[Antreas P. Hatzipolakis]:
Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of IΒC, ICA, IAB, resp.
A'B'C' = the pedal triangle of P wrt triangle NaNbNc.
(Oa), (Ob), (Oc) = the circles (A', A'Na), (B', B'Nb), (C', C'Nc), resp.
Ra = the radical axis of (Ob), (Oc)
Rb = the radical axis of (Oc), (Oa)
Rc = the radical axis of (Oa), (Ob)
Which is the locus of P such that the reflections of Ra, Rb, Rc in AI, BI, CI are concurrent?
Partly the IN line ( = Euler line of NaNbNc) ?
And which is the locus of the point of concurrence?
Partly the OI line?
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[Ercole Suppa]
*** the locus of P such that the reflections of Ra, Rb, Rc in AI, BI, CI are concurrent is: {line IN}, where I,N are the incenter and the NPC center of ABC
--- ETC points X(i) ∈ IN for these i: 1,5,11,12,80,119,355,495,496,952,1317,1387,1411,1421,1483,1484,1807,1837,2006,2594,2596,2606,3614,4551,5219,5252,5396,5399,5400,5443,5531,5533,5534,5587,5660,5718,5719,5720,5721,5722,5723,5724,5725,5726,5727,5881,5886,5901,6127,6264,6265,6326,7173,7741,7951,7958,7972,7988,7989,7993,8068,8070,8227,9578,9581,9624,9817,9897,10057,10073,10283,10523,10592,10593,10826,10827,10886,10887,10942,10943,10944,10948,10949,10950,10954,10955,10956,10957,10958,10959,11373,11374,11375,11376,11698,11729,12019,12025,12433,12550,12735,12737,12738,12739,12740,12749,12750,12751,13244,14204,14584,14679,15017,15251,15252,15253,15888,15935,15943,15950,16173,17602,17717,17718,17719,17720,17721,17722,17723,17724,17725,17726,17857,18357,19372,19907,20586,22392,23477,23513,23517,23708,24217,24222
*** for P ∈ IN let Q=Q(P) be the point of concurrence. The locus of Q is the line IO, where I,O are the incenter and the circumcenter of ABC
*** pairs {P=X(i) ∈ IN, Q=X(j)} for these {i,j}: {1,517},{5,3},{355,1385},{495,11249},{496,11248},{952,1},{1483,1482},{1484,11849},{5400,14131},{5534,5045},{5587,13624},{5719,5709},{5720,9940},{5881,15178},{5886,3579},{5901,40},{6326,5885},{10283,12702},{10942,56},{10943,55},{12737,10284},{17857,13373},{18357,3576},{19907,25413}
*** some points:
Q(X(11)) = COMPLEMENT OF X(10525)
= a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+2 a^3 b c+a^2 b^2 c-2 a b^3 c-2 a^3 c^2+a^2 b c^2+b^3 c^2+2 a^2 c^3-2 a b c^3+b^2 c^3+a c^4-c^5) : : (barys)
= 3*X[2]-X[10525], X[20]+X[10526], X[30]-X[6796], X[511]-X[5495], X[528]-X[10943], X[550]-X[5841], X[952]-X[5450], X[1158]-X[2771], X[2829]-X[10942], X[3189]+3*X[5770], 3*X[4421]+X[12114], X[5844]-X[8666], X[14988]-X[22836]
= lies on these lines: {1,3}, {2,10525}, {5,3035}, {8,6950}, {10,6914}, {12,24466}, {20,10526}, {21,25005}, {24,1872}, {30,6796}, {78,5694}, {100,355}, {104,3871}, {140,3816}, {404,5886}, {405,11231}, {474,11230},{496,6713}, {497,6961}, {498,6923}, {511,5495}, {528,10943}, {549,10199}, {550,5841}, {601,5396}, {603,5399}, {946,6924}, {952,5450}, {962,6942}, {993,5690}, {1012,11499}, {1030,1766}, {1158,2771}, {1376,3560}, {1479,6958}, {1483,25439}, {1490,12660}, {1538,3149}, {1621,6940}, {1698,7489}, {1837,10058}, {2550,6892}, {2829,10942}, {2932,4855}, {3085,6948}, {3189,5770}, {3434,6977}, {3474,5761}, {3526,5259}, {3583,6971}, {3651,5812}, {3654,17549}, {3656,13587}, {3811,12341}, {3885,12737}, {4188,5603}, {4189,5657}, {4276,15952}, {4294,6891}, {4302,6928}, {4421,12114}, {4640,14454}, {4848,17010}, {4996,14923}, {5218,6850}, {5225,6978}, {5250,19524}, {5267,11362}, {5310,16434}, {5432,6842}, {5440,5887}, {5552,6938}, {5587,13743}, {5687,22758}, {5691,18524}, {5777,11517}, {5844,8666}, {5881,12331}, {6265,17100}, {6284,6882}, {6831,18407}, {6847,18517}, {6876,9778}, {6905,12699}, {6909,11491}, {6911,9955}, {6921,10531}, {6952,13199}, {6966,12116}, {6972,20066}, {7491,15338}, {7701,13146}, {7741,10738}, {8553,21853}, {9817,13222}, {10090,11376}, {10785,20075}, {11929,12943}, {12528,12738}, {12611,12775}, {12645,18515}, {14988,22836}, {15171,15845}, {17662,18976}, {19525,19860}
= midpoint of X(i) and X(j) for these {i,j}: {3,11248}, {3811,24467}, {5450,8715}
= reflection of X1482) in X(11567)
= complement of X(10525)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,40,25413}, {3,55,1385}, {3,56,23961}, {3,1482,36}, {3,3295,10269}, {3,10267,13624}, {3,10306,11249}, {3,10310,3579}, {3,10679,56}, {3,10680,5204}, {3,11508,18857}, {3,11849,1}, {3,12702,11012}, {3,22765,7280}, {35,2077,3}, {40,5010,3}, {55,8071,9957}, {56,10679,10222}, {100,6906,355}, {1012,11499,18480}, {1376,3560,9956}, {1385,10222,25405}, {1385,10284,1}, {1470,11508,24928}, {3295,10269,15178}, {5217,10310,3}, {5432,11826,6842}, {5537,11012,12702}, {6909,11491,18481}, {6911,11496,9955}, {7280,7982,22765}, {8069,11509,942}, {10222,23961,56}, {11248,11249,10306}
= (6-8-13) search numbers [-4.38833141232552100, -3.32898337616424600, 7.97072900955602064]
Q(X(12)) = COMPLEMENT OF X(10526)
= a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+2 a^3 b c-a^2 b^2 c-2 a b^3 c+2 b^4 c-2 a^3 c^2-a^2 b c^2+4 a b^2 c^2-b^3 c^2+2 a^2 c^3-2 a b c^3-b^2 c^3+a c^4+2 b c^4-c^5) : : (barys)
= 3*X[2]-X[10526], X[20]+X[10525], X[30]-X[3829], X[529]-X[10942], X[952]-X[6796], X[2771]-X[6261], X[2818]-X[10282], X[5842]-X[10943], X[5844]-X[8715], X[6985]+X[12114], 3*X[11194]+X[11500]
= lies on these lines: {1,3}, {2,10526}, {5,993}, {8,6942}, {10,6924}, {11,7491}, {20,10525}, {21,5886}, {30,3829}, {48,5755}, {63,5694}, {78,22935}, {84,6597}, {104,411}, {140,25466}, {355,2975}, {378,1872}, {382,18515}, {388,6954}, {405,11230}, {474,11231}, {495,15865}, {499,6928}, {529,10942}, {548,12511}, {549,10197}, {550,1484}, {573,7113}, {946,5267}, {952,6796}, {956,11499}, {958,6911}, {962,6950}, {1006,5253}, {1012,22793}, {1193,5398}, {1437,3417}, {1468,5396}, {1478,6863}, {1656,5251}, {1699,13743}, {1766,5124}, {2551,6970}, {2771,6261}, {2818,10282}, {2915,8279}, {3086,6868}, {3149,18480}, {3218,21740}, {3436,6880}, {3560,9955}, {3583,15446}, {3585,6980}, {3616,6875}, {3632,12331}, {3653,21161}, {3654,13587}, {3656,17549}, {3869,4996}, {3916,5887}, {4188,5657}, {4189,5603}, {4278,15952}, {4293,6825}, {4299,6923}, {4973,5884}, {5080,6949}, {5248,5901}, {5250,19525}, {5258,5790}, {5260,6946}, {5265,6987}, {5288,12645}, {5303,6906}, {5322,19544}, {5428,11281}, {5433,6882}, {5690,25440}, {5731,6876}, {5842,10943}, {5844,8715}, {5881,18524}, {6326,6763}, {6713,6922}, {6827,7288}, {6842,7354}, {6848,18516}, {6881,24953}, {6883,25524}, {6885,19843}, {6910,10532}, {6934,10527}, {6960,20067}, {6962,12115}, {6985,12114}, {7489,8227}, {10058,12701}, {10090,12619}, {10786,20076}, {10913,18763}, {11194,11500}, {11483,11512}, {11928,12953}, {12053,17010}, {12515,18861}, {12556,12913}, {15326,15908}, {15844,18990}, {15888,21155}, {17734,19550}, {18761,19541}, {19524,19860}, {19861,21165}
= midpoint of X(i) and X(j) for these {i,j}: {3,11249}, {20,10525}, {6261,24467}, {6796,8666}, {6985,12114}, {11248,22770}
= complement of X(10526)
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,56,1385}, {3,999,10267}, {3,1482,35}, {3,3428,3579}, {3,5204,23961}, {3,10246,10902}, {3,10269,13624}, {3,10679,5217}, {3,10680,55}, {3,11849,5010}, {3,12702,2077}, {3,22765,1}, {3,22770,11248}, {36,11012,3}, {40,7280,3}, {55,10680,10222}, {56,5204,7742}, {56,7742,5126}, {56,8071,942}, {104,411,18481}, {484,11014,25413}, {946,5267,6914}, {958,6911,9956}, {999,10267,15178}, {2975,6905,355}, {3149,22758,18480}, {3428,5204,3}, {3560,22753,9955}, {3579,23961,3}, {5010,7982,11849}, {5433,11827,6882}, {5563,10902,10246}, {5901,7508,5248}, {8069,10966,9957}, {11248,11249,22770}, {13373,13624,1385}
= (6-8-13) search numbers [17.9530572007182121, 15.1188852296270523, -15.1122824620119857]
Q(X(80)) = X(1)X(3) ∩ X(5)X(214)
= -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+8 a^4 b c-3 a^3 b^2 c-6 a^2 b^3 c+6 a b^4 c-2 b^5 c-3 a^4 c^2-3 a^3 b c^2+10 a^2 b^2 c^2-3 a b^3 c^2-b^4 c^2+6 a^3 c^3-6 a^2 b c^3-3 a b^2 c^3+4 b^3 c^3+6 a b c^4-b^2 c^4-3 a c^5-2 b c^5+c^6) : : (barys)
= X[2771]-X[5450], X[5840]-X[5901]
= lies on these lines: {1,3}, {5,214}, {355,6224}, {631,2320}, {944,6972}, {1389,13587}, {1483,11715}, {2475,5886}, {2476,11230}, {2771,5450}, {3616,6951}, {3871,12737}, {4511,5694}, {5443,12119}, {5693,18515}, {5731,6903}, {5840,5901}, {6261,12524}, {6265,6906}, {6830,18480}, {6840,18481}, {10950,12619}, {11231,25005}, {18357,20400}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,35,25414}, {1,11849,10284}, {3,1482,484}, {3,5903,10225}, {1385,10222,1319}, {1385,24929,15178}, {6224,6952,355}, {10222,10225,5903}, {13624,15178,9940}
= (6-8-13) search numbers [-1.34901145143424392, -0.819337433353606423, 4.83051875950652690]
Q(X(119)) = MIDPOINT OF X(3) AND X(56)
= a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+4 a^3 b c-a^2 b^2 c-4 a b^3 c+2 b^4 c-2 a^3 c^2-a^2 b c^2+4 a b^2 c^2-b^3 c^2+2 a^2 c^3-4 a b c^3-b^2 c^3+a c^4+2 b c^4-c^5) : : (barys)
= X[5]-X[2829], X[30]-X[7681], X[182]-X[8679], X[496]-X[5840], X[515]-X[6924], X[529]-X[549], 5*X[631]-X[3436], X[952]-X[8256], X[971]-X[15297], X[1483]-X[5854], X[2390]-X[11202], 7*X[3523]+X[20076], X[4299]+X[6928], X[5428]-X[17768], X[5841]-X[6922], X[11499]-3*X[16371]
= lies on these lines: {1,3}, {4,10584}, {5,2829}, {12,21154}, {24,1828},{30,7681}, {104,355}, {119,13747}, {140,993}, {182,8679}, {214,5884}, {388,6961}, {474,9956}, {496,5840}, {499,6923}, {515,6924}, {529,549}, {631,3436}, {944,4188}, {952,8256}, {958,11231}, {971,15297}, {997,5694}, {1006,5303}, {1012,9955}, {1125,6914}, {1158,22775}, {1478,6958}, {1483,5854}, {1656,18515}, {1766,21773}, {1837,10090}, {2390,11202}, {2975,6940}, {3035,10942}, {3086,6948}, {3523,20076}, {3526,5251}, {3560,3824}, {3585,6971}, {3616,6950}, {3624,7489}, {3653,17549}, {3655,11491}, {4190,10785}, {4293,6891}, {4299,6928}, {4881,21740}, {5229,6978}, {5253,5886}, {5267,10165}, {5298,15908}, {5322,16434}, {5428,17768}, {5433,6842}, {5690,8666}, {5731,6942}, {5818,17572}, {5841,6922}, {5881,12773}, {5887,17614}, {6256,6959}, {6265,18861}, {6850,7288}, {6875,11415}, {6882,7354}, {6885,18517}, {6905,18481}, {6909,12699}, {6911,12114}, {6918,18761}, {6921,12115}, {6929,10200}, {6944,18516}, {6955,10527}, {6966,10532}, {6970,12667}, {6980,15446}, {7491,15326}, {7741,12764}, {8227,13743}, {8583,22936}, {8703,12511}, {9657,11929}, {10058,11376}, {10943,20418}, {11263,17009}, {11499,16371}, {12737,14923}, {15325,15866}, {19525,19861}, {22753,22793}, {22836,24475}
= midpoint of X(i) and X(j) for these {i,j}: {3,56}, {4299,6928}, {10310,10680}
= reflection of X(i) in X(j) for these {i,j}: {5,6691}, {1329,140}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,999,11248}, {3,1482,2077}, {3,10246,35}, {3,10269,1385}, {3,10680,10310}, {3,11249,3579}, {3,16202,5217}, {3,16203,55}, {3,22765,40}, {36,14800,1}, {55,16203,15178}, {56,8069,24928}, {56,10310,10680}, {104,404,355}, {474,22758,9956}, {997,24467,5694}, {999,11248,10222}, {1319,25414,1}, {1385,23961,3}, {1470,22766,942}, {2077,5563,1482}, {3086,6948,10525}, {3560,25524,11230}, {3576,7280,3}, {4293,6891,10526}, {5253,6906,5886}, {6911,12114,18480}, {8071,22768,24929}, {9940,13624,1385}, {13528,20323,23340}
= (6-8-13) search numbers [3.48474095869173549, 3.17201829119025209, -0.163690162158904110]
Best regards
Ercole Suppa
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