[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point on the IN line.
Denote:
Lp = the perpendicular to IN at P
Lpa, Lpb, Lpc = the reflections of Lp in AI, BI, CI, resp.
A*B*C* = the triangle bounded by Lpa, Lpb, Lpc
ABC, A*B*C* are parallelogic.
The parallelogic center (ABC, A*B*C*) is a fixed point (on the circumcircle)
Which is the other parallelogic center (A*B*C*, ABC) for some simple points on the IN line: X(80)), X(11), X(5) ?
And which is its locus as P moves on the IN line?
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[Ercole Suppa]
*** The parallelogic center (ABC, A*B*C*) is X(901)
*** The locus of the parallelogic center Q(ABC,A*B*C*) is the trilenar polar of X(3257)
*** pares {P=X(i) ∈ IN, Q=X(j)} for these {i,j}: {1,1},{5,8715},{11,100},{12,3871},{80,5541},{952,2802},{1317,1320},{1387,214},{5252,3895},{5718,2177},{7972,12653},{10944,3885},{10950,14923},{10956,13278},{11376,4855},{12433,3754},{16173,15015},{17724,3722}
*** some points Q(X(i)) with X(i) ∈ IN
Q(X(119)) = MIDPOINT OF X(3189) AND X(12247)
= a (a^6-a^5 b-2 a^4 b^2+2 a^3 b^3+a^2 b^4-a b^5-a^5 c+a^4 b c+4 a^3 b^2 c-2 a^2 b^3 c-3 a b^4 c+b^5 c-2 a^4 c^2+4 a^3 b c^2-10 a^2 b^2 c^2+6 a b^3 c^2+2 a^3 c^3-2 a^2 b c^3+6 a b^2 c^3-2 b^3 c^3+a^2 c^4-3 a b c^4-a c^5+b c^5) : : (barys)
= 2*X[5]-X[13271], X[1768]+X[6765], X[2136]+X[6264], 3*X[3158]-X[6326], X[3189]+X[12247], X[10698]-2*X[22836], X[10742]-2*X[12607], 2*X[10915]-X[12751], 3*X[11236]-2*X[22799]
= lies on these lines: {1,88},{3,5854},{5,13271},{8,10058},{11,5687},{55,1145},{78,12758},{80,6735},{104,519},{119,381},{145,10074},{149,5154},{200,18254},{518,12515},{952,3913},{997,15558},{1317,1470},{1376,1387},{1519,14217},{1768,6765},{2136,6264},{2800,3811},{2801,2950},{2829,10306},{2900,12691},{3035,3295},{3158,6326},{3189,12247},{3359,15528},{3434,8068},{3689,17638},{3870,11570},{3880,12737},{3935,12532},{4421,10269},{5440,12740},{5531,7995},{5840,6256},{5853,10265},{6154,10956},{6224,12648},{6667,9709},{8069,18802},{10528,20095},{10609,11509},{10698,22836},{10742,12607},{10915,12751},{10942,13272},{11236,22799},{11362,17009},{12115,13199},{12514,14740},{12764,17757}
= midpoint of X(i) and X(j) for these {i,j}: {3189,12247}
= reflection of X(i) in X(j) for these {i,j}: {100,8715},{10698,22836},{10742,12607},{12751,10915},{13271,5}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {100,1320,10090},{100,3871,10087},{100,13278,1},{145,17100,10074},{5440,17652,12740},{10679,12331,119}
= (6-8-13) search numbers [-1.39749132198761125, 12.4526399608029297, -4.33539795773105932]
Q(X(495)) = MIDPOINT OF X(1) AND X(3895)
= a (a^3 - a b^2 - 4 a b c + b^2 c - a c^2 + b c^2) : : (barys)
= X[1478]-3*X[11239], 2*X[2886]-3*X[10197], X[3434]-2*X[3822], X[12703]+X[18446]
= lies on these lines: {1,88},{3,3244},{8,3746},{10,1001},{21,3632},{35,145},{36,3241},{40,3243},{41,1023},{46,3881},{55,519},{56,3635},{57,3892},{78,3884},{149,7951},{165,4973},{200,10176},{392,3689},{405,3626},{411,11531},{474,3636},{484,3873},{495,528},{496,6667},{497,3814},{515,10679},{535,4302},{551,1376},{595,2209},{643,9275},{758,3870},{942,15570},{943,12625},{946,18491},{950,10915},{958,3625},{997,3158},{999,4421},{1018,2280},{1125,3303},{1329,15172},{1478,11239},{1479,10528},{1482,6796},{1621,3679},{1697,3811},{1698,17546},{2077,7967},{2136,11525},{2241,20691},{2646,22837},{2886,10197},{2975,3633},{3035,10199},{3057,22836},{3058,17757},{3208,4251},{3240,5315},{3256,3476},{3293,3915},{3305,3956},{3336,3889},{3339,11526},{3421,10385},{3434,3822},{3436,4309},{3584,11680},{3617,5259},{3621,5258},{3623,5563},{3634,16856},{3656,18524},{3678,4917},{3748,3753},{3816,15170},{3825,5552},{3828,4423},{3833,4666},{3868,11010},{3919,15934},{3935,5692},{3938,4424},{3950,4254},{3957,5902},{4084,12702},{4125,4387},{4188,20057},{4189,5288},{4297,10306},{4301,11500},{4326,5223},{4428,4669},{4430,4880},{4649,5145},{4668,5260},{4757,11520},{4848,11510},{4857,11681},{5082,10198},{5123,18527},{5267,12513},{5276,9331},{5284,19875},{5290,16133},{5291,10987},{5310,20020},{5440,5919},{5450,11849},{5493,12631},{5537,5731},{5726,8543},{5727,15863},{5775,12630},{5882,11248},{5886,12331},{6049,13370},{6154,11112},{6681,10072},{6684,16202},{6905,16200},{7982,11491},{7991,12511},{8256,12433},{8668,12437},{9668,11236},{9709,19862},{10267,11362},{10483,20066},{10624,21077},{10902,12245},{10965,12053},{11499,12000},{12575,21616},{12607,15171},{12632,19843},{12703,18446},{15808,16408},{16474,17126},{16784,17756},{18483,18518}
= midpoint of X(i) and X(j) for these {i,j}: {1,3895},{1478,20075},{3870,5119},{10087,13278},{12703,18446}
= reflection of X(i) in X(j) for these {i,j}: {993,55},{3434,3822}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,3871,8715},{8,3746,5248},{35,145,8666},{1376,6767,551},{1697,3811,3878},{3295,3913,10},{3303,5687,1125},{3434,10056,3822},{3871,13278,3895},{3895,10087,8715},{4189,20050,5288},{4428,8168,9708},{8168,9708,4669},{11239,20075,1478},{11499,12000,13464}
= (6-8-13) search numbers [1.48833429080961504, 2.39427690260677202, 1.29616464588294538]
Q(X(496)) = MIDPOINT OF X(3) AND X(11499)
= a (a^3 - a b^2 + b^2 c - a c^2 + b c^2) : : (barys)
= X[46]+X[78], X[4311]+X[6736], X[12953]-3*X[17556]
= lies on these lines: {1,88},{2,35},{3,10},{4,2077},{5,3035},{6,9679},{8,36},{9,2173},{11,13747},{12,6174},{20,8165},{21,1698},{24,1861},{30,1329},{31,3216},{32,1575},{39,4386},{40,997},{41,16549},{43,58},{46,78},{55,474},{56,519},{57,3811},{63,3678},{65,5440},{72,1155},{75,1078},{80,17100},{81,5312},{99,6376},{101,3501},{104,5881},{140,2886},{145,5563},{165,411},{171,386},{182,17792},{183,20888},{187,1574},{191,3876},{198,17355},{200,4973},{210,3916},{224,17700},{226,7702},{228,3980},{238,17749},{318,4242},{333,4278},{376,2551},{377,498},{405,3634},{442,5432},{443,5218},{480,5850},{484,3869},{495,17563},{496,528},{497,10200},{499,3434},{516,3149},{517,6924},{535,3436},{548,9711},{549,4999},{550,3820},{551,3295},{574,1107},{595,978},{601,5150},{603,4551},{631,2550},{662,17104},{851,3454},{899,1724},{908,1770},{946,6911},{950,11502},{952,8256},{956,3626},{960,3579},{962,5537},{975,3743},{976,3670},{995,5255},{999,3244},{1001,16408},{1004,1259},{1010,4276},{1012,19925},{1018,9310},{1030,17303},{1145,10944},{1151,1377},{1152,1378},{1158,5720},{1193,5264},{1203,17126},{1210,8069},{1319,10914},{1385,5836},{1387,13463},{1402,17733},{1403,8669},{1444,17270},{1454,15556},{1465,4347},{1466,4298},{1468,3293},{1470,10106},{1478,4190},{1490,10270},{1532,11826},{1621,3624},{1697,3898},{1699,6915},{1706,3576},{1739,3924},{1740,4279},{1754,3682},{1768,12528},{1771,22350},{1816,2328},{1837,2932},{1975,6381},{2049,19760},{2057,20588},{2078,5082},{2178,2321},{2209,18792},{2222,2757},{2223,16825},{2241,16604},{2242,20691},{2276,5277},{2475,7951},{2478,4302},{2646,3753},{2801,17857},{2975,3679},{3008,21477},{3057,17614},{3061,5011},{3085,6904},{3086,17784},{3158,3333},{3218,4420},{3256,3485},{3303,3636},{3304,3635},{3336,3868},{3337,3873},{3338,3870},{3339,12559},{3359,6261},{3361,6765},{3416,5096},{3419,5172},{3430,20368},{3452,6985},{3523,15931},{3530,9710},{3555,3689},{3560,10175},{3583,4193},{3585,11681},{3612,3918},{3616,3746},{3617,5258},{3625,12513},{3661,19308},{3681,5131},{3684,4253},{3724,4647},{3741,4191},{3749,11512},{3752,5266},{3771,16056},{3788,20541},{3813,15325},{3816,15171},{3817,6918},{3826,6675},{3828,16370},{3831,13738},{3836,16415},{3840,16059},{3846,19543},{3877,11010},{3884,5119},{3911,10916},{3912,11329},{3923,19548},{3925,7483},{3927,4134},{3930,17736},{3938,3953},{3940,4067},{3941,4974},{3951,3988},{3984,4127},{4015,4652},{4023,15447},{4084,12635},{4090,20805},{4187,6284},{4189,5251},{4203,16569},{4251,17754},{4255,5711},{4257,5247},{4293,7080},{4301,10306},{4304,8582},{4306,9364},{4308,13370},{4311,6736},{4314,9843},{4324,11114},{4384,21495},{4423,16862},{4428,19883},{4511,5903},{4557,22458},{4640,5044},{4660,19513},{4669,11194},{4689,6051},{4745,19705},{4847,7742},{4861,4881},{5019,21857},{5030,21384},{5047,9342},{5080,10483},{5086,18395},{5087,22793},{5123,18480},{5124,17275},{5126,11260},{5205,7283},{5260,17549},{5280,17756},{5281,17580},{5284,17535},{5289,12702},{5293,17596},{5322,10327},{5493,6244},{5542,6600},{5554,14803},{5584,12447},{5587,6906},{5657,6942},{5691,6909},{5694,10225},{5705,6986},{5777,18232},{5818,6950},{5842,6922},{5880,11263},{5882,10269},{5886,11849},{6224,14800},{6256,6948},{6358,22342},{6516,9312},{6644,9712},{6667,10593},{6679,16298},{6685,11358},{6690,8728},{6702,10058},{6713,10943},{6835,12558},{6845,18406},{6876,7688},{6910,19854},{6912,7989},{6914,9956},{6946,7704},{6955,10786},{6959,10525},{7295,17353},{7330,15064},{7354,17757},{7411,8580},{7580,12512},{7786,20179},{7815,21264},{7824,17030},{7987,9623},{8053,16286},{8167,16863},{8668,21627},{8671,17793},{9367,16283},{9458,13589},{9655,11236},{9701,22115},{10039,14793},{10165,10267},{10479,16451},{10527,14798},{10609,10950},{10679,13464},{10948,15866},{11113,15338},{11249,11362},{11285,20172},{11347,20106},{11349,17284},{11373,13205},{11375,14882},{11507,12609},{11599,13173},{11680,17566},{11813,12699},{12178,21636},{12332,21635},{12436,13405},{12607,18990},{12629,13462},{12672,13528},{12688,17613},{12751,18861},{12953,17556},{13204,13605},{13607,16203},{14377,20335},{15326,21031},{15587,18233},{15625,16414},{15888,17583},{16062,19842},{16342,16828},{16347,19874},{16611,16968},{16819,17684},{16858,19876},{16865,19877},{17279,20872},{17281,19297},{17299,21773},{17308,21511},{17536,19872},{17750,18755},{17886,20926},{18492,21669},{19270,19858},{19278,19853},{20833,20989}
= midpoint of X(i) and X(j) for these {i,j}: {3,11499},{46,78},{56,5687},{100,10090},{3149,10310},{3436,4299},{4311,6736}
= reflection of X(i) in X(j) for these {i,j}: {496,6691},{1479,3825},{10199,17564},{12053,1125},{12059,3678},{21616,6700}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,100,8715},{1,5541,3885},{2,35,5248},{2,1479,3825},{3,10,993},{3,355,5450},{3,958,5267},{3,1376,10},{3,9709,958},{3,11500,4297},{3,18524,18481},{8,36,8666},{8,4188,36},{10,5267,958},{40,997,3878},{40,5438,997},{55,474,1125},{57,3811,3874},{65,5440,22836},{100,404,1},{100,5253,3871},{165,411,12511},{165,936,12514},{187,1574,4426},{377,498,3822},{404,3871,5253},{405,4413,3634},{443,5218,10198},{496,6691,10199},{496,17564,6691},{497,17567,10200},{499,6921,6681},{936,12514,10176},{956,19537,5204},{958,1376,9709},{958,5267,993},{958,9709,10},{975,17594,3743},{978,3550,595},{999,3913,3244},{1001,16408,19862},{1004,1259,4292},{1319,10914,22837},{1470,11501,10106},{1621,17531,3624},{1698,5010,21},{2975,13587,7280},{3218,4420,5904},{3338,3870,3881},{3434,6921,499},{3679,7280,2975},{3868,9352,3336},{3871,5253,1},{4189,9780,5251},{4190,5552,1478},{4292,6745,21077},{4413,5217,405},{4421,16417,551},{4423,16862,19878},{4861,4881,21842},{5119,19861,3884},{5657,6942,11012},{5687,16371,56},{5880,11374,11263},{6911,11248,946},{6918,11496,3817},{6940,11491,3576},{10306,22753,4301},{11681,17579,3585},{12512,20103,12572}
= (6-8-13) search numbers [2.01152949926159850, 0.570713020720732597, 2.31715723713390338]
Best regards
Ercole Suppa
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