474), (31730,31760), (31737,19925), (31738,18483), (31817,31871), (31834,12102), (34224,10112), (36987,51), (37484,1216), (38738,39835), (38749,39806), (43581,15800)
X(45186) = anticomplement of X(15644)
X(45186) = crosssum of X(3)and X(11411)
X(45186) = barycentric product X(343)*X(19173)
X(45186) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3527,10601),(3,5446,51),(3,44413,11424),(4,5562,15030),(4,11412,5907),(5,10625,3917),(6,11414,10984),(20,3060,389),(23,34148,10282),(25,37498,1092),(26,13352,13367),(52,185,14831),(52,10263,21969),(52,10575,6102),(143,550,9730),(155,18534,26883),(185,21969,52),(376,3567,9729),(376,21849,16226),(381,37484,1216),(546,6101,5891),(568,1657,40647),(631,9781,5943),(1112,16163,16223),(1147,7517,1495),(1351,39568,1181),(1498,11477,12160),(1656,5447,5650),(1656,13340,5447),(1993,6759,43844),(2070,37495,12038),(2937,37472,18475),(2979,3091,11793),(3522,11002,15043),(3522,15043,16836),(3523,5640,11695),(3528,15045,17704),(3530,13451,15026),(3543,12111,13474),(3567,9729,16226),(3627,12162,32062),(3853,5876,16194),(5650,27355,1656),(5907,11412,5562),(5943,13348,631),(6102,10263,14449),(6102,10575,185),(6102,14449,52),(6642,37483,43652),(7387,36747,184),(7530,16266,10539),(9729,21849,3567),(9777,37198,37514),(9927,31723,11572),(10110,15644,2),(10539,16266,3292),(12002,13340,27355),(12085,37489,1204),(13630,15704,14855),(14516,34603,13419),(23335,41587,125),(34417,43652,6642)

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X45168, X45178, X45181 = X 44802, X44803, X11799

X(45168) =  EULER LINE INTERCEPT OF X(52)X(323)

Barycentrics    a^2 (a-b-c)^2 (a+b-c)^2 (a-b+c)^2 (a+b+c)^2 (a^4+5 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+3 a^4 b^2 c^2-4 a^2 b^4 c^2+3 b^6 c^2-4 a^2 b^2 c^4-4 b^4 c^4+2 a^2 c^6+3 b^2 c^6-c^8) : :
Barycentrics    (SB+SC) (S^2-8 R^2 SA+SA^2-4 SB SC) (3 S^2+7 SC^2) : :
X(45168) = X(3)+3*X(13621),2*X(3)+3*X(34484),X(3)-3*X(43809),X(34484)+2*X(43809)

As a point on the Euler line, X(45168) has Shinagawa coefficients (E-4*F,2*E+4*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45168) lies on these lines: {2,3}, {6,9545}, {32,15355}, {49,1199}, {51,34148}, {52,323}, {54,5462}, {96,11815}, {110,389}, {143,22115}, {156,15032}, {161,18928}, {182,15028}, {184,15043}, {185,15053}, {195,16881}, {390,10046}, {511,43811}, {567,15026}, {569,11464}, {575,6467}, {578,5640}, {1092,3060}, {1125,9590}, {1147,1994}, {1173,5504}, {1181,35264}, {1204,15305}, {1493,11597}, {1495,9729}, {1511,10095}, {1588,9682}, {1614,9730}, {2079,7745}, {2393,43815}, {2888,3580}, {2917,37649}, {2929,13568}, {2931,9820}, {3043,16222}, {3047,20771}, {3066,11425}, {3284,8882}, {3292,15801}, {3357,11439}, {3410,12359}, {3448,12134}, {3574,5972}, {3581,11591}, {3600,10037}, {3618,15577}, {3620,37488}, {3815,44523}, {4993,19185}, {5012,10282}, {5254,44537}, {5286,44527}, {5422,19357}, {5446,38848}, {5449,41171}, {5609,11561}, {5622,43130}, {5651,11444}, {5731,8185}, {5866,32819}, {5889,9306}, {5890,10539}, {5907,43614}, {5926,10280}, {5943,13367}, {5944,13353}, {6000,43601}, {6102,18350}, {6146,43816}, {6193,37644}, {6403,8538}, {6684,9625}, {6696,10117}, {6699,18488}, {6759,10574}, {6800,37514}, {7585,8276}, {7586,8277}, {7592,9544}, {7689,15058}, {7691,11793}, {7735,9608}, {7738,44524}, {7999,37478}, {8907,11427}, {9538,11399}, {9541,35777}, {9591,10164}, {9626,10165}, {9659,10588}, {9672,10589}, {9706,13366}, {9707,11003}, {9713,30478}, {9723,32835}, {9780,15177}, {9781,13352}, {9786,11441}, {9833,18911}, {9924,15582}, {10314,26216}, {10316,10986}, {10540,13630}, {10541,35707}, {10545,11430}, {10546,11438}, {10601,17821}, {10610,32205}, {10982,38942}, {10984,26881}, {10985,22401}, {11002,36747}, {11004,37493}, {11064,11745}, {11202,11451}, {11264,23236}, {11440,15030}, {11468,16261}, {11477,20806}, {11562,14094}, {12038,15033}, {12112,13491}, {12278,18390}, {12282,41619}, {12300,12358}, {12370,12383}, {12584,22330}, {13142,22550}, {13289,15059}, {13346,34417}, {13347,35268}, {13364,43394}, {13392,22051}, {13445,13474}, {13472,15317}, {13567,14516}, {13598,44106}, {13754,43598}, {14061,39854}, {14157,40647}, {14627,32609}, {14652,34837}, {14671,14675}, {14683,18932}, {14915,43804}, {15019,37505}, {15045,26882}, {15049,43610}, {15054,21650}, {15062,21663}, {15066,17834}, {15068,37490}, {15072,26883}, {15080,37515}, {15107,15644}, {15139,41589}, {15531,44489}, {15647,32184}, {16194,43604}, {17701,41671}, {17704,32237}, {17810,35602}, {18381,26913}, {18474,26917}, {18475,43651}, {18912,34799}, {18931,40914}, {19459,33748}, {20190,43129}, {20304,22804}, {20791,44082}, {20987,25406}, {22800,34563}, {22948,44573}, {25739,43817}, {30482,30504}, {30522,43821}, {31834,32608}, {33884,37486}, {34782,37648}, {34835,40604}

X(45168) = midpoint of X(13621) and X(43809)
X(45168) = reflection of X(34484) in X(13621)
X(45168) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,24,7488), (2,7487,37444), (2,7488,37126), (2,31304,6643), (3,4,12086), (3,5,35500), (3,378,35494), (3,546,14865), (3,1995,3091), (3,3091,7527), (3,3518,23), (3,3525,7496), (3,3627,7464), (3,3628,7550), (3,7506,13861), (3,7530,3529), (3,7545,3627), (3,10594,3146), (3,12086,2071), (3,12088,16661), (3,12106,3518), (3,13861,4), (3,18369,546), (3,31861,35475), (3,35494,35497), (3,35500,14118), (3,37440,12088), (3,37924,12103), (4,631,44441), (4,3548,31074), (4,6644,22467), (4,7506,13595), (4,22467,2071), (5,186,14118), (5,10018,2), (5,16532,140), (5,34330,1656), (5,38321,4), (5,44211,10018), (23,16661,12088), (24,6642,2), (24,7488,37940), (24,7509,14070), (24,10018,186), (25,17928,20), (26,631,6636), (49,5946,1199), (54,5462,34545), (110,16223,40640), (140,2070,7512), (140,7512,15246), (140,13163,33332), (156,37481,15032), (186,14118,38448), (186,35500,3), (376,7517,12087), (378,7529,3832), (381,37814,3520), (403,31833,34007), (468,9825,13160), (550,18378,37925), (569,15024,15018), (631,35482,23336), (632,12107,3), (1147,3567,1994), (1511,10095,37472), (1594,16238,2), (1598,11413,3543), (1656,1658,35921), (2070,16532,186), (2072,31830,4), (3146,14002,10594), (3147,7401,2), (3147,14940,10018), (3292,16625,15801), (3515,5020,7503), (3515,7503,10298), (3518,12088,37440), (3520,37814,37941), (3525,7556,3), (3530,37936,13564), (3545,21844,7526), (3546,37122,7391), (3627,7545,26863), (3628,7575,3), (3843,11250,13596), (3850,15646,14130), (5004,5005,7667), (5020,7503,5056), (5056,10298,7503), (5576,44452,6143), (5640,11449,578), (5890,10539,43605), (5943,13367,13434), (5944,13363,13353), (6636,44441,2071),(6644,7506,4),(6644,13595,2071),(6644,13861,3),(6644,38321,186),(7464,26863,3627),(7528,37119,5169),(7542,10127,14788),(7542,14788,2), (7542,44234,10018), (7555,14869,3), (7667,40132,7570), (9707,36752,11003), (9714,10323,37913), (9786,35259,11441), (10024,44232,37943), (10303,38435,3), (10546,11438,15052), (11464,15024,569), (12086,22467,3), (12088,37440,23), (12134,26879,3448), (13595,22467,4), (14130,21308,3850), (14157,43597,40647), (15026,32171,567), (15717,37913,10323), (16881,40111,195), (21451,34007,403), (22462,34864,547), (22462,37922,34864), (26863,35500,18403), (37126,37940,7488), (38848,43574,5446)

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X(45178) =  EULER LINE INTERCEPT OF X(113)X(11817)

Barycentrics    (a-b-c)^2 (a+b-c)^2 (a-b+c)^2 (a+b+c)^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4+5 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+10 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) : :
Barycentrics    SB SC (3 S^2+7 SC^2) (14 R^2-SA-SW) : :
X(45178) = X(1657)-5*X(43809),3*X(3830)+5*X(13621)

As a point on the Euler line, X(45178) has Shinagawa coefficients (-2*F,5*E-2*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45178) lies on these lines: {2,3}, {113,11817}, {133,11792}, {389,32111}, {1503,43812}, {1514,11745}, {1533,9729}, {1614,16657}, {2883,3567}, {5254,33885}, {5890,15873}, {6152,22970}, {11381,26879}, {11439,12359}, {11457,15811}, {12022,26883}, {12112,18914}, {12241,14157}, {12290,13567}, {12300,41598}, {13366,14862}, {13568,38848}, {15033,16252}, {15063,16625}, {15305,41587}, {16621,25739}, {16659,18390}, {22660,36852}, {22802,34417}, {33880,39565}

X(45178) = midpoint of X(4) and X(34484)
X(45178) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,25,18560), (4,186,13488), (4,235,1594), (4,403,15559), (4,1598,7576), (4,3089,378), (4,3518,1885), (4,3542,35502), (4,6623,7547), (4,7487,35490), (4,7505,1597), (4,7577,1907), (4,10594,6240), (4,16868,1595), (4,26863,6756), (4,37119,11403), (4,37122,35480), (235,1594,403), (235,1595,16868), (428,44226,4), (546,11799,13160), (1595,16868,1594), (1596,1906,4), (1885,3518,10295), (3542,35502,37118), (3843,15761,5133), (3861,11563,5576), (7487,35490,6240), (10594,35490,7487), (11403,37119,35484), (13861,31725,38323), (18535,37197,4)

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X(45181) =  EULER LINE INTERCEPT OF X(11)X(4351)

Barycentrics   (a^2-b^2-c^2) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+6 a^6 b^2 c^2-2 a^4 b^4 c^2-8 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-2 a^4 b^2 c^4+12 a^2 b^4 c^4-2 b^6 c^4-8 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10) (a^10+2 a^8 b^2-3 a^6 b^4-3 a^4 b^6+2 a^2 b^8+b^10-3 a^8 c^2-3 a^6 b^2 c^2+8 a^4 b^4 c^2-3 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-3 a^4 b^2 c^4-3 a^2 b^4 c^4+2 b^6 c^4+2 a^4 c^6+7 a^2 b^2 c^6+2 b^4 c^6-3 a^2 c^8-3 b^2 c^8+c^10) : :
Barycentrics    (SC^2 (18 R^2-5 SW)+S^2 (6 R^2-SW)) (S^2 (3 R^2-SW)+SB SC (9 R^2-SW)) : :
X(45181) = 2*X(3)-3*X(44214),3*X(4)-X(10296),X(4)+2*X(16619),2*X(5)-3*X(403),4*X(5)-3*X(2072),X(5)-3*X(11563),5*X(5)-3*X(37938),X(5)+3*X(43893),X(20)-3*X(186),X(20)-5*X(37760),2*X(20)-3*X(44246),3*X(23)+X(10296),2*X(23)+X(18323),2*X(140)-3*X(44282),3*X(186)-5*X(37760),X(376)-3*X(37907),3*X(376)-5*X(37952),3*X(381)-X(7574),3*X(381)-2*X(10297),3*X(381)+X(37924),3*X(381)-4*X(37984),X(382)+3*X(2070),X(382)-3*X(31726),X(382)+4*X(37897),X(382)+6*X(37971),3*X(403)-X(858),5*X(403)-2*X(37938),X(403)+2*X(43893),2*X(468)+X(18325),4*X(468)-3*X(44214),4*X(546)+X(37900),2*X(546)+X(37967),X(548)-3*X(10096),2*X(548)-3*X(15646),4*X(548)-3*X(16386),2*X(550)-3*X(44280),5*X(631)-3*X(2071),5*X(631)-9*X(37943),5*X(631)-6*X(44452),X(691)-3*X(38227),2*X(858)-3*X(2072),X(858)-6*X(11563),5*X(858)-6*X(37938),X(858)+6*X(43893),2*X(1514)+X(3581),5*X(1656)-4*X(5159),5*X(1656)-X(35001),X(1657)-4*X(37934),X(1657)-5*X(37958),3*X(2070)-4*X(37897),X(2071)-3*X(37943),X(2072)-4*X(11563),5*X(2072)-4*X(37938),X(2072)+4*X(43893),7*X(3090)-5*X(30745),5*X(3091)-X(5189),5*X(3091)+X(37946),X(3146)+5*X(37953),3*X(3153)-7*X(3832),3*X(3153)+X(20063),7*X(3526)-6*X(10257),7*X(3526)-3*X(18859),7*X(3526)-8*X(37911),7*X(3528)-9*X(37941),X(3529)-7*X(37957),4*X(3530)-3*X(34152),2*X(3530)-3*X(44234),X(3543)+3*X(37909),3*X(3545)-X(10989),X(3627)+2*X(12105),X(3830)+2*X(37904),3*X(3830)+5*X(37923),7*X(3832)+X(20063),7*X(3832)-6*X(23323),7*X(3832)+3*X(37925),3*X(3839)+X(37901),5*X(3843)+3*X(5899),5*X(3843)-6*X(10151),5*X(3843)-3*X(18403),5*X(3843)+2*X(37899),X(3853)-3*X(11558),2*X(3853)+3*X(37936),2*X(3853)-3*X(44283),4*X(3861)+3*X(37947),4*X(5159)-X(35001),X(5899)+2*X(10151),3*X(5899)-2*X(37899),3*X(7426)-2*X(7575),3*X(7426)-X(10295),3*X(7426)-4*X(25338),9*X(7426)-8*X(44264),3*X(7426)+2*X(44267),X(7574)-4*X(37984),3*X(7575)-4*X(44264),4*X(7575)-3*X(44265),X(7575)-3*X(44266),X(7728)+2*X(32269),4*X(10096)-X(16386),3*X(10096)-2*X(22249),3*X(10151)+X(37899),3*X(10257)-4*X(37911),X(10295)-4*X(25338),3*X(10295)-8*X(44264),2*X(10295)-3*X(44265),X(10295)-6*X(44266),X(10295)+2*X(44267),X(10296)+6*X(16619),2*X(10296)-3*X(18323),2*X(10297)+X(37924),3*X(10540)-X(23236),2*X(11064)-3*X(14643),2*X(11558)+X(37936),5*X(11563)-X(37938),X(12121)-4*X(15448),X(12295)+2*X(32237),X(12383)-3*X(35265),3*X(13445)-7*X(15057),2*X(13473)+3*X(37956),3*X(13619)+X(33703),X(13619)-3*X(37940),3*X(14643)-X(37477),3*X(15362)-X(20126),3*X(15362)-2*X(44569),3*X(15646)-4*X(22249),5*X(15696)-9*X(37955),3*X(16222)-4*X(44084),6*X(16227)-5*X(37481),3*X(16386)-8*X(22249),4*X(16531)-3*X(37941),3*X(16532)-2*X(37968),4*X(16619)+X(18323),5*X(17578)+9*X(37939),X(17800)-9*X(37922),X(17800)-6*X(37931),2*X(18325)+3*X(44214),3*X(18374)-X(32233),3*X(18403)+2*X(37899),4*X(18571)-3*X(44280),2*X(18572)+X(37900),2*X(18579)-3*X(37907),6*X(18579)-5*X(37952),3*X(18859)-8*X(37911),X(18859)-4*X(37942),X(20063)+6*X(23323),X(20063)-3*X(37925),2*X(20725)-3*X(38788),2*X(23323)+X(37925),3*X(25338)-2*X(44264),8*X(25338)-3*X(44265),2*X(25338)-3*X(44266),2*X(25338)+X(44267),3*X(31726)+4*X(37897),X(31726)+2*X(37971),X(33703)+9*X(37940),10*X(37760)-3*X(44246),2*X(37897)-3*X(37971),6*X(37904)-5*X(37923),9*X(37907)-5*X(37952),2*X(37911)-3*X(37942),3*X(37922)-2*X(37931),X(37924)+4*X(37984),4*X(37934)-5*X(37958),4*X(37935)-3*X(37955),X(37938)+5*X(43893),3*X(37943)-2*X(44452),X(37950)-3*X(44282),4*X(44264)-9*X(44266),4*X(44264)+3*X(44267),X(44265)-4*X(44266),3*X(44265)+4*X(44267),3*X(44266)+X(44267)

As a point on the Euler line, X(45181) has Shinagawa coefficients (E+4*F,-5*E+4*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45181) lies on these lines: {2,3}, {11,4351}, {12,4354}, {49,16252}, {94,10688}, {113,511}, {115,3003}, {125,1533}, {127,18809}, {133,38971}, {195,13142}, {265,1177}, {339,44138}, {343,18435}, {399,3564}, {495,9642}, {517,41607}, {523,19912}, {524,5655}, {541,32225}, {567,16657}, {691,38227}, {1154,41596}, {1495,17702}, {1498,25738}, {1499,36255}, {1514,3581}, {1614,12370}, {2393,15125}, {2777,32110}, {2781,8262}, {2883,34783}, {3292,16534}, {3448,12112}, {3521,13568}, {3580,5663}, {3767,16306}, {5063,5475}, {5139,42424}, {5254,16308}, {5446,43831}, {5449,11381}, {5480,8705}, {5486,20423}, {5511,42422}, {5512,16188}, {5656,18917}, {5891,40107}, {5965,6053}, {5972,10564}, {6000,16003}, {6243,22660}, {6530,34334}, {6759,44076}, {6795,16324}, {7687,29012}, {7706,34417}, {8758,11809}, {9644,37719}, {9704,43595}, {9820,37495}, {9927,26883}, {10112,14862}, {10149,15888}, {10293,15362}, {10540,23236}, {10610,15807}, {10706,15360}, {10752,41721}, {11064,14643}, {11455,23293}, {11472,37638}, {11645,20301}, {11649,18388}, {12121,15448}, {12188,38595}, {12279,26917}, {12295,32237}, {12359,18439}, {12383,35265}, {12420,12429}, {12897,13367}, {12918,20957}, {13394,14805}, {13445,15057}, {13470,43865}, {13491,26879}, {13556,38953}, {13754,14448}, {13851,44407}, {14094,41724}, {14672,42426}, {14693,38611}, {14934,16319}, {15060,37636}, {15069,18451}, {15305,38397}, {16194,21243}, {16222,44084}, {16227,37481}, {16658,34514}, {16760,18860}, {18374,32233}, {18445,34117}, {18474,44470}, {18914,22533}, {20725,38788}, {24855,40115}, {29181,32218}, {32137,34826}, {32358,43605}, {34150,40352}, {36982,41725}, {37648,40280}, {38577,38583}, {39899,41719}

X(45181) = midpoint of X(i) and X(j) for these {i,j}: {3,18325}, {4,23}, {125,1533}, {1514,32269}, {2070,31726}, {3153,37925}, {3448,12112}, {3580,32111}, {3581,7728}, {5189,37946}, {5899,18403}, {7574,37924}, {7575,44267}, {10706,15360}, {10752,41721}, {11563,43893}, {11676,36174}, {14094,41724}, {15063,41586}, {18572,37967}, {37936,44283}
X(45181) = reflection of X(i) in X(j) for these (i,j): (3,468), (23,16619), (376,18579), (403,11563), (548,22249), (550,18571), (858,5), (2070,37971), (2071,44452), (2072,403), (3153,23323), (3292,16534), (3581,32269), (6795,16324), (7426,44266), (7464,15122), (7472,37459), (7574,10297), (7575,25338), (7728,1514), (10257,37942), (10295,7575), (10297,37984), (10564,5972), (14934,16319), (15646,10096), (15980,14120), (16386,15646), (18323,4), (18403,10151), (18572,546), (18859,10257), (18860,16760), (20126,44569), (32110,32223), (34152,44234), (37477,11064), (37900,37967), (37950,140), (38611,14693), (44246,186), (44265,7426), (44283,11558)
X(45181) = reflection of X(2) in X(523)X(44204)
X(45181) = complement of X(7464)
X(45181) = anticomplement of X(15122)
+ X(45181) = complementary conjugate of the complement of X(10293)
X(45181) = circumperp conjugate of the ctic conjugate of X(40914)
X(45181) = X(5475)-line conjugate of X(5063)
X(45181) = X(523)-vertex conjugate of X(6644)
X(45181) = crossdifference of every pair of points on line X(647)-X(5063)
X(45181) = crosssum of X(i)and X(j) for these {i,j}: {3,15136},{6,40114}
X(45181) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,31861), (2,7464,15122), (2,13596,44236), (3,468,44214), (3,5073,34622), (3,5899,37928), (3,37973,2070), (4,26,18563), (4,403,37981), (4,7530,7540), (4,7552,7527), (4,10024,5576), (4,15761,10024), (5,20,37452), (5,427,7579), (5,858,2072), (5,1906,3843), (5,3853,15559), (20,37760,186), (23,186,26), (23,11563,10024), (186,3542,468), (235,468,403), (235,43893,18325), (376,37907,18579), (381,7574,10297), (381,12083,18531), (381,15154,10751), (381,15155,10750), (381,15760,37347), (381,18534,31723), (381,37924,7574), (403,858,5), (403,18323,5576), (468,1885,44281), (546,7553,31724), (546,16618,34664), (548,10096,22249), (548,22249,15646), (550,18571,44280), (1113,1114,6644), (1312,1313,381), (1346,1347,39484), (1596,15760,381), (1658,44271,18560), (1885,13383,3), (2041,2042,12084), (2070,7517,23), (2071,37943,44452), (2072,37981,5576), (2883,41587,34783), (3091,14789,5), (3146,16868,13371), (3627,13406,1594), (3830,10254,427), (3832,20063,3153), (5073,10255,23335), (5169,7493,30739), (5189,37943,13154), (6623,18531,381), (7387,37197,18404), (7426,10295,7575), (7527,7552,140), (7542,13488,14130), (7575,10295,44265), (7575,25338,7426), (7575,44266,25338), (7579,10254,5), (10024,18323,2072), (10096,16238,468), (10201,44276,378), (10257,37911,3526), (10297,37984,381), (10750,10751,31723), (12086,14940,23336), (13160,18572,2072), (14120,36169,381), (14643,37477,11064), (15154,15155,18534), (15362,20126,44569), (15761,43893,23), (16618,34664,3), (18323,44246,18563), (18325,31726,31725), (20408,20409,382), (25338,44267,10295), (34007,34484,31830), (37440,44279,6240), (37950,44282,140), (44266,44267,7575)

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HATZIPOLAKIS - SUPPA CIRCLE

Locus Problem:

Let ABC be a triangle, A'B'C' the pedal triangle of H and P a point.
Denote: (Oa), (Ob), (Oc) = the circumcircles of (AA'A"), (BB'B|), (CC'C") resp. (ie the circles with diameters AA", BB", CC", resp)

(O'a), (O'b), (O'c) = the reflections of (Oa), (Ob), (Oc) in AA', BB'. CC', resp.

R'a = the radical axis of (O'b), (O'c)
R'b = the radical axis of (O'c), (O'a)
R'c = the radical axis of (O'a), (O'b)

R'1, R'2, R'3 = the reflections of R'a, R'b, R'c in BC, CA, AB, resp.
The locus of P such that the parallels to R'1, R'2, R'3 through A',B',C', resp. is the circle (X(382), R), the reflection of the circumcircle in the orthocenter H.

The circle is named HATZIPOLAKIS - SUPPA circle

Let Q5=Q5(P) be the concurrency point.

*** ETC points on the circle:
X(i), i=10152,10721,10722,10723,10724,10725,10726,10727,10728,10729,10730,10731,10732,10733,10734,10735,10736,10737,14989

*** ETC pairs (P=X(i),Q5=X(j)): {10152,133},{10721,125},{10722,115},{10723,114},{10724,119},{10725,118},{10726,124},{10727,116},{10728,11},{10729,5511},{10730,5510},{10731,25640},{10732,117},{10733,113},{10734,5512},{10735,132},{10736,1312},{10737,1313},{14989,3258}

*** Q5(P)=image of P under the homotety with center H and factor k=-1/2
*** The locus of points Q5(P) with P∈(X(382),R) is the ninepoint circle of ABC.

Reference: Euclid 2592