HYACINTHOS 28285

 

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

 

Which is the locus of P such that the reflections of PNa, PNb, PNc in AI, BI, CI, resp are concurrent?

The Euler line of NaNbNc?

 

[César Lozada]:

Locus: {Euler line of NaNbNc=Line IN of ABC} ∪ {circumcircle of NaNbNc=circle (N, NX(10)) of ABC }

 

IF P ∈ {Euler line of NaNbNc=Line IN of ABC} then the point of concurrence Q(P) ∈ X(1)X(3)-of-ABC = Line X(4)X(94) of NaNbNc.

For P such that IP/IN=t,

Q(P) = a*( (2*a^3-3*(b+c)*a^2-2*(b^2-3*b*c+c^2)*a+3*(b^2-c^2)*(b-c))*t+2*(-a+b+c)*(a-b+c)*(a+b-c)) : :  (barys)

 

and IQ/IO = (t/2)/(1-t)

 

ETC-pairs (P,Q(P)): (1,1), (5,517), (11,65), (12,3057), (80,11009), (355,1482), (495,9957), (496,942), (1484,6583), (1837,2099), (5219,1697), (5252,2098), (5443,35), (5587,7982), (5881,16200), (5886,3), (5901,1385), (7741,5903), (7951,5697), (7958,7957), (7988,7991), (7989,11531), (8227,40), (9578,7962), (9581,3340), (9624,3576), (10283,15178), (10886,12435), (10944,5048), (10948,5570), (10950,11011), (11373,999), (11374,3295), (11375,55), (11376,56), (15888,5919), (15950,2646), (16173,5563), (17718,3303), (17720,5710), (18357,11278), (19907,11567)

 

If P lies on the circumcircle of NaNbNc (through ETC’s centers of ABC 10, 502, 946, 11798, 13604, 15529), then Q(P) moves on the circle with radius=(R-2*r)/2 and center Oq given below. This circle passes through ETC’s 946, 3244

 

ETC pairs (P,Q(P)) = (10, 3244), (946,946)

 

Oq = midpoint of X(1) and X(1482)

= a*(2*a^3-3*(b+c)*a^2-2*(b^2-3*b*c+c^2)*a+3*(b^2-c^2)*(b-c)) : : (barys)

= 3*X(1)-X(3), 5*X(1)-X(40), 11*X(1)-3*X(165), 7*X(1)-3*X(3576), 4*X(1)-X(3579), 3*X(1)+X(7982), 13*X(1)-5*X(7987), 9*X(1)-X(7991), 5*X(1)+X(8148), 5*X(1)-3*X(10246), X(1)-3*X(10247), 5*X(1)+3*X(11224), 2*X(1)+X(11278), 7*X(1)+X(11531), 7*X(1)-X(12702), 5*X(1)-2*X(13624), 3*X(1)-2*X(15178), 3*X(1)+5*X(16189)

= on lines: {1, 3}, {4, 1392}, {5, 519}, {8, 3090}, {10, 3628}, {19, 23073}, {20, 3655}, {30, 4301}, {42, 19546}, {72, 1173}, {79, 14217}, {140, 551}, {145, 355}, {381, 5881}, {388, 10525}, {392, 5047}, {497, 10526}, {515, 1483}, {516, 13607}, {518, 576}, {546, 946}, {547, 4669}, {548, 5493}, {573, 3723}, {575, 1386}, {631, 3654}, {632, 1125}, {758, 11260}, {912, 15083}, {936, 11530}, {944, 3146}, {956, 3951}, {960, 18233}, {962, 3529}, {1000, 5703}, {1012, 11520}, {1056, 4323}, {1058, 4345}, {1210, 1387}, {1320, 1389}, {1457, 5399}, {1656, 3679}, {1657, 9589}, {1699, 18525}, {1766, 16884}, {1829, 10594}, {1837, 7743}, {1870, 1872}, {1953, 22356}, {2102, 15157}, {2103, 15156}, {2771, 7984}, {2800, 3881}, {2802, 19907}, {3058, 7491}, {3083, 21549}, {3084, 21546}, {3242, 11477}, {3243, 18761}, {3419, 6984}, {3434, 10597}, {3436, 10596}, {3485, 6982}, {3488, 5812}, {3518, 11363}, {3523, 3653}, {3525, 3616}, {3544, 20050}, {3555, 5887}, {3560, 12513}, {3577, 18491}, {3584, 5559}, {3585, 7972}, {3621, 5818}, {3622, 5657}, {3625, 10175}, {3632, 5079}, {3633, 5072}, {3636, 6684}, {3680, 6918}, {3751, 11482}, {3753, 17531}, {3811, 10912}, {3817, 12811}, {3827, 15581}, {3857, 19925}, {3874, 14988}, {3877, 16865}, {3880, 13374}, {3892, 5884}, {3893, 11524}, {3913, 6911}, {3915, 5398}, {3940, 4853}, {3957, 21740}, {3962, 5288}, {3991, 4919}, {4004, 5253}, {4297, 12103}, {4342, 15172}, {4511, 6946}, {4658, 15952}, {4663, 22330}, {4677, 5055}, {4691, 10172}, {4701, 10171}, {4745, 15699}, {4848, 15325}, {4864, 15310}, {4870, 6980}, {4930, 6913}, {5044, 5289}, {5054, 9588}, {5068, 20049}, {5070, 19875}, {5076, 5691}, {5198, 11396}, {5250, 19526}, {5258, 7489}, {5396, 19646}, {5440, 14923}, {5497, 10700}, {5609, 11699}, {5722, 5761}, {5727, 9669}, {5731, 17538}, {5763, 15935}, {5840, 12735}, {5853, 20330}, {5854, 10915}, {5855, 10916}, {6419, 7969}, {6420, 7968}, {6427, 18991}, {6428, 18992}, {6447, 9583}, {6519, 9616}, {6797, 12740}, {6833, 11240}, {6834, 11239}, {6842, 15888}, {6860, 12649}, {6863, 10056}, {6865, 15933}, {6914, 8666}, {6924, 8715}, {6958, 10072}, {6978, 11373}, {6988, 7320}, {7377, 17389}, {7419, 18180}, {7680, 10943}, {7681, 10942}, {7978, 15054}, {7983, 23235}, {9619, 22332}, {9708, 15829}, {10039, 15950}, {10165, 12108}, {10446, 17393}, {10573, 11376}, {10696, 19904}, {10895, 11928}, {10896, 11929}, {10944, 12047}, {11041, 14986}, {11272, 22475}, {11375, 12647}, {11496, 12559}, {11551, 11826}, {11552, 12119}, {11705, 20415}, {11706, 20416}, {11707, 21401}, {11708, 21402}, {11717, 13497}, {11724, 20399}, {11725, 20398}, {11728, 20401}, {11735, 20397}, {12053, 18527}, {12104, 22937}, {12331, 12653}, {12610, 17390}, {12773, 13253}, {12778, 15034}, {13211, 15027}, {13743, 16126}, {14563, 21625}, {16173, 19914}, {16239, 19883}, {16842, 19860}, {16862, 19861}, {17018, 19647}, {17438, 22357}, {17572, 17614}

= midpoint of X(i) and X(j) for these {i,j}: {1, 1482}, {145, 355}, {944, 12699}, {962, 18481}, {1320, 6265}, {1657, 9589}, {3555, 5887}, {3633, 12645}, {3811, 10912}, {11496, 12559}, {12331, 12653}, {12773, 13253}, {13743, 16126}

= reflection of X(i) in X(j) for these (i,j): (5, 13464), (8, 9956), (10, 5901)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7982, 3), (1, 8148, 13624), (1, 11224, 40), (1, 11278, 3579), (1, 11531, 3576), (40, 11224, 8148), (65, 11567, 1385), (1482, 8148, 11224), (1482, 10246, 8148), (1482, 10247, 1), (3576, 11531, 12702), (7373, 10306, 10269), (7982, 16189, 1482), (7982, 16200, 16189), (8148, 10246, 40), (10680, 12000, 55)

= [ -0.8557186829126229, -0.4120126991801520, 4.3208511272995400 ]

 

Some others Q(P):

Q( X(119) ) = midpoint of X(4) and X(3885)

= a*((b+c)*a^5-(b^2+6*b*c+c^2)*a^4-2*(b+c)*(b^2-5*b*c+c^2)*a^3+2*(b^4+c^4+2*(b^2-4*b*c+c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-8*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : : (barys)

= 4*X(1)-3*X(10202), 5*X(1)-4*X(13373), 7*X(1)-5*X(15016), 3*X(65)-4*X(6583), 3*X(392)-2*X(5690), 2*X(942)-3*X(10247), 4*X(942)-3*X(10273), 3*X(944)-X(9961), 8*X(3628)-7*X(4002), 3*X(3655)-2*X(9943), 3*X(3656)-2*X(7686), 5*X(3698)-6*X(11230), 3*X(3753)-4*X(5901), 3*X(3877)-X(12245), 5*X(3890)-3*X(5657)

= on lines: {1, 3}, {4, 3885}, {5, 6735}, {8, 6893}, {72, 5844}, {119, 946}, {145, 912}, {355, 3880}, {392, 5690}, {496, 17622}, {519, 5887}, {944, 9961}, {952, 12672}, {962, 12115}, {971, 18526}, {1000, 5555}, {1210, 15558}, {1320, 12775}, {1519, 10942}, {1699, 11929}, {1872, 1877}, {2136, 5720}, {2800, 3244}, {2950, 12773}, {3555, 14988}, {3585, 12749}, {3625, 20117}, {3628, 4002}, {3633, 5693}, {3655, 9943}, {3656, 7686}, {3698, 11230}, {3753, 5901}, {3869, 6930}, {3877, 5084}, {3878, 5795}, {3881, 15528}, {3884, 11362}, {3890, 5657}, {3898, 6684}, {4301, 12608}, {5053, 21853}, {5252, 10525}, {5439, 10283}, {5552, 5603}, {5587, 11928}, {5734, 6970}, {5761, 6848}, {5777, 12625}, {5836, 5886}, {6256, 12699}, {6827, 9785}, {6882, 12053}, {6923, 12700}, {6958, 11373}, {6971, 7743}, {7330, 12629}, {7491, 10624}, {7680, 13463}, {7967, 13369}, {7970, 13189}, {7978, 13217}, {7983, 12189}, {7984, 12381}, {9856, 18525}, {10526, 12701}, {10595, 17567}, {10698, 13278}, {10705, 13118}, {10738, 12751}, {10866, 18527}, {10912, 11496}, {12650, 12686}, {12705, 18519}, {13099, 13313}

= midpoint of X(i) and X(j) for these {i,j}: {4, 3885}, {3633, 5693}

= reflection of X(i) in X(j) for these (i,j): (1071, 1483), (3625, 20117), (7491, 10624)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 40, 10269), (1, 2077, 1385), (1, 3359, 16203), (1, 5903, 18838), (1, 11010, 14803), (1, 12703, 11248), (942, 9957, 20789), (946, 10915, 119), (1482, 10679, 1), (2098, 11509, 1), (2099, 10965, 1), (3746, 11014, 1385), (10596, 12245, 5554), (10942, 22791, 1519), (12702, 16203, 3359)

= [ -2.2411636793303000, -1.5560108497491040, 5.7522859991169750 ]

 

Q( X(13604) ) = X(35)X(60) ∩ X(758)X(3057)

= ((b-c)^2*a^6+(b+c)*b*c*a^5-(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^4-(b+c)*(3*b^2-4*b*c+3*c^2)*b*c*a^3+(3*b^4+3*c^4+(7*b^2+9*b*c+7*c^2)*b*c)*(b-c)^2*a^2+(b+c)*(2*b^4+2*c^4-(2*b^2-b*c+2*c^2)*b*c)*b*c*a-(b^2-c^2)^2*(b^4+c^4+(b^2+b*c+c^2)*b*c))*a^2 : : (barys)

= on lines: {35, 60}, {758, 3057}, {946, 6003}, {5563, 6584}, {8674, 13604}, {11009, 23153}

= [ -1.7062356821329990, 0.7822171051486402, 3.8866229547120880 ]

 

César Lozada

 

Note: I made some attempts for taking NaNbNc as the reference triangle. In fact, ABC can be obtained from NaNbNc by drawing two centers and a pair of lines in this latter. Unfortunately, the resulting algebraic expressions for A,B,C are not nice to work with.