[Antreas P. Hatzipolakis]:
Another VARIATION (of Hyacinthos 28285, 28430. Variation: Hyacinthos 28433):
Let ABC be a triangle, A'B'C' the pedal triangle of I and P a point.
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
MaMbMc = the medial triangle of NaNbNc
La =: MaP, Lb =: MbP, Lc =: McP
L1, L2, L3 = the reflections of La, Lb, Lc in IA, IB, IC, resp.
Which is the locus of P such that: L1, L2, L3 are concurrent ?
Is the locus the Euler line of NaNbNc [ = IN line of ABC]?
And which is the locus of the point of concurrence as P moves on the locus (is it the OI line?) ?
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[Ercole Suppa]
*** locus of point P such that L1, L2, L3 are concurrent is {line IN} U {c1=circunference with center X(5901) and radius (1/4)*sqrt(R(R-2r))} , where R,r are the circumradius and the inradius of ABC resp.
c1: ∑ [(a^3+3 a^2 b-a b^2-3 b^3+3 a^2 c+3 a b c-9 b^2 c-a c^2-9 b c^2-3 c^3) x^2+2 (5 a^3+3 a^2 b-3 a b^2-b^3+3 a^2 c+3 a b c+b^2 c-3 a c^2+b c^2-c^3) y z] = 0
c1: pass through points X(i) for these i: {1125,13464,15308}
*** The locus of the point of concurrence Q=Q(P):
-- if P ∈ IN then Q(P) ∈ IO
-- if P ∈ c1 then Q(P) ∈ c2=circunference with center X(15178) and radius (1/4)(R-2r)
*** pairs (P,Q(P))
-- pairs (P ∈ IN, Q(P)) : {1,1},{5,1385},{11,2646},{12,1319},{355,10246},{952,15178},{1387,9957},{5219,1420},{5252,1388},{5443,36},{5719,5045},{5886,3},{7951,21842},{8227,3576},{9581,13384},{9624,40},{10283,10222},{11373,3295},{11374,999},{11375,56},{11376,55},{15888,20323},{15950,65},{16173,3746},{17718,3304}
-- pairs (P ∈ c1, Q(P)) : {1125, 1125}, {13464, 13607}
*** some points Q=Q(P) :
P1= Q(X(80)) = X(1)X(3) ∩ X(10)X(5444)
Barycentrics a (3 a^3-2 a^2 b-3 a b^2+2 b^3-2 a^2 c+3 a b c-2 b^2 c-3 a c^2-2 b c^2+2 c^3) : :
= lies on these lines : {1,3},{10,5444},{79,21578},{80,1125},{214,3918},{498,7967},{515,5443},{551,10572},{944,7951},{950,16173},{1387,10543},{1476,5424},{1479,3622},{1483,5432},{2320,5248},{2975,4067},{3583,5901},{3584,10944},{3585,15950},{3616,3822},{3636,5441},{3655,11375},{3678,4511},{3723,5356},{3754,4881},{3897,5251},{3901,11194},{4188,21398},{4302,10595},{4324,22791},{4848,5442},{5288,22836},{5445,10165},{5494,6126},{5557,12563},{5727,15079},{5731,10483},{5836,15015},{6284,10283},{7294,11545},{7972,10039},{9897,9956},{11715,12691},{18393,18481},{18493,18514}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3,11009},{1,56,5425},{1,484,11011},{1,1385,36},{1,1420,18398},{1,2646,3746},{1,3576,5903},{1,3612,5697},{1,5010,1482},{1,7280,2099},{1,11010,10222},{1,21842,5563},{3,11009,3245},{1385,11567,3},{1420,18398,5563},{2646,15178,1},{3612,5697,35},{3746,14803,35},{10039,13607,7972},{11011,13624,484},{18398,21842,1420}
= (6-8-13) search numbers [2.67072812625401267, 2.49986661627279633, 0.677343843139821950]
P2=Q(X(119)) = X(1)X(3) ∩ X(119)X(1125)
Barycentrics a (2 a^6-3 a^5 b-3 a^4 b^2+6 a^3 b^3-3 a b^5+b^6-3 a^5 c+14 a^4 b c-8 a^3 b^2 c-12 a^2 b^3 c+11 a b^4 c-2 b^5 c-3 a^4 c^2-8 a^3 b c^2+20 a^2 b^2 c^2-8 a b^3 c^2-b^4 c^2+6 a^3 c^3-12 a^2 b c^3-8 a b^2 c^3+4 b^3 c^3+11 a b c^4-b^2 c^4-3 a c^5-2 b c^5+c^6) : :
= lies on these lines : {1,3},{119,1125},{140,6735},{214,13607},{355,10200},{551,12608},{631,12648},{944,6944},{952,17614},{1483,5440},{1519,5901},{2320,5553},{3616,6893},{3897,5084},{4308,6827},{4311,7491},{5554,7967},{5731,10531},{5777,12773},{5836,12737},{5886,6256},{6265,12675},{6713,10039},{6848,10586},{6882,10106},{6923,11373},{9956,12751},{10165,10915},{10942,17527}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3359,1482},{1,3576,11248},{1,5193,942},{1,7987,12703},{1,16203,10202},{1,16209,7982},{1385,15178,2646},{1388,22768,1},{10246,16203,1}
= (6-8-13) search numbers [3.23560160941711564, 2.96629741772282740, 0.0937193729837529161]
P3=Q(X(495)) = MIDPOINT OF X(1) AND X(56)
= X[10]-2*X[6691], X[1837]-3*X[10072], X[4299]+X[12701], X[10090]+X[20586], X[10573]-3*X[17728]
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) : :
= lies on these 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}
= 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}
= complement of isogonal conjugate of X15617
= {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]
P4=Q(X(496)) = MIDPOINT OF X(1) AND X(55)
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}
= 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}
= complement of X3419
= {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]
**** Note: sampled points X(i) with 1 <= i <= 23110
Best regards
Ercole Suppa
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