HYACINTHOS 26809

[Antreas P. Hatzipolakis]:

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

Denote:

Oa, Ob, Oc = the circumcenters of PBC, PCA, PAB, resp.

OaaOabOac = the midway triangle of Oa
(ie Oaa, Oab, Oac = the midpoints of AOa, BOa, COa, resp.)
ObaObbObc = the midway triangle of Ob
OcaOcbOcc = the midway triangle of Oc.

Q1, Q2, Q3 = same points Q of  OaaOabOac, ObaObbObc., OcaOcbOcc, resp.

For P = I

 

A'B'C', Q1Q2Q3  are homothetic

Q = H : The homothetic center of A'B'C', H1H2H3  is the Feuerbach point.

Which is the locus of the homothetic center as Q moves on the Euler line?

 

 

[César Lozada]:

 

>>For P = I: A'B'C', Q1Q2Q3  are homothetic

 

A'B'C' and Q1Q2Q3  are homothetic if a) P lies on the angle bisectors of ABC, or b) P or Q lies in the infinity. So, P=I is the only finite center such that A'B'C' and Q1Q2Q3 are homothetic.

 

>> Which is the locus of the homothetic center as Q moves on the Euler line?

The asked locus is the line X(11)X(36) which is parallel to the Euler line. If OQ=t*OH and Z(Q) is the  homothetic center then X(11)Z(Q)=(1-t)*X(11)X(36) and

 

Z(Q) = (2*a^4-(b^2+c^2)*a^2-(b^2-c^2) ^2)*t-2*a^2*(a^2-b^2+b*c-c^2) : :  (barycentrics)

= t*X(11)-(t-1)*X(36)

 

ETC pairs (Q,Z(Q)):

(2,5298), (3,36), (4,11), (21,5427), (381,3582), (382,3583), (1657,4316), (11251,11913)

 

Others:

 

Z(X(5)) = midpoint of X(11) and X(36)

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

= X(11)-3*X(3582) = 3*X(11)-X(3583) = 5*X(11)+X(4316) = X(11)+3*X(5298) = X(36)+3*X(3582) =    3*X(36)+X(3583) = 5*X(36)-X(4316) = X(36)-3*X(5298) = X(149)+3*X(13587) = 9*X(3582)-X(3583) = 15*X(3582)+X(4316) = 5*X(3583)+3*X(4316) = X(3583)+9*X(5298) = X(4316)-15*X(5298) = 2*X(5126)+X(12019)

= On lines: {1, 140}, {2, 495}, {3, 496}, {4, 5265}, {5, 56}, {8, 13747}, {10, 6691}, {11, 30}, {12, 3628}, {20, 9669}, {21, 13100}, {35, 3530}, {40, 11373}, {46, 11376}, {55, 549}, {57, 5886}, {65, 5901}, {104, 1532}, {119, 5193}, {149, 13587}, {202, 396}, {203, 395}, {226, 11230}, {230, 1015}, {241, 15251}, {350, 6390}, {354, 5719}, {355, 1420}, {376, 5274}, {381, 4293}, {382, 10591}, {388, 1656}, {390, 3524}, {442, 5253}, {468, 1870}, {474, 10527}, {498, 632}, {515, 5126}, {516, 5122}, {517, 1387}, {519, 3035}, {523, 8043}, {529, 3814}, {546, 7354}, {547, 5434}, {548, 6284}, {550, 1479}, {551, 6690}, {614, 1060}, {631, 3295}, {758, 942}, {912, 3660}, {944, 5704}, {950, 13624}, {952, 1319}, {954, 6878}, {958, 10200}, {993, 3816}, {1012, 7956}, {1058, 3523}, {1111, 7181}, {1124, 8981}, {1329, 8666}, {1335, 13966}, {1388, 1483}, {1398, 3542}, {1428, 3564}, {1447, 1565}, {1470, 6914}, {1484, 5172}, {1595, 11399}, {1617, 6911}, {1657, 5225}, {1749, 3337}, {2067, 7584}, {2093, 3656}, {2099, 10283}, {2242, 3815}, {2275, 5305}, {2646, 12433}, {2975, 4187}, {3028, 10272}, {3085, 3526}, {3090, 3600}, {3091, 9655}, {3149, 10785}, {3216, 5399}, {3297, 5418}, {3298, 5420}, {3303, 14869}, {3333, 3624}, {3338, 6147}, {3339, 9624}, {3361, 8227}, {3476, 5790}, {3485, 5708}, {3487, 5550}, {3584, 5326}, {3585, 3850}, {3614, 5270}, {3616, 7483}, {3627, 4299}, {3653, 13384}, {3654, 7962}, {3655, 5727}, {3712, 4975}, {3720, 12081}, {3746, 12108}, {3845, 12943}, {3851, 5229}, {3853, 10483}, {3861, 4325}, {4292, 9955}, {4302, 8703}, {4308, 5818}, {4315, 10175}, {4870, 11551}, {4881, 10609}, {4973, 11813}, {4995, 11812}, {5010, 12100}, {5045, 13411}, {5049, 13405}, {5054, 5218}, {5055, 10590}, {5067, 5261}, {5070, 10588}, {5194, 14693}, {5217, 10386}, {5272, 6677}, {5435, 5603}, {5442, 11010}, {5450, 7681}, {5542, 20001}, {5587, 13462}, {5687, 6921}, {5763, 12704}, {5791, 8583}, {6001, 13226}, {6502, 7583}, {6592, 7159}, {6644, 10832}, {6684, 9957}, {6705, 9856}, {6905, 7677}, {6907, 10269}, {6910, 10586}, {6922, 11249}, {6924, 10943}, {6935, 8732}, {6958, 10680}, {6961, 10306}, {7051, 11543}, {7508, 14793}, {7516, 10831}, {9956, 10106}, {10035, 13391}, {10056, 11539}, {10074, 11698}, {10091, 10264}, {10165, 11019}, {10543, 11277}, {10959, 14798}, {11112, 11680}, {11544, 12047}, {11729, 14988}, {12374, 14677}

= midpoint of X(i) and X(j) for these {i,j}: {11, 36}, {104, 1532}, {1319, 1737}, {1749, 3649}, {3582, 5298}, {4973, 11813}, {5122, 7743}

= reflection of X(i) in X(j) for these (i,j): (3035, 6681), (3814, 6667), (11545, 1737)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3, 3086, 496), (11, 5298, 36), (36, 3582, 11), (56, 499, 5), (1388, 10573, 1483), (1479, 5204, 550), (3086, 7288, 3), (4299, 10896, 3627), (6921, 10529, 5687)

= [ 0.033924167292768, 0.77042275576474, 3.091637573781360 ]

 

Z( X(20) ) = reflection of X(11) in X(36)

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

 

= 5*X(11)-6*X(3582) = 3*X(11)-2*X(3583) = X(11)+2*X(4316) = 2*X(11)-3*X(5298) = 5*X(36)-3*X(3582) = 3*X(36)-X(3583) = 4*X(36)-3*X(5298) = X(80)-3*X(5131) = 2*X(3035)-3*X(13587) = 9*X(3582)-5*X(3583) = 3*X(3582)+5*X(4316) = 4*X(3582)-5*X(5298) = X(3583)+3*X(4316) = 4*X(3583)-9*X(5298) = 4*X(4316)+3*X(5298) = X(5080)-3*X(13587)

 

= On lines: {1, 550}, {2, 12943}, {3, 12}, {4, 5204}, {5, 7280}, {11, 30}, {20, 56}, {21, 12615}, {35, 548}, {40, 10944}, {46, 10950}, {55, 376}, {65, 4297}, {80, 5131}, {100, 529}, {104, 5842}, {140, 3585}, {154, 12940}, {165, 5252}, {354, 4304}, {371, 9649}, {372, 9647}, {382, 499}, {388, 3522}, {442, 5267}, {468, 5370}, {484, 952}, {495, 4995}, {515, 1155}, {516, 1319}, {517, 1317}, {519, 5183}, {535, 6174}, {549, 5326}, {609, 15048}, {631, 10895}, {758, 10609}, {942, 10543}, {958, 4190}, {962, 1388}, {993, 3925}, {999, 3058}, {1012, 7965}, {1015, 6781}, {1092, 9652}, {1329, 4188}, {1358, 5088}, {1368, 5345}, {1385, 1770}, {1420, 12701}, {1457, 3000}, {1470, 7580}, {1479, 1657}, {1737, 5122}, {1836, 3576}, {1870, 7286}, {2077, 10956}, {2098, 6361}, {2099, 3474}, {2475, 4999}, {2646, 3649}, {2829, 6905}, {3035, 5080}, {3053, 9597}, {3057, 4311}, {3068, 9663}, {3085, 3528}, {3086, 3529}, {3100, 10149}, {3146, 7288}, {3184, 3324}, {3245, 5844}, {3295, 4317}, {3303, 3600}, {3304, 4294}, {3320, 14689}, {3337, 5441}, {3428, 6948}, {3434, 11194}, {3476, 9778}, {3486, 5221}, {3488, 4860}, {3516, 11392}, {3523, 5229}, {3524, 10590}, {3543, 10589}, {3560, 7958}, {3601, 10404}, {3627, 7741}, {3816, 11114}, {4081, 10538}, {4296, 9630}, {4309, 7373}, {4312, 13384}, {4315, 5919}, {4324, 5563}, {4330, 15172}, {4333, 12699}, {4423, 11111}, {4534, 5011}, {4652, 5794}, {4872, 7181}, {4881, 5057}, {5046, 6691}, {5059, 5225}, {5172, 6909}, {5206, 9651}, {5218, 10304}, {5322, 7667}, {5535, 12119}, {5538, 5762}, {5855, 6224}, {5894, 6285}, {5925, 12950}, {6200, 13901}, {6253, 6934}, {6396, 13958}, {6449, 9648}, {6450, 13963}, {6840, 13273}, {6899, 10953}, {6950, 7680}, {6977, 10894}, {7296, 9607}, {7727, 14677}, {7765, 9341}, {7987, 9579}, {8162, 10385}, {9580, 13462}, {9656, 10588}, {9659, 10323}, {9666, 13346}, {9668, 10072}, {9672, 11413}, {9673, 12082}, {9833, 10076}, {10081, 12121}, {10106, 12512}, {10535, 15311}, {10949, 11249}, {10957, 11012}, {11001, 11238}, {12047, 13624}, {12373, 15035}

= midpoint of X(i) and X(j) for these {i,j}: {1, 15228}, {36, 4316}, {5535, 12119}

= reflection of X(i) in X(j) for these (i,j): (11, 36), (1737, 5122), (5080, 3035)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (11, 36, 5298), (5059, 5265, 5225), (5080, 13587, 3035), (6934, 12114, 6253)

= [ -7.793708694564541, -7.03656061295784, 12.109225842215750 ]

 

César Lozada