[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
A', B', C' = the orthogonal projections of Na, Nb, Nc on BC, CA, AB, resp.
(Oab), (Oac) = the ciecles (Na, NaB'), (Na, NaC')
(Obc), (Oba) = the ciecles (Nb, NbC'), (Nb, NbA')
(Oca), (Ocb) = the ciecles (Nc, NcA'), (Nc, NcB')
Ra = the radical axis of (Oba), (Oca)
Rb = the radical axis of (Ocb), (Oab)
Rc = the radical axis of (Oac), (Obc)
Sa = the radical axis of (Obc), (Ocb)
Sb = the radical axis of (Oca), (Oac)
Sc = the radical axis of (Oab), (Oba)
Which is the locus of P such that:
1 Ra, Rb, Rc are concurrent ?
O, I lie on the locus
2. Sa, Sb, Sc are concurrent ?
O, I lie on the locus
(for P = O, the point of concurrence is the N)
[Ercole Suppa]:
1. Let P(x:y:z) be the barycentric coordinates of P and Q1= Q1(P) the concurrency point.
*** Locus of P such that Ra, Rb, Rc are concurrent = {Linf} U {circumcircle} U {K003 McCay cubic}
*** if P lies on the circumcircle Q1(P) = midpoint of P and O=X(3)
*** Pairs {P = X(i),Q1 = X(j)} for these {i,j}: {3, 1511},{74, 12041}, {98, 12042}, {110, 1511}, {111, 14650}
*** Some points:
Q1(X(1)) = MIDPOINT OF X(1) AND X(33337)
= (b + c-2 a) (3 a^3-a^2 b-3 a b^2+b^3-a^2 c+5 a b c-b^2 c-3 a c^2-b c^2+c^3) : : (barys)
= 5*X[1]-X[149], X[10]+X[7972], X[11]-2*X[3636], X[80]-3*X[551], X[100]+X[3244], X[145]+3*X[15015], X[944]+X[21635], X[1483]+X[22935], 2*X[3035]-X[3626], 2*X[3036]-3*X[3828], 3*X[3241]+X[5541], 5*X[3616]-X[9897], 5*X[3623]-X[12653], 2*X[3634]-X[15863], X[3878]+X[17660], 3*X[3898]-X[17638], X[4297]+X[10698], X[4301]+X[12119], 3*X[5731]+X[13253], X[5882]+X[6265], 3*X[5883]-X[17636], X[6326]+3*X[7967], 3*X[10165]-X[19914], 3*X[10246]-X[10265], 2*X[11729]-X[19925], X[13199]+3*X[16200], 7*X[15808]-5*X[31272], 7*X[20057]+X[20095]
= lies on these lines: {1,149}, {10,7972}, {11,3636}, {80,551}, {100,3244}, {119,28236}, {142,12737}, {145,15015}, {214,519}, {515,12611}, {516,25485}, {942,2802}, {944,21635}, {952,1125}, {1320,5425}, {1385,32905}, {1483,22935}, {1484,3822}, {1537,28164}, {2771,3884}, {2800,13369}, {2932,25439}, {3035,3626}, {3036,3828}, {3241,5541}, {3616,9897}, {3623,12653}, {3634,15863}, {3825,11698}, {3878,17660}, {3898,17638}, {4297,10698}, {4301,12119}, {5248,12773}, {5731,13253}, {5882,6265}, {5883,17636}, {6264,30147}, {6326,7967}, {10074,14798}, {10165,19914}, {10246,10265}, {11715,24299}, {11729,19925}, {13199,16200}, {15808,31272}, {20057,20095}, {24466,28228}
= midpoint of X(i) and X(j) for these {i,j}: {1,33337}, {10,7972}, {100,3244}, {214,1317}, {551,10031}, {944,21635}, {1483,22935}, {3878,17660}, {4301,12119}, {5882,6265}, {6224,21630}
= reflection of X(i) in X(j) for these {i,j}: {11,3636}, {3626,3035}, {3635,12735}, {15863,3634}, {19925,11729}
= barycentric quotient of X(44)/X(1343)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,6224,21630},{214,11274,1317},{21630,33337,6224}
= (6-9-13) search numbers: [1.4417771099426264477, 0.0548194877267187891, 2.9372769396615923502]
Q1(X(99)) = MIDPOINT OF X(3) AND X(99)
= 2 a^8-5 a^6 b^2+5 a^4 b^4-2 a^2 b^6-5 a^6 c^2+4 a^4 b^2 c^2-a^2 b^4 c^2+b^6 c^2+5 a^4 c^4-a^2 b^2 c^4-2 b^4 c^4-2 a^2 c^6+b^2 c^6 : : (barys)
= 7 S^4 + (-9 SB SC+2 SB SW+2 SC SW-3 SW^2)S^2 + SB SC SW^2 : : (barys)
= 3*X[2]-X[6321], X[4]-3*X[15561], X[5]-2*X[620], X[20]+X[6033], X[115]-2*X[140], X[147]+3*X[376], X[148]-5*X[631], 3*X[381]-X[10723], 2*X[547]-3*X[9167], 2*X[548]+X[14981], 5*X[632]-4*X[6722], X[671]-3*X[5054], X[1350]+X[12177], X[1351]-3*X[5182], 5*X[1656]-3*X[14639], X[1657]+X[10722], 5*X[3522]-X[9862], 7*X[3523]-3*X[14651], 3*X[3524]+X[8591], 7*X[3526]-5*X[14061], X[3534]+X[6054], 3*X[3576]+X[13174], X[3627]-4*X[20399], 4*X[3628]-3*X[23514], X[3655]+X[9881], X[3830]-3*X[23234], 3*X[5050]-X[10754], 2*X[5461]-3*X[11539], X[5984]-9*X[10304], X[6055]-2*X[12100], 2*X[6699]-X[15535], 3*X[7709]+X[8782], X[7970]+X[12702], X[7983]-3*X[10246], X[8586]-3*X[22525], X[8596]-9*X[15708], 3*X[9166]-X[12355], X[9864]+X[18481], 4*X[10124]-3*X[14971], 3*X[10165]-X[11599], X[11005]+X[12121], X[11177]-5*X[19708], 2*X[11623]-5*X[15712], X[11710]-2*X[13624], 2*X[11724]-X[22791], X[12243]-5*X[15692], X[12383]+X[15545], X[13178]-3*X[26446], X[13179]-3*X[26451], 7*X[14869]-4*X[20398], 3*X[15035]-X[18332], 5*X[15051]-X[22265], X[15342]-3*X[32609], 3*X[15699]-4*X[22247], X[21636]+X[31730]
= lies on these lines: {1,15452}, {2,6321}, {3,76}, {4,15561}, {5,620}, {20,6033}, {24,5186}, {30,114}, {35,3023}, {36,3027}, {55,10089}, {56,10086}, {115,140}, {147,376}, {148,631}, {182,5969}, {187,1569}, {378,12131}, {381,10723}, {384,11272}, {498,13182}, {499,13183}, {511,5026}, {517,11711}, {542,8703}, {543,549}, {547,9167}, {548,14981}, {550,2794}, {574,2023}, {632,6722}, {671,5054}, {690,1511}, {804,11616}, {1003,10352}, {1350,12177}, {1351,5182}, {1353,14645}, {1656,14639}, {1657,10722}, {1916,8350}, {2080,4027}, {2452,4558}, {2795,5428}, {2967,4235}, {3095,3552}, {3398,7783}, {3522,9862}, {3523,14651}, {3524,8591}, {3526,14061}, {3534,6054}, {3576,13174}, {3582,12354}, {3584,18969}, {3587,24469}, {3627,20399}, {3628,23514}, {3655,9881}, {3734,9734}, {3830,23234}, {3972,32447}, {4226,9155}, {4299,12184}, {4302,12185}, {5050,10754}, {5149,9737}, {5204,10069}, {5217,10053}, {5461,11539}, {5965,14148}, {5984,10304}, {5985,17549}, {5987,6636}, {6055,12100}, {6179,32520}, {6194,8289}, {6319,26341}, {6320,26348}, {6390,32458}, {6642,13175}, {6699,15535}, {7417,31128}, {7502,14655}, {7709,8782}, {7757,11842}, {7863,32152}, {7970,12702}, {7983,10246}, {8290,11676}, {8586,22525}, {8589,15819}, {8596,15708}, {9166,12355}, {9864,18481}, {10124,14971}, {10165,11599}, {10267,13173}, {10269,22514}, {10734,16281}, {11005,12121}, {11156,11288}, {11177,19708}, {11623,15712}, {11710,13624}, {11724,22791}, {12054,12176}, {12243,15692}, {12251,33014}, {12383,15545}, {13178,26446}, {13179,26451}, {13180,26492}, {13181,26487}, {13189,16203}, {13190,16202}, {13335,32448}, {13479,22143}, {13496,18570}, {13860,22681}, {14033,14494}, {14570,22085}, {14869,20398}, {15035,18332}, {15051,22265}, {15342,32609}, {15357,32423}, {15699,22247}, {21636,31730}, {24472,24929}
= midpoint of X(i) and X(j) for these {i,j}: {3,99}, {20,6033},{98,13188},{115,10992},{376,8724},{381,12117},{1350,12177},{1657,10722}, {1916,19910}, {3534,6054}, {3655,9881}, {5984,14692}, {6055,15300}, {6321,13172}, {7970,12702}, {8591,11632}, {9864,18481}, {11005,12121}, {12188,23235}, {12383,15545}, {21636,31730}
= reflection of X(i) in X(j) for these {i,j}: {5,620}, {115,140}, {6055,12100}, {9880,547}, {11710,13624}, {12042,3},{15535,6699}, {22505,114}, {22515,5}, {22791,11724}
= complement of X(6321)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,13172,6321}, {3,1975,10104}, {3,13188,98}, {98,99,13188}, {99,21166,3}, {114,22505,22566}, {187,1569,12829}, {1656,14639,15092}, {3523,20094,14651}, {3524,8591,11632}, {9167,9880,547}, {11171,19910,1916}, {12121,14850,11005}, {12355,15694,9166}, {14692,14830,5984}, {23514,31274,3628}, {32481,32482,13108}
= (6-9-13) search numbers: [5.1105461739222289993, 2.1890589422603534517, -0.2335514814676635939]
Q1(X(100)) = MIDPOINT OF X(3) AND X(100)
= a (2 a^6-2 a^5 b-4 a^4 b^2+4 a^3 b^3+2 a^2 b^4-2 a b^5-2 a^5 c+4 a^4 b c+2 a^3 b^2 c-5 a^2 b^3 c+b^5 c-4 a^4 c^2+2 a^3 b c^2+2 a b^3 c^2+4 a^3 c^3-5 a^2 b c^3+2 a b^2 c^3-2 b^3 c^3+2 a^2 c^4-2 a c^5+b c^5) : : (barys)
= 5 R S^3 - (4 a R^2-4 b R^2-2 a SB+2 b SB+c SB-2 a SC+b SC+2 c SC+b SW)S^2 -3 R S SB SC-b SB SC^2 + c SB SC^2 + b SB SC SW : : (barys)
= 3*X[2]-X[10738], X[5]-2*X[3035], X[20]+X[10742], X[40]+X[6265], X[149]-5*X[631], X[153]+3*X[376], 3*X[165]+X[6326], X[355]+X[12119], 3*X[381]-X[10724], X[1320]-3*X[10246], X[1657]+X[10728], X[1768]+X[12738], 5*X[3522]-X[12248], 7*X[3523]+X[20095], 7*X[3526]-5*X[31272], 4*X[3530]+X[6154], X[3534]+X[10711], 3*X[3576]+X[5541], X[3627]-4*X[20400], 4*X[3628]-3*X[23513], 3*X[5050]-X[10755], 3*X[5054]-X[10707], X[5528]+3*X[21153], X[5531]+7*X[16192], 3*X[5660]-X[16128], 3*X[5886]-X[14217], X[6246]-2*X[9956], X[6264]-5*X[7987], 2*X[6702]-3*X[11231], X[6909]+X[18524], 3*X[10164]-X[10265], 3*X[10165]-X[21630], X[10767]-3*X[14643], X[10768]-3*X[15561], X[10778]-3*X[15061], 3*X[11171]-X[32454], 3*X[11230]-2*X[16174], X[11362]+X[33337], X[11715]-2*X[13624], X[12751]+X[18481], X[13268]-3*X[26451], 3*X[13587]-X[22765], 5*X[15712]-2*X[20418], X[21635]+X[31730]
= lies on these lines: {1,24302}, {2,10738}, {3,8}, {5,3035}, {10,26086}, {11,35}, {20,10742}, {24,1862}, {30,119}, {36,1317}, {40,6265}, {46,12739}, {55,1387}, {56,10087}, {80,5010}, {149,631}, {153,376}, {165,6326}, {182,9024}, {214,517}, {355,12119}, {378,12138}, {381,10724}, {404,5901}, {498,13273}, {499,13274}, {516,12611}, {519,23961}, {528,549}, {550,2829}, {632,5248}, {758,10225}, {971,6594}, {993,3036}, {1006,12690}, {1155,11570}, {1320,10246}, {1376,6914}, {1385,2802}, {1482,4188}, {1483,5854}, {1511,8674}, {1537,6905}, {1657,10728}, {1737,12743}, {1768,12738}, {2080,13194}, {2771,9943}, {2783,12042}, {2800,3579}, {2805,14650}, {3149,12775}, {3522,12248}, {3523,20095}, {3526,31272} ,{3530,6154}, {3534,10711}, {3576,5541}, {3627,20400}, {3628,23513}, {3651,13257}, {3871,25416}, {3880,18857}, {4189,12747}, {4242,21664}, {4299,12763}, {4302,12764}, {4421,10269}, {4640,18254}, {5050,10755}, {5054,10707}, {5119,12740}, {5172,12832}, {5204,10074}, {5217,10058}, {5259,16239}, {5267,15863}, {5428,6684}, {5432,8068}, {5433,5533}, {5440,14988}, {5499,31659}, {5528,21153}, {5531,16192}, {5537,28212}, {5660,16128}, {5762,10427} ,{5790,6950}, {5843,6068}, {5856,31657}, {5886,14217}, {6147,10044}, {6221,19082}, {6246,9956}, {6264,7987}, {6398,19081}, {6551,29348}, {6642,13222}, {6702,11231}, {6906,18357}, {6909,18524}, {6924,11248}, {6942,10698}, {6970,8166}, {7280,7972}, {7427,31073}, {10039,18976}, {10073,24914}, {10164,10265}, {10165,21630}, {10267,13205}, {10679,16371}, {10680,19537}, {10767,14643}, {10768,15561}, {10778,15061}, {10956,18990}, {11171,32454}, {11230,16174}, {11362,33337}, {11495,12332}, {11715,13624}, {12054,12199}, {12100,15931}, {12736,24929}, {12751,18481}, {13235,26316}, {13268,26451}, {13269,26341}, {13270,26348}, {13271,26492}, {13272,26487}, {13278,16203}, {13279,16202}, {13587,22765}, {15325,32760}, {15712,20418}, {19919,31835}, {21635,31730}, {31789,32554}
= midpoint of X(i) and X(j) for these {i,j}: {3,100}, {11,10993}, {20,10742}, {40,6265}, {104,12331}, {119,24466}, {355,12119}, {550,11698}, {1657,10728}, {1768,12738}, {3534,10711}, {3579,22935}, {5541,12737}, {6224,19914}, {6326,12515}, {6909,18524}, {10698,12702}, {10738,13199}, {11362,33337}, {12751,18481}, {21635,31730}, {22560,25438}
= reflection of X(i) in X(j) for these {i,j}: {5,3035}, {11,140}, {1484,6713}, {6246,9956}, {11715,13624}, {12619,6684}, {19907,214}, {22791,11729}, {22799,119}, {22938,5}
= complement of X(10738)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,13199,10738}, {3,5687,32153}, {3,12331,104}, {40,15015,6265}, {55,10090,1387}, {56,10087,12735}, {100,104,12331}, {100,2932,9945}, {100,4996,1145}, {100,17100,10609}, {165,6326,12515}, {404,11849,5901}, {549,1484,6713}, {3576,5541,12737}, {4421,22560,25438}, {5010,26446,7508}, {5657,6224,19914}, {6174,24466,119}, {6924,11248,22791}, {8715,32612,1483}, {23513,31235,3628}, {25440,26285,5}
= (6-9-13) search numbers: [4.6920995145860464520, 2.2040937105154880299, -0.0508309397968263570]
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2. Let P(x:y:z) be the barycentric coordinates of P and Q2 = Q2(P) the concurrency point.
*** Locus of P such that Sa, Sb, Sc are concurrent = {Linf} U {circumcircle} U {K003 McCay cubic}
*** Pairs {P=X(i),Q2=X(j)} for these {i,j}: {3,5}
*** Some points:
Q2(X(1)) = MIDPOINT OF X(65) AND X(1125)
= a (2 a^2 b-2 b^3+2 a^2 c+2 a b c+3 b^2 c+3 b c^2-2 c^3) : : (barys)
= 3*X[2]+X[4084], 7*X[10]-3*X[3681], 5*X[65]+3*X[392], X[72]-3*X[3828], 3*X[354]-X[3635], 9*X[551]-5*X[3890], X[960]-3*X[3833], 5*X[1698]-X[4067], X[3244]-5*X[18398], X[3555]+7*X[3922], 5*X[3617]+3*X[3894], X[3625]+3*X[3873], X[3626]-3*X[3753], 5*X[3698]-3*X[4745], 3*X[3740]-X[4127], 3*X[3742]-X[3884], X[3869]+3*X[4744], 3*X[3877]-7*X[15808], 3*X[3892]+X[10914], 3*X[3899]-11*X[5550], X[3901]+3*X[4134], X[4018]+3*X[10176], X[4297]-5*X[15016], 3*X[4430]+5*X[4668], X[5694]-3*X[10172], X[5784]+3*X[30329], X[5884]+X[19925], X[5887]-3*X[10171]
= lies on these lines: {1,1392}, {2,4084}, {8,17146}, {10,3681}, {46,30143}, {57,30147}, {65,392}, {72,3828}, {354,3635}, {404,5425}, {515,5885}, {516,31870}, {517,3530}, {518,3918}, {519,942}, {535,24470}, {551,3890}, {758,3634}, {960,3833}, {993,5221}, {1159,25524}, {1193,24168}, {1698,4067}, {1737,11263}, {1739,2650}, {2802,5045}, {3035,16137}, {3244,18398}, {3306,30144}, {3333,22837}, {3336,5267}, {3339,31424}, {3555,3922}, {3617,3894}, {3625,3873}, {3626,3753}, {3649,3814}, {3698,4745}, {3740,4127}, {3742,3884}, {3869,4744}, {3877,15808}, {3892,10914}, {3899,5550}, {3901,4134}, {3997,20271}, {4018,10176}, {4297,15016}, {4298,18838}, {4430,4668}, {4867,17531}, {4880,5260}, {5046,11552}, {5270,26842}, {5290,18419}, {5438,11529}, {5542,10915}, {5690,25557}, {5694,10172}, {5708,8666}, {5784,30329}, {5884,19925}, {5887,10171}, {6001,12571}, {6583,28234}, {6738,10122}, {7686,28164}, {8715,15934}, {9943,28158}, {10129,15079}, {11375,20107}, {11551,24982}, {12005,28236}, {13145,28194}, {17049,28512}, {18395,31019}, {20104,24914}, {28228,31788}
= midpoint of X(i) and X(j) for these {i,j}: {65,1125}, {942,3754}, {960,4757}, {3555,4701}, {3626,3874}, {3812,31794},{3881,5836}, {4745,24473}, {5045,10107}, {5884,19925}
= reflection of X(i) in X(j) for these {i,j}: {960,19878}, {3634,3812}, {4691,3918}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {65,5439,3878}, {65,5883,1125}, {942,5836,3881}, {960,3833,19878}, {3753,3874,3626}, {3754,3881,5836}, {3833,4757,960}, {3878,5439,1125}, {3878,5883,5439}, {3901,9780,4134}, {4744,19862,3869}, {12736,13750,6738}
= (6-9-13) search numbers: [1.4621650176179250476, 1.2346234874987090459, 2.1110797516616151263]
Best regards,
Ercole Suppa
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