HYACINTHOS 26257

[Telv Cohl] (*)

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the reflections of N in BC, CA, AB, resp.

D = AO /\ NbNc
E = BO /\ NcNa
F = CO /\ NaNb

1. NaNbNc, DEF are perspective at T.

2. The isogonal conjugate of T wrt triangle NaNbNc lies on the Euler line of ABC

(*) Isogonal conjugate of Perspector lies on Euler line

[Peter Moses]:

Hi Antreas,

 

1).

a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+2 a^4 b^2 c^2-4 a^2 b^4 c^2+b^6 c^2-3 a^4 c^4-4 a^2 b^2 c^4+3 a^2 c^6+b^2 c^6-c^8):: 

on lines {{2,5876},{3,54},{4,3521},{5, 113},{6,12084},{20,568},{26, 9786},{30,143},{49,1511},{51, 3627},{52,550},{74,13434},{ 140,9729},{156,1181},{182, 7689},{186,5944},{376,6243},{ 381,6241},{382,3567},{511,548} ,{546,5462},{547,11695},{549, 5562},{567,1986},{569,1204},{ 575,2781},{578,11250},{632, 5891},{973,11750},{1199,2071}, {1216,3530},{1656,12111},{ 1657,3060},{1658,11438},{1853, 7564},{2779,5885},{2807,5901}, {3448,6288},{3526,11459},{ 3528,13340},{3581,7512},{3628, 5892},{3819,12108},{3830,9781} ,{3843,5640},{3845,11381},{ 3850,5943},{3853,10110},{3861, 13474},{3917,11592},{5054, 11444},{5072,11451},{5073, 13321},{5079,11465},{5447, 12100},{5498,6699},{5878,7729} ,{5899,8718},{6143,7722},{ 6240,6746},{6688,12812},{6759, 12106},{7502,10984},{7506, 11456},{7514,12163},{7526, 10605},{7529,12174},{8254, 10628},{8548,12301},{8703, 10625},{9704,11449},{9705, 12284},{10116,11802},{10226, 11430},{11245,12370},{11432, 12085},{12022,12236},{12233, 13371},{12254,13368}}.

complement X(5876).

midpoint of X(i) and X(j) for these {i,j}: {{3, 6102}, {4, 13491}, {5, 185}, {20, 10263}, {52, 550}, {1986, 12041}, {3627, 10575}, {5889, 6101}, {9729, 13382}, {10264, 11562}, {12254, 13368}}.

reflection of X(i) in X(j) for these {i,j}: {{4, 10095}, {5, 12006}, {140, 9729}, {143, 389}, {546, 5462}, {1216, 3530}, {3853, 10110}, {5907, 3628}, {10627, 3}, {11591, 140}, {13363, 9730}, {13421, 52}, {13474, 3861}}.

7 X[5] - 9 X[373], 7 X[185] + 9 X[373], X[20] + 3 X[568], 7 X[3] - 3 X[2979], X[1657] + 3 X[3060], X[382] - 5 X[3567], 3 X[51] - X[3627], 3 X[143] - 2 X[5446], 3 X[389] - X[5446], 3 X[549] - X[5562], 5 X[3843] - 9 X[5640], 3 X[3] + X[5889], 9 X[2979] + 7 X[5889], X[5889] - 9 X[5890], X[3] + 3 X[5890], X[2979] + 7 X[5890], 5 X[632] - 3 X[5891], 2 X[3628] - 3 X[5892], 3 X[5892] - X[5907], 2 X[3850] - 3 X[5943], X[4] - 3 X[5946], 9 X[2979] - 7 X[6101], 3 X[3] - X[6101], 9 X[5890] + X[6101], X[5889] - 3 X[6102], 3 X[5890] - X[6102], X[6101] + 3 X[6102], 3 X[2979] + 7 X[6102], 3 X[381] + X[6241], 3 X[376] + X[6243], X[5878] + 3 X[7729], 3 X[373] - 7 X[9730], X[5] - 3 X[9730], X[185] + 3 X[9730], 3 X[3830] - 7 X[9781], 3 X[5946] - 2 X[10095], 3 X[568] - X[10263], X[6101] - 15 X[10574], X[3] - 5 X[10574], 3 X[5890] + 5 X[10574], X[6102] + 5 X[10574], X[5889] + 15 X[10574], 3 X[51] + X[10575], 3 X[8703] - X[10625], 6 X[2979] - 7 X[10627], 2 X[6101] - 3 X[10627], 10 X[10574] - X[10627], 6 X[5890] + X[10627], 2 X[6102] + X[10627], 2 X[5889] + 3 X[10627], 5 X[5] - 4 X[11017], 15 X[9730] - 4 X[11017], 5 X[185] + 4 X[11017], 3 X[3845] - X[11381], 15 X[2979] - 7 X[11412], 5 X[6101] - 3 X[11412], 5 X[10627] - 2 X[11412], 5 X[3] - X[11412], 15 X[5890] + X[11412], 5 X[6102] + X[11412], 5 X[5889] + 3 X[11412], 9 X[381] - 5 X[11439], 3 X[6241] + 5 X[11439], 9 X[5054] - 5 X[11444], 11 X[5072] - 15 X[11451], 7 X[3526] - 3 X[11459], 13 X[5079] - 17 X[11465], 2 X[974] + X[11561], X[6240] + 2 X[11565], 4 X[9729] - X[11591], 3 X[3917] - 4 X[11592], 3 X[547] - 4 X[11695], 3 X[11591] - 4 X[11793], 3 X[140] - 2 X[11793], 3 X[9729] - X[11793], 9 X[373] - 14 X[12006], 2 X[11017] - 5 X[12006], 3 X[9730] - 2 X[12006], X[185] + 2 X[12006], 7 X[11017] - 10 X[12046], 7 X[5] - 8 X[12046], 9 X[373] - 8 X[12046], 7 X[12006] - 4 X[12046], 7 X[185] + 8 X[12046], 2 X[5447] - 3 X[12100], 3 X[3819] - 4 X[12108], 5 X[1656] - X[12111], 12 X[11017] - 5 X[12162], 3 X[5] - X[12162], 9 X[9730] - X[12162], 6 X[12006] - X[12162], 3 X[185] + X[12162], 3 X[3830] + X[12279], 7 X[9781] + X[12279], 5 X[3843] - X[12290], 9 X[5640] - X[12290], 3 X[11245] - X[12370], 3 X[10610] - X[12606], 6 X[6688] - 5 X[12812], X[5073] - 9 X[13321], 7 X[3528] - 3 X[13340], 8 X[11017] - 15 X[13363], 2 X[12162] - 9 X[13363], 6 X[373] - 7 X[13363], 2 X[5] - 3 X[13363], 4 X[12006] - 3 X[13363], 2 X[185] + 3 X[13363], 2 X[546] - 3 X[13364], 4 X[5462] - 3 X[13364], X[140] + 2 X[13382], X[11793] + 3 X[13382], X[11591] + 4 X[13382], 2 X[550] + X[13421], 3 X[5946] + X[13491], 2 X[10095] + X[13491], X[13470] + 2 X[13568], 5 X[5446] - 3 X[13598], 5 X[143] - 2 X[13598], 5 X[389] - X[13598].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5012,10610),(3,5889,6101),( 3,5890,6102),(4,5946,10095),( 5,373,12046),(5,9730,12006),( 5,12006,13363),(20,568,10263), (51,10575,3627),(185,9730,5),( 546,5462,13364),(1181,6644, 156),(5640,12290,3843),(5890, 10574,3),(5892,5907,3628),( 5946,13491,4),(6101,6102,5889) ,(9781,12279,3830).

crosssum of X(3) and X(5448).

Searches: {6. 13399365580523289611373458561, 6. 13787064942937559990864517617, -3. 43970457806914226425550675129} .

 

2). 

(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^12-10 a^10 b^2+9 a^8 b^4+4 a^6 b^6-11 a^4 b^8+6 a^2 b^10-b^12-10 a^10 c^2+16 a^8 b^2 c^2-6 a^6 b^4 c^2+10 a^4 b^6 c^2-16 a^2 b^8 c^2+6 b^10 c^2+9 a^8 c^4-6 a^6 b^2 c^4+5 a^4 b^4 c^4+10 a^2 b^6 c^4-15 b^8 c^4+4 a^6 c^6+10 a^4 b^2 c^6+10 a^2 b^4 c^6+20 b^6 c^6-11 a^4 c^8-16 a^2 b^2 c^8-15 b^4 c^8+6 a^2 c^10+6 b^2 c^10-c^12):: 

on lines {{2,3},{143,1263}}.

reflection of X(5) in X(13362).

Searches: {2. 76319663981997650414732090399, 1. 88638733441670154950994035753, 1. 05938249354821195154347982237} .

 

Best regards,

Peter Moses.