[Antreas P. Hatzipolakis]:
3Yet Another VARIATION (2yet variation: Hyacinthos 28455. 1yet variation. Hyacinthos 28443)
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
Ra = the radical axis of the circles (Nb, NbP), (Nc, NcP)
Rb = the radical axis of the circles (Nc, NcP), (Na, NaP)
Rc = the radical axis of the circles (Na, NaP), (Nb, NbP)
Which is the locus of P such that the reflections of Ra, Rb, Rc in AI, BI, CI, resp are concurrent?
Part of the locus is the Euler line of NaNbNc [ = IN line], and when P moves on the IN line the point of concurrence lies on the OI line.
APH
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[Ercole Suppa]
*** locus of point P such that the reflections of Ra, Rb, Rc in AI, BI, CI, resp are concurrent is : {line IN}
IN pass through X(i) for these i: {1,5,11,12,80,119,355,495,496,952,1317,1387,1411,1421,1483,1484,1807,1837,2006,2594,2596,2606,3614,4551,5219,5252,5396,5399,5400,5443,5531,5533,5534,5587,5660,5718,5719,5720,5721,5722,5723,5724,5725,5726,5727,5881,5886,5901,6127,6264,6265,6326,7173,7741,7951,7958,7972,7988,7989,7993,8068,8070,8227,9578,9581,9624,9817,9897,10057,10073,10283,10523,10592,10593,10826,10827,10886,10887,10942,10943,10944,10948,10949,10950,10954,10955,10956,10957,10958,10959,11373,11374,11375,11376,11698,11729,12019,12025,12433,12550,12735,12737,12738,12739,12740,12749,12750,12751,13244,14204,14584,14679,15017,15251,15252,15253,15888,15935,15943,15950,16173,17602,17717,17718,17719,17720,17721,17722,17723,17724,17725,17726,17857,18357,19372,19907,20586,22392}
*** locus of point Q=Q(X(P)) is the line IO where O is the circumcenter of ABC
*** pairs {P=X(i),Q=X(j)} for these {i,j}: {1,1},{5,1385},{11,1319},{12,2646},{80,36},{355,3},{952,517},{1317,5048},{1483,10222},{1837,56},{5219,13384},{5252,55},{5531,5538},{5587,3576},{5722,999},{5724,3666},{5727,57},{5881,40},{5886,10246},{5901,15178},{7741,21842},{9578,3601},{9581,1420},{9897,484},{10827,3612},{10944,3057},{10950,65},{11376,1388},{12019,5126},{12433,5045},{12751,2077},{18357,13624}
*** some points Q(X(P)):
Q(X(495)) = COMPLEMENT X(3419)
Barycentrics a (2 a^3-a^2 b-2 a b^2+b^3-a^2 c-2 a b c-b^2 c-2 a c^2-b c^2+c^3) : :
= 3*X[2]-X[3419], X[63]-3*X[16370], X[956]+X[3870], X[1478]-3*X[17718], X[1836]+X[4302], X[5252]-3*X[10056]
= lies on these lines: {1,3},{2,3419},{4,4313},{5,950},{7,376},{8,5791},{9,3940},{10,6675},{11,6881},{12,6841},{20,3487},{21,72},{28,1255},{30,226},{33,7497},{37,101},{41,16601},{42,8731},{63,16370},{73,500},{74,934},{78,405},{79,4324},{80,3584},{100,3753},{104,2346},{105,20219},{140,1210},{142,214},{145,3897},{169,4258},{191,3962},{200,9708},{210,5251},{212,5398},{228,859},{329,11111},{355,3085},{377,3824},{378,1876},{381,3586},{382,9612},{386,1104},{388,4305},{390,5603},{392,1621},{404,5439},{443,1058},{474,4855},{495,515},{496,1125},{497,5886},{498,1837},{516,8255},{518,993},{519,5745},{548,4114},{549,3911},{550,3982},{553,8703},{579,1100},{582,1451},{610,3247},{631,938},{674,1386},{758,4640},{855,21319},{894,4234},{908,11113},{910,4262},{912,6914},{936,2900},{944,5787},{946,4314},{951,7100},{954,971},{956,3870},{958,3811},{960,5248},{975,7535},{976,10448},{991,6610},{995,1279},{997,1001},{1000,2320},{1006,5728},{1026,16443},{1043,5295},{1056,5731},{1064,2293},{1071,6906},{1212,4251},{1400,14636},{1426,7414},{1447,13634},{1464,4337},{1478,17718},{1479,9955},{1656,9581},{1657,9579},{1699,9668},{1737,5432},{1766,21848},{1770,3649},{1785,7510},{1817,17019},{1829,14017},{1836,4302},{1858,5694},{1864,7489},{1870,4219},{1892,18533},{1895,7531},{2003,23071},{2256,3211},{2269,5755},{2271,16968},{2329,3991},{2475,11015},{2649,9367},{2650,9340},{2687,14733},{2771,10058},{2807,11700},{2808,15730},{2975,3555},{2999,16485},{3024,11699},{3058,15950},{3086,6989},{3146,5714},{3149,5806},{3158,9623},{3189,19843},{3218,17549},{3244,5855},{3306,16371},{3475,4293},{3476,3655},{3485,4294},{3522,11036},{3524,5435},{3525,5704},{3534,4654},{3560,5777},{3582,5444},{3583,17605},{3585,5441},{3622,6904},{3636,12436},{3647,4067},{3656,10385},{3671,16137},{3678,5302},{3679,3689},{3683,5692},{3693,16788},{3697,4420},{3720,16056},{3752,4256},{3820,6745},{3868,3916},{3871,10914},{3874,5267},{3876,16865},{3892,15570},{3920,4224},{3927,11523},{3984,15650},{3996,16821},{4134,15481},{4276,18165},{4297,18990},{4299,10404},{4309,12701},{4428,5289},{4649,5429},{4652,11520},{4857,5443},{4870,18393},{4999,10916},{5088,14828},{5110,20227},{5175,6856},{5218,18391},{5249,11112},{5250,5730},{5252,10056},{5256,21483},{5262,7523},{5281,5657},{5287,11347},{5290,9655},{5325,15673},{5341,16777},{5396,14547},{5428,10122},{5434,21578},{5437,16417},{5438,16408},{5450,12675},{5453,13754},{5529,17123},{5687,19860},{5691,9654},{5705,12625},{5720,6913},{5727,5790},{5761,5812},{5768,6935},{5771,5844},{5794,10198},{5804,6927},{5836,8715},{5882,6245},{5887,12711},{5901,12053},{5927,6912},{6175,9963},{6261,9856},{6284,12047},{6326,14100},{6684,6738},{6692,17564},{6700,17527},{6705,13607},{6734,7483},{6762,7160},{6796,7686},{6909,10167},{6910,12649},{6920,9844},{7098,16139},{7308,16857},{7354,13407},{7428,22345},{7508,17010},{7520,9538},{8068,12743},{8227,9669},{9578,18525},{9605,16780},{9614,18493},{9619,16781},{9785,10595},{9946,15558},{9947,17857},{10039,10950},{10165,11019},{10175,12019},{10176,15254},{10304,21454},{10360,18931},{10386,10624},{10436,19276},{10592,19925},{11246,11551},{11281,12609},{11552,15228},{11712,15746},{11715,12735},{12109,15489},{12114,12260},{12512,12563},{12514,12635},{12575,13464},{12672,21740},{12690,17530},{13726,19767},{15008,15299},{15677,17484},{15837,18412},{16054,16826},{16457,19859},{16583,18755},{16843,19753},{17525,17781},{17558,20007},{17576,20214},{17637,22936},{18482,21617},{19520,19861}
= complement of X3419
= midpoint of X(i) and X/j) for these {i,j}: {1,55},{72,16465},{226,4304},{954,7675},{956,3870},{1012,18446},{1836,4302},{2099,5119},{10058,12739}
= reflection of X(i) in X(j) for these {i,j}: {10,6690},{226,5719},{495,13405},{942,11018},{2886,1125},{5173,5045},{18407,9955}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,942},{1,35,65},{1,36,354},{1,56,5045},{1,57,15934},{1,165,11529},{1,484,5425},{1,999,5049},{1,1420,7373},{1,1697,1482},{1,2646,1385},{1,3057,10222},{1,3295,9957},{1,3576,999},{1,3601,3},{1,3612,56},{1,3746,3057},{1,5010,5902},{1,5119,2099},{1,5563,17609},{1,5697,11011},{1,7280,18398},{1,7962,10247},{1,7987,3333},{1,9819,16200},{1,10383,18443},{1,10389,6767},{1,13384,10246},{1,14793,5570},{1,14800,13751},{1,15803,11518},{1,21842,20323},{1,22766,16193},{1,22768,13373},{2,3488,5722},{3,57,5122},{3,1482,5709},{3,5708,15803},{3,10246,18443},{3,10247,2095},{3,15934,57},{3,18443,11227},{4,5703,11374},{8,6857,5791},{12,10543,10572},{12,10572,18480},{35,65,3579},{35,5425,484},{55,2099,5119},{55,5172,35},{56,3612,13624},{57,15934,942},{78,405,5044},{140,12433,1210},{388,4305,18481},{484,5425,65},{497,5886,7743},{498,1837,9956},{550,6147,4292},{936,5436,11108},{942,5122,57},{944,6847,5787},{946,4314,15171},{950,13411,5},{999,3576,5126},{1058,3616,11373},{1319,3748,1},{1479,11375,9955},{1621,4511,392},{1737,5432,11231},{2646,3748,1319},{3085,3486,355},{3485,4294,12699},{3576,6282,3},{3586,5219,381},{3601,13384,10383},{3649,15338,1770},{3868,4189,3916},{3940,16418,9},{4297,21620,18990},{4313,5703,4},{4420,5260,3697},{5010,5902,1155},{5045,13624,56},{5049,5126,999},{5119,13462,3428},{5248,22836,960},{5697,11011,11278},{5708,11518,942},{5720,6913,10157},{5731,10578,1056},{5761,6868,5812},{5901,15172,12053},{6261,11496,9856},{6284,12047,22793},{6767,10246,1},{6767,10383,942},{6909,18444,10167},{6935,7967,5768},{10165,11019,15325},{10386,22791,10624},{10389,13384,1},{11230,18527,11},{11518,15803,5708},{14547,22350,5396}
= (6-8-13) search numbers [2.16288896279710855, 2.08053054354282155, 1.20204073816376024]
Q(X(496)) =
X(24928) = MIDPOINT OF X(1) AND X(56)
Barycentrics a (2 a^3-a^2 b-2 a b^2+b^3-a^2 c+6 a b c-b^2 c-2 a c^2-b c^2+c^3) : :
= X[10]-2*X[6691], X[1837]-3*X[10072], X[4299]+X[12701], X[10090]+X[20586], X[10573]-3*X[17728]
= lies on the lines: {1,3},{4,4308},{5,10106},{7,10595},{8,17567},{10,6691},{11,18480},{12,11230},{21,15179},{30,4311},{37,5053},{58,18211},{72,17624},{104,1476},{145,5440},{210,5288},{214,3244},{219,1732},{226,5901},{355,3086},{376,9785},{381,9613},{382,9614},{388,5886},{392,2975},{404,10914},{405,11035},{452,3487},{474,3872},{495,1125},{496,515},{497,18481},{499,5252},{519,8256},{529,551},{550,10624},{938,6049},{944,5722},{946,1387},{952,1210},{956,5044},{958,12128},{960,8666},{997,12513},{998,3445},{1056,3436},{1058,5731},{1064,4322},{1066,1201},{1100,4266},{1108,1731},{1191,3157},{1193,5399},{1317,22935},{1386,8679},{1412,15952},{1475,17439},{1478,9955},{1519,22792},{1538,6256},{1621,10569},{1656,9578},{1657,9580},{1699,9655},{1706,16417},{1737,10944},{1743,22147},{1776,5887},{1828,1870},{1836,4317},{1837,10072},{1872,15500},{2257,20818},{2320,3296},{2649,9444},{2771,10074},{2841,11700},{3028,11699},{3035,10915},{3073,9363},{3218,5330},{3419,10529},{3485,6930},{3486,3655},{3555,4511},{3582,17606},{3585,12764},{3600,5603},{3623,11041},{3633,3689},{3636,12577},{3656,4295},{3680,17573},{3752,15854},{3753,4861},{3813,17647},{3871,4881},{3877,3916},{3884,4640},{3885,4188},{3898,5267},{3911,5690},{3927,15829},{3940,6762},{4002,17531},{4253,6603},{4292,22791},{4293,12699},{4297,15171},{4298,13464},{4299,12701},{4304,15172},{4314,15170},{4848,5844},{4853,9709},{5030,21872},{5083,19907},{5176,17619},{5248,10179},{5265,5657},{5270,17605},{5284,14150},{5290,9624},{5427,22937},{5433,10039},{5434,12047},{5435,12245},{5438,12629},{5533,18976},{5542,5625},{5550,8164},{5691,9669},{5727,18526},{5836,22837},{5882,11019},{6147,10283},{6284,21578},{6377,9434},{6735,13747},{6738,13607},{6797,12737},{6921,12648},{6948,12700},{6970,18391},{7320,10299},{7354,22793},{8227,9654},{8583,9708},{9259,16583},{9581,18525},{9612,18493},{9623,16408},{9856,12114},{10090,20586},{10572,18527},{10573,17728},{10593,19925},{11036,17576},{11037,11111},{11194,12514},{12573,20330},{12616,20418},{12650,19541},{13407,15950},{14100,16132},{15180,15446},{17100,17652},{17644,18908}
= complement of isogonal conjugate of X(15617)
= midpoint of X(i) and X/j) for these {i,j}: {1,56},{46,2098},{4299,12701},{4311,12053},{10074,12740},{10090,20586}
= reflection of X(i) in X(j) for these {i,j}: {10,6691},{1329,1125},{9957,20789}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,9957},{1,35,5919},{1,36,3057},{1,46,2098},{1,57,1482},{1,65,10222},{1,999,942},{1,1319,1385},{1,1388,15178},{1,1420,3},{1,3304,5045},{1,3337,11009},{1,3338,2099},{1,3339,16200},{1,3340,10247},{1,3361,7982},{1,3576,3295},{1,3601,6767},{1,3612,3303},{1,5563,65},{1,5902,11011},{1,5903,5048},{1,7373,5049},{1,13462,40},{1,15803,7962},{1,21842,2646},{3,1420,5126},{4,11373,7743},{36,3057,3579},{56,2098,46},{499,5252,9956},{938,6049,7967},{944,14986,5722},{946,4315,18990},{956,19861,5044},{1056,3616,11374},{1125,5795,17527},{1319,2646,21842},{1319,20323,1},{1387,18990,946},{1388,3304,1},{1478,11376,9955},{2646,21842,1385},{3086,3476,355},{3337,11009,65},{4861,5253,3753},{5045,9940,16218},{5045,15178,1},{5048,5903,11278},{5126,9957,3},{5433,10039,11231},{5563,11009,3337},{5708,10247,3340},{6583,11567,10222},{6583,15178,11567},{7373,10246,1},{7962,15803,12702}
= (6-8-13) search numbers [0.938713766322079298, 1.06969708255306711, 2.46685245568360374]
Q(X(1387)) = MIDPOINT OF X(1) AND X(1319)
Barycentrics a (4 a^3-3 a^2 b-4 a b^2+3 b^3-3 a^2 c+10 a b c-3 b^2 c-4 a c^2-3 b c^2+3 c^3) : :
= 2*X[1125]-X[5123], X[1317]+X[1737], X[1320]+3*X[4881], X[1785]+X[3319], 3*X[3582]+X[7972], 5*X[3616]-X[5176], 4*X[3636]-X[5087]
= lies on these lines: {1,3},{4,6049},{30,15368},{214,3880},{226,10283},{355,6981},{376,4345},{495,551},{496,5882},{497,3655},{515,1387},{519,3035},{912,19907},{944,5274},{1056,5080},{1125,5123},{1201,5399},{1210,1483},{1317,1737},{1318,1870},{1320,4881},{1391,4511},{1785,3319},{2718,8699},{3476,5886},{3582,7972},{3616,5176},{3622,6919},{3636,5087},{3653,5218},{3656,4293},{3897,17624},{3911,5844},{3916,5330},{4308,10595},{4311,22791},{4861,17614},{5083,14988},{5180,11037},{5252,11230},{5284,17615},{5326,10039},{5722,6969},{5901,10106},{6001,11715},{8609,17455},{9581,18526},{9613,18493},{9624,9654},{9655,11522},{9956,10944},{10786,14986},{11231,12647},{11376,18480},{11813,21620},{12577,16137},{13464,18990},{20586,22935}
= midpoint of X(i) and X/j) for these {i,j}: {1,1319},{36,5048},{1317,1737},{1785,3319},{12735,15325}
= reflection of X(i) in X(j) for these {i,j}: {5122,5126},{5123,1125},{5126,1319},{5570,5045},{7743,1387}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,36,5048},{1,56,10222},{1,57,10247},{1,1385,9957},{1,1388,1385},{1,1420,1482},{1,5563,11011},{1,13384,6767},{1,13462,16200},{1,20323,5045},{1,21842,3057},{36,9819,13528},{1319,5048,36},{1385,3660,5126},{3057,21842,13624},{5049,9957,12915},{6767,10246,13384},{7373,8171,999}
= (6-8-13) search numbers [0.623909685427857788, 0.809755123640411171, 2.79210646457431393]
Best regards
Ercole Suppa
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