[Antreas P. Hatzipolakis]:
4Yet Variation (3yet variation Hyacinthos 28477)
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.
D = the Poncelet point of ABCI = Feuerbach point X(11)
Da, Db, Dc = the antipodes of D in (Na), (Nb), (Nc), resp.
Which is the locus of P such that the reflections of PDa, PDb, PDc in AI, BI, CI, resp. are concurrent?
The IN line? And which is the locus of the point of concurrence? (the OI line?)
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[Ercole Suppa]
**** locus of point P such that the reflections of PDa, PDb, PDc in AI, BI, CI, resp. are concurrent : {line IN} U {c2=circunference with center X(119) and radius sqrt(R(R-2r))}
where R,r are the circumradius and the inradius of ABC resp.
** properties of 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}
-- if P ∈ IN the locus of point Q=Q(P) is the line IO, where O is the circumcenter of ABC
-- pares {P=X(i)∈IN, Q=X(j)} for these {i,j}: {1,1},{11,65},{952,1482},{1317,5048},{1387,999},{10956,3057},{11729,3},{12751,7982},{15017,7991},{16173,3338}
-- some points Q(X(i)):
Q(X(5)) = X(1)X(3) ∩ X(5)X(1537)
= -a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-3 a^4 b c+4 a^3 b^2 c+a^2 b^3 c-5 a b^4 c+2 b^5 c-a^4 c^2+4 a^3 b c^2-8 a^2 b^2 c^2+4 a b^3 c^2+b^4 c^2-2 a^3 c^3+a^2 b c^3+4 a b^2 c^3-4 b^3 c^3+2 a^2 c^4-5 a b c^4+b^2 c^4+a c^5+2 b c^5-c^6) : : (barys)
= 4*X[140]-3*X[3877], 3*X[381]-2*X[12672], 6*X[392]-7*X[3526], 2*X[1071]-X[18526], 2*X[1483]-X[3885], 5*X[1656]-6*X[3753], 3*X[3679]-2*X[5694], 4*X[3754]-3*X[5886], 5*X[3843]-4*X[9856], X[3869]-2*X[5690], 3*X[3919]-2*X[13464], 3*X[5790]-4*X[5836], 8*X[10107]-5*X[18493], X[12747]-2*X[17636], X[12773]-2*X[17654]
= lies on these lines: {1,3},{5,1537},{8,6923},{10,6980},{119,8256},{140,3877},{145,6948},{355,2800},{381,12672},{392,3526},{404,10698},{912,10914},{946,6971},{952,14923},{962,6928},{1071,18526},{1145,10942},{1483,3885},{1656,3753},{1837,10738},{2771,5881},{2802,5884},{3617,6982},{3656,10199},{3679,5694},{3754,5886},{3843,9856},{3868,5844},{3869,5690},{3919,13464},{5330,6940},{5450,12515},{5554,6929},{5603,6958},{5657,6863},{5790,5836},{5840,10950},{6001,18525},{6261,18524},{6797,9581},{6850,12245},{6868,20070},{6882,22791},{6961,10595},{7489,19860},{7741,12619},{10052,12647},{10107,18493},{10404,13375},{10525,10573},{10826,17638},{10948,12832},{11362,21077},{11373,12736},{11376,12758},{11729,13747},{12747,17636},{12773,17654}
= reflection of X(i) in X(j) for these {i,j}: {1482,65},{3869,5690},{3885,1483},{5697,1385},{5887,5836},{10247,10273},{10284,5885},{12645,10914},{12747,17636},{12773,17654},{18526,1071}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {40,7982,5538},{65,3057,5570},{65,20323,5902},{1482,5708,12001},{1482,12702,10306},{5836,5887,5790},{5885,7957,11849},{5885,10284,1},{8148,12001,1482},{11009,14800,1}
= (6-8-13) search numbers [4.46119850229238548, 3.97830481059493818, -1.17256123417783294]
Q(X(12)) = X(1)X(3) ∩ X(5)X(12758)
= a (b + c - a) (a^4 b - 2 a^2 b^3 + b^5 + a^4 c - 2 a^3 b c + 3 a^2 b^2 c + 3 a b^3 c - 3 b^4 c + 3 a^2 b c^2 - 6 a b^2 c^2 + 2 b^3 c^2 - 2 a^2 c^3 + 3 a b c^3 + 2 b^2 c^3 - 3 b c^4 + c^5) : : (barys)
= lies on these lines: {1,3},{5,12758},{11,17665},{12,1537},{149,1837},{355,17638},{404,12740},{1145,10958},{1317,5884},{1858,3880},{2800,10944},{2802,10950},{3754,15558},{3878,17757},{3884,5432},{3885,17637},{5836,11680},{6917,10043},{8275,12409},{10199,17622},{10573,10947},{10948,20118}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,5119,11849},{1,5697,10284},{55,5697,3057},{65,3057,5048},{999,5903,65},{1837,14923,17636},{5885,9957,1},{5919,13751,1}
= (6-8-13) search numbers [2.32185038816594536, 2.21178914789302291, 1.03780258498185000]
Q(X(80)) = X(1)X(3) ∩ X(8)X(6871)
= a (a^3 - 3 a^2 b - a b^2 + 3 b^3 - 3 a^2 c + 4 a b c - 3 b^2 c - a c^2 - 3 b c^2 + 3 c^3): : (barys)
= 2*X[226]-X[12647], 4*X[2886]-3*X[3679], 2*X[3419]-X[3632]
= lies on these lines: {1,3},{8,6871},{9,3899},{10,6933},{11,3656},{33,1845},{79,3633},{80,1537},{84,21398},{90,1389},{145,4295},{200,4867},{226,12647},{392,8167},{474,10107},{497,11041},{498,11362},{499,4848},{516,11526},{519,1478},{528,3243},{674,16496},{758,3872},{944,1770},{946,10573},{952,1836},{962,10572},{968,17461},{1000,3475},{1022,4905},{1056,11551},{1317,11246},{1320,3873},{1387,17728},{1479,4301},{1572,5332},{1698,5443},{1709,2800},{1723,1953},{1728,15556},{1737,5603},{1756,7174},{1788,10595},{1837,22791},{2161,16670},{2364,16548},{2802,3870},{2807,15430},{2886,3679},{2900,3880},{3085,4323},{3086,5734},{3158,5541},{3241,4293},{3244,4292},{3306,3919},{3419,3632},{3474,7967},{3485,8164},{3555,10912},{3583,5727},{3585,5881},{3624,5445},{3626,3984},{3635,4311},{3654,5432},{3655,15326},{3753,5289},{3754,19861},{3811,14923},{3877,5284},{3878,19860},{3898,4666},{3901,6762},{3922,16408},{4018,12513},{4084,22837},{4299,5882},{4305,20070},{4333,18481},{4338,7354},{4677,11525},{4845,10697},{4853,5904},{5252,5844},{5258,12526},{5274,18391},{5326,15950},{5441,8000},{5554,21616},{5559,5665},{5561,13143},{5587,18393},{5690,11375},{5691,7971},{5692,9623},{5730,5836},{5790,17605},{5837,19854},{5842,10483},{5854,12749},{6264,11571},{6834,15867},{6950,14497},{7160,15173},{7162,17097},{7672,12758},{7680,7741},{8168,10914},{8227,18395},{9814,15733},{9897,10742},{10390,13602},{10525,18962},{10950,12699},{10954,15908},{12650,15071},{12943,18499},{17606,18493}
= reflection of X(i) in X(j) for these {i,j}: {1,2099},{3632,3419},{5119,1},{7991,3428},{10679,10222},{12647,226}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,40,3612},{1,65,3338},{1,484,3576},{1,2093,36},{1,3336,1420},{1,3339,5563},{1,5010,13384},{1,5903,46},{1,7991,35},{1,11010,3601},{1,11280,7982},{1,11531,5697},{1,15803,21842},{1,18421,5902},{3,11011,1},{8,12047,10827},{36,2093,46},{36,5903,2093},{40,13384,5010},{55,5173,10980},{56,10222,1},{57,16200,1},{65,1482,1},{65,5048,999},{65,5570,5902},{65,11011,18967},{65,20323,5708},{942,2098,1},{942,11278,2098},{946,10573,10826},{999,1482,5048},{999,5048,1},{1319,10247,1},{1482,12001,10222},{3339,16189,1},{3340,7962,11529},{3340,7982,1},{3474,7967,21578},{3485,12245,10039},{4848,13464,499},{5010,13384,3612},{5903,11009,1},{5919,15934,1},{7962,11529,1},{7982,11529,7962},{10306,22766,35},{11224,18421,1}
= (6-8-13) search numbers [-0.200013303903268424, 0.129420373113879569, 3.64337959463012483]
** properties of circunference c2:
-- c2 pass through X(i) for these i: {1145, 1537}
-- if P ∈ c2 the locus of point Q=Q(P) is the circunference with center X(1482) and radius R-r
-- pares {P=X(i)∈IN, Q=X(j)} for these {i,j}: {1537, 1537}
-- some points Q(X(i)):
Q(X(1145)) = MIDPOINT OF X(80) AND X(3633)
= 4 a^4 - 6 a^3 b - 3 a^2 b^2 + 6 a b^3 - b^4 - 6 a^3 c + 18 a^2 b c - 8 a b^2 c - 3 a^2 c^2 - 8 a b c^2 + 2 b^2 c^2 + 6 a c^3 - c^4: : (barys)
= 2*X[1]-X[1145], X[8]-2*X[1387], X[72]-2*X[15558], X[80]+X[3633], X[119]-2*X[10222], X[214]-2*X[3635], 3*X[392]-2*X[14740], 2*X[3036]-X[3632], X[3625]-2*X[6702], 3*X[3679]-4*X[6667], 3*X[3753]-4*X[18240], 3*X[10031]-X[20095], 3*X[10247]-2*X[11729], X[10914]-2*X[12736], 2*X[11362]-3*X[21154], 2*X[12019]-X[12531], X[12630]+X[20119], X[12751]-3*X[16200]
= lies on these lines: {1,1145},{3,13278},{4,145},{8,1387},{11,519},{65,1317},{72,15558},{80,3633},{100,999},{104,10306},{119,10222},{214,3635},{392,14740},{517,3937},{518,12758},{528,3243},{1385,22082},{1392,11681},{1478,13271},{1483,3885},{2800,3555},{2829,7982},{3036,3632},{3338,5541},{3625,6702},{3679,6667},{3753,18240},{3811,12740},{3813,8068},{3870,6265},{3880,5570},{3913,10090},{4345,17556},{4999,5559},{5708,9945},{6049,19537},{7962,11113},{7993,12127},{9797,9803},{10031,20095},{10058,12513},{10074,13205},{10087,22560},{10247,11729},{10914,12736},{10956,11011},{11362,21154},{12001,12331},{12019,12531},{12630,20119},{12649,19914},{12751,16200},{21077,21630}
= midpoint of X(i) and X(j) for these {i,j}: {80,3633},{145,1320},{3555,17652},{7972,12653},{12531,20050},{12630,20119}
= reflection of X(i) in X(j) for these {i,j}: {8,1387},{72,15558},{100,12735},{119,10222},{1145,1},{1317,3244},{1537,1482},{3625,6702},{3632,3036},{10609,1317},{10914,12736},{12531,12019},{12732,10609},{13257,10698},{13996,214},{17757,5048}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {100,3241,12735},{3632,16173,3036}
= (6-8-13) search numbers [-0.174045727849851581, 3.18493848873259476, 1.51603586410020654]
Best regards
Ercole Suppa
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