HYACINTHOS 29622

[Kadir Atintas]:

 

Let ABC be a triangle, P be a point and A'B'C' the pedal triangle of P.

Denote:

Oa, Ob, Oc = the circumcenters of PBC, PCA, PAB, resp.
Ga, Gb, Gc = the  centroids of PObOc, POcOa, POaOb, resp.

Which is the locus of P such that A'B'C' and GaGbGc are perspective?  

Some perspectors?

 

[César Lozada]:

 

Locus = {Linf} ∪ {circumcircle} ∪ {q6=excentral-circum-degree-6 through ETCs 1, 3, 4}

 

q6: ∑ [ y*z*(-2*(b^2-c^2)*b^2*c^2*x^4+(-c^2*(3*a^4-2*(b^2+3*c^2)*a^2+b^4-2*b^2*c^2+3*c^4)*y+b^2*(3*a^4-2*(3*b^2+c^2)*a^2+3*b^4-2*b^2*c^2+c^4)*z)*x^3+2*a^4*y^3*z*c^2+2*(b^2-c^2)*a^4*y^2*z^2-2*a^4*y*z^3*b^2)] = 0

 

ETC-pairs (P,Q(P)=perspector): {4, 4}, {74, 15055}, {99, 21166}, {107, 23239}, {110, 15035}

 

If P lies on the circumcircle then OQ=(1/3)*OP, ie, Q lies on the circle (O, R/3).

 

Q( X(1) ) = MIDPOINT OF X(1) AND X(3612)

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

= 2*X(1)+X(5217)

= lies on these lines: {1, 3}, {2, 10950}, {4, 15950}, {6, 17440}, {8, 4999}, {11, 2476}, {12, 944}, {21, 2320}, {33, 4214}, {37, 2261}, {45, 22356}, {78, 3711}, {80, 1656}, {140, 10573}, {145, 5218}, {214, 474}, {226, 9657}, {244, 8572}, {381, 5443}, {382, 18393}, {388, 6840}, {390, 25557}, {392, 1858}, {405, 30144}, {442, 26475}, {497, 2475}, {498, 952}, {515, 10895}, {551, 950}, {632, 11545}, {946, 12953}, {956, 22836}, {958, 3715}, {962, 15338}, {993, 5730}, {1001, 10394}, {1056, 6903}, {1058, 6951}, {1125, 1837}, {1201, 14547}, {1317, 12247}, {1329, 10955}, {1387, 13274}, {1389, 6942}, {1437, 4653}, {1468, 2361}, {1479, 5901}, {1483, 12647}, {1486, 18614}, {1831, 17523}, {1836, 4297}, {1854, 10535}, {1864, 5436}, {2170, 4258}, {2256, 17438}, {2268, 4287}, {2269, 5036}, {2293, 19945}, {2330, 3242}, {2886, 10959}, {2975, 12635}, {3035, 5554}, {3058, 4313}, {3085, 6952}, {3086, 6853}, {3158, 3893}, {3207, 17451}, {3241, 4995}, {3243, 15837}, {3474, 4323}, {3475, 4308}, {3476, 5703}, {3485, 5731}, {3487, 5434}, {3488, 6937}, {3526, 5444}, {3560, 6265}, {3583, 18493}, {3586, 9624}, {3623, 5281}, {3624, 5727}, {3636, 12053}, {3649, 4293}, {3655, 11237}, {3683, 15829}, {3689, 4853}, {3698, 5438}, {3754, 16371}, {3816, 10958}, {3868, 11194}, {3870, 11260}, {3871, 10912}, {3878, 16370}, {3890, 4428}, {3895, 33895}, {3913, 4861}, {3927, 4867}, {3940, 5258}, {4294, 10595}, {4295, 15326}, {4302, 22791}, {4304, 12701}, {4305, 5603}, {4311, 10404}, {4317, 6147}, {4325, 18541}, {4413, 19860}, {4421, 14923}, {4423, 19861}, {4640, 11682}, {4855, 5836}, {4863, 12437}, {4870, 9612}, {5054, 5445}, {5252, 5882}, {5283, 11998}, {5326, 9780}, {5426, 17637}, {5433, 18391}, {5441, 9668}, {5499, 15174}, {5592, 23761}, {5691, 17605}, {5736, 17221}, {5794, 24541}, {5818, 20400}, {5886, 10572}, {6049, 10578}, {6738, 17728}, {6827, 18962}, {6882, 10954}, {7052, 22236}, {7082, 31435}, {7221, 11997}, {7770, 30140}, {7866, 30120}, {7951, 18525}, {7968, 19038}, {7969, 19037}, {7983, 15452}, {9581, 25055}, {9670, 30384}, {9673, 11365}, {9844, 10393}, {10058, 19907}, {10072, 12433}, {10106, 17718}, {10165, 24914}, {10200, 34123}, {10283, 15171}, {10592, 28224}, {10826, 11230}, {10827, 28204}, {11285, 30136}, {11502, 25524}, {11715, 12739}, {11723, 12374}, {11724, 12185}, {11725, 13183}, {11729, 12764}, {11735, 12904}, {12047, 12943}, {12114, 21740}, {12743, 16173}, {13463, 20075}, {13607, 31397}, {13901, 19066}, {13902, 19030}, {13958, 19065}, {13959, 19029}, {15015, 17636}, {15228, 15696}, {17044, 26101}, {17662, 31480}, {18526, 31479}, {21031, 27383}, {21677, 30478}, {22238, 33655}, {23846, 28348}, {24558, 26105}, {28922, 30847}, {28924, 30826}, {30124, 32954}, {31165, 31424}

= midpoint of X(1) and X(3612)

= reflection of X(i) in X(j) for these (i,j): (5217, 3612), (10895, 11375)

= X(3612)-of-anti-Aquila triangle

= X(5217)-of-Mandart-incircle triangle

= X(7547)-of-2nd circumperp triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 3304, 10966), (65, 3576, 5204), (999, 7742, 56), (3057, 17609, 18839), (3338, 5126, 56), (3340, 7987, 1155), (3576, 16193, 56), (5010, 11009, 12702), (8071, 16203, 56), (10267, 22766, 5172), (16193, 31786, 65)

= [ 2.3795882908092240, 2.2594648434366430, 0.9781480714624456 ]

 

Q( X(3) ) = X(5)X(11202) ∩ X(6)X(3515)

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

= (SB+SC)*(16*R^2+SA-5*SW)*(S^2+(16*R^2+SA-6*SW)*SA) : : (barys)

= lies on these lines: {5, 11202}, {6, 3515}, {1147, 18324}, {1493, 23358}, {3292, 22333}, {4550, 10282}, {11821, 15035}, {17821, 33537}

= [ 3.5496134467512130, 2.0629597352178810, 0.5741784590252865 ]

 

Q( X(98) ) = X(3)X(76) ∩ X(20)X(115)

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

= 2*X(3)+X(98), 4*X(3)-X(99), X(3)+2*X(12042), 5*X(3)+X(12188), 7*X(3)-X(13188), 10*X(3)-X(23235), 5*X(3)-2*X(33813), X(4)-4*X(6036), 2*X(4)-5*X(14061), 2*X(98)+X(99), X(98)-4*X(12042), 5*X(98)-2*X(12188), 7*X(98)+2*X(13188), 5*X(98)+X(23235), 5*X(98)+4*X(33813), X(99)+8*X(12042), 5*X(99)+4*X(12188), 7*X(99)-4*X(13188), 5*X(99)-2*X(23235), 8*X(6036)-5*X(14061), 5*X(14061)-4*X(23514)

= lies on these lines: {2, 2794}, {3, 76}, {4, 6036}, {5, 10722}, {20, 115}, {30, 9166}, {35, 10069}, {36, 10053}, {40, 7983}, {74, 15342}, {83, 13335}, {114, 631}, {140, 6033}, {147, 620}, {148, 3522}, {182, 10753}, {186, 30716}, {187, 5999}, {262, 12150}, {315, 8781}, {371, 19055}, {372, 19056}, {376, 671}, {381, 34127}, {385, 18860}, {511, 21445}, {542, 3524}, {543, 10304}, {549, 6054}, {550, 6321}, {648, 14060}, {690, 15055}, {962, 11725}, {1003, 9756}, {1092, 3044}, {1151, 19109}, {1152, 19108}, {1350, 10754}, {1352, 7835}, {1385, 7970}, {1569, 15515}, {1587, 8980}, {1588, 13967}, {1656, 22505}, {1657, 22515}, {1916, 5188}, {2023, 3053}, {2077, 13189}, {2407, 13479}, {2482, 11177}, {2784, 10164}, {2790, 20792}, {3023, 5204}, {3027, 5217}, {3091, 6722}, {3515, 12131}, {3516, 5186}, {3525, 6721}, {3528, 13172}, {3529, 20398}, {3543, 5461}, {3564, 7799}, {3785, 32458}, {3788, 9863}, {3839, 14971}, {3843, 15092}, {3972, 13860}, {4027, 33004}, {4188, 5985}, {4297, 13178}, {5010, 10086}, {5013, 12829}, {5054, 23234}, {5055, 26614}, {5085, 5182}, {5092, 12177}, {5171, 12176}, {5432, 12184}, {5433, 12185}, {5569, 9877}, {5584, 22514}, {5969, 31884}, {5984, 14981}, {5986, 15246}, {6034, 29181}, {6194, 9888}, {6308, 8295}, {6684, 9864}, {6699, 11005}, {6713, 10768}, {7280, 10089}, {7603, 10486}, {7709, 9734}, {7710, 33216}, {7757, 9755}, {7760, 9737}, {7793, 30270}, {7824, 10333}, {7894, 10983}, {7911, 32152}, {7987, 9860}, {8703, 11632}, {8721, 32964}, {8724, 12100}, {9167, 15708}, {9747, 15078}, {9880, 11001}, {10267, 12190}, {10269, 12189}, {10303, 31274}, {10347, 26316}, {10516, 33220}, {10733, 15359}, {10769, 24466}, {10992, 21735}, {11012, 13190}, {11599, 12512}, {12041, 18332}, {12121, 15535}, {12243, 19708}, {12355, 15695}, {13174, 16192}, {13182, 15326}, {13183, 15338}, {14223, 18556}, {14532, 15655}, {15694, 22566}, {15721, 22247}, {15803, 24472}, {16111, 33511}, {20094, 21734}, {21163, 33273}

= midpoint of X(i) and X(j) for these {i,j}: {98, 21166}, {376, 14651}, {14830, 15561}

= reflection of X(i) in X(j) for these (i,j): (4, 23514), (99, 21166), (381, 34127), (671, 14651), (3839, 14971), (5055, 26614), (5182, 5085), (6054, 15561), (14651, 6055), (15561, 549), (21166, 3), (23234, 5054), (23514, 6036)

= circumperp conjugate of X(23235)

= X(5085)-of-1st anti-Brocard triangle

= X(21166)-of-ABC-X3 reflections triangle

= X(23514)-of-anti-Euler triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 98, 99), (3, 12042, 98), (3, 12188, 33813), (4, 6036, 14061), (20, 115, 10723), (40, 11710, 7983), (98, 23235, 12188), (631, 9862, 114), (12188, 33813, 23235), (23235, 33813, 99)

= [ 7.8969073743790900, 8.3655455830454370, -5.7955935560681970 ]

 

Q( X(100) ) = X(3)X(8) ∩ X(20)X(119)

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

= 2*X(3)+X(100), 4*X(3)-X(104), 5*X(3)+X(12331), 7*X(3)-X(12773), X(3)+2*X(33814), X(4)-4*X(3035), X(4)+2*X(24466), 4*X(5)-X(10724), 2*X(10)+X(12119), 2*X(100)+X(104), 5*X(100)-2*X(12331), 7*X(100)+2*X(12773), X(100)-4*X(33814), 5*X(104)+4*X(12331), 7*X(104)-4*X(12773), X(104)+8*X(33814), X(944)+2*X(1145), 2*X(3035)+X(24466), 2*X(4996)+X(11491), 4*X(5690)-X(12531)

= lies on these lines: {2, 5840}, {3, 8}, {4, 3035}, {5, 10724}, {10, 12119}, {11, 631}, {20, 119}, {21, 11231}, {35, 6940}, {36, 10087}, {40, 214}, {80, 6684}, {140, 10738}, {149, 3523}, {153, 3522}, {165, 2800}, {182, 10755}, {371, 19112}, {372, 19113}, {376, 2829}, {404, 5886}, {516, 1519}, {517, 4881}, {528, 3524}, {548, 11698}, {549, 10707}, {550, 10742}, {620, 10768}, {962, 11729}, {1006, 3586}, {1125, 14217}, {1151, 19082}, {1152, 19081}, {1155, 12739}, {1156, 31658}, {1317, 5204}, {1320, 1385}, {1350, 10759}, {1376, 6950}, {1387, 9785}, {1484, 3530}, {1537, 6361}, {1587, 13922}, {1588, 13991}, {1656, 22938}, {1657, 22799}, {1698, 6246}, {1768, 16192}, {1783, 22055}, {1811, 3417}, {1862, 3515}, {2771, 15055}, {2787, 21166}, {2801, 21165}, {2802, 3576}, {2803, 23239}, {3090, 31235}, {3487, 24465}, {3516, 12138}, {3525, 6667}, {3528, 12248}, {3529, 20400}, {3579, 6265}, {3601, 12736}, {3624, 16174}, {3651, 5660}, {3654, 10031}, {3871, 32612}, {4188, 11248}, {4293, 10956}, {4297, 12751}, {4302, 6963}, {4421, 5854}, {5054, 34126}, {5083, 15803}, {5085, 9024}, {5122, 14151}, {5171, 13194}, {5218, 6955}, {5253, 10283}, {5432, 6951}, {5433, 13274}, {5541, 7987}, {5584, 22775}, {5587, 6906}, {5603, 16371}, {5732, 6594}, {5759, 10427}, {5848, 10519}, {5851, 21168}, {5856, 21151}, {6036, 10769}, {6049, 12735}, {6154, 10299}, {6326, 12520}, {6699, 10778}, {6702, 31423}, {6710, 10772}, {6711, 10777}, {6712, 10770}, {6718, 10771}, {6868, 32554}, {6876, 12332}, {6902, 12764}, {6909, 28160}, {6911, 9779}, {6920, 10172}, {6937, 8068}, {6942, 10310}, {6946, 7988}, {6949, 11826}, {6986, 33862}, {7280, 10074}, {7489, 9342}, {7972, 11362}, {7991, 25485}, {8104, 8127}, {8128, 13267}, {8674, 15035}, {9588, 9897}, {10164, 21161}, {10175, 28461}, {10267, 13279}, {10269, 13278}, {10306, 19537}, {10525, 17566}, {11012, 12776}, {11571, 31806}, {12333, 12868}, {12512, 21635}, {12515, 22935}, {12532, 31837}, {12653, 30389}, {12702, 19907}, {12737, 13624}, {12738, 13243}, {12743, 24914}, {12749, 21578}, {12763, 15326}, {13334, 32454}, {13607, 26726}, {13912, 19078}, {13975, 19077}, {15528, 16209}, {15717, 20095}, {16370, 34122}, {17549, 26446}, {17654, 31787}, {24042, 31263}

= midpoint of X(165) and X(15015)

= reflection of X(i) in X(j) for these (i,j): (5603, 34123), (16173, 10165)

= anticomplement of X(23513)

= X(15035)-of-1st circumperp triangle

= X(15055)-of-2nd circumperp triangle

= X(23515)-of-excentral triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 100, 104), (3, 5690, 5303), (3, 33814, 100), (20, 119, 10728), (40, 214, 10698), (100, 5303, 12531), (631, 13199, 11), (3035, 24466, 4), (5541, 7987, 11715), (22935, 31663, 12515)

= [ 5.3888539744561470, 3.4343794492541270, -1.2241462019405450 ]

 

César Lozada