HYACINTHOS 26559

 

[César Lozada]:

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

BP, CP intersect B'C' at Ab, Ac, resp.

CP, AP intersect C'A' at Bc, Ba, resp.
AP, BP intersect A'B' at Ca, Cb, resp.

Which is the locus of P such that the perpendicular bisectors of AbAc, BcBa, CaCb are concurrent at Q(P)?

 

The locus is = {Line at infinity} \/ { circles with segment-sides of ABC as diameters} \/ { Gibert’s curve Q030, through ETC’s 1, 4, 13, 14, 74, 80}  

 

ETC pairs (P,Q(P)) = (1,40), (13, 13), (14,14), (74, 9934)

 

Q( X(4) ) = X(3)X(66) ∩ X(4)X(51)

= a^10-3*(b^2+c^2)*a^8+4*(b^4-b^ 2*c^2+c^4)*a^6-4*(b^4-c^4)*(b^ 2-c^2)*a^4+3*(b^4-c^4)^2*a^2-( b^4-c^4)*(b^2-c^2)^3 : : (barycentrics)

= 4*cos(B-C)+cos(A)*(2*cos(2*(B- C))-7)+cos(3*A) : : (trilinears)

= 3*X(3)-4*X(6696) = 3*X(4)-X(6225) = 2*X(5)-3*X(1853) = 3*X(64)-X(5925) = 4*X(140)-3*X(154) = X(1498)-3*X(1853) = 3*X(5878)-2*X(6225) = X(5878)+2*X(12324) = X(6225)+3*X(12324) = 3*X(6247)-2*X(6696)

= On lines:

{2, 6759}, {3, 66}, {4, 51}, {5, 1498}, {6, 1595}, {20, 2888}, {30, 64}, {70, 74}, {125, 3542}, {133, 6526}, {140, 154}, {182, 5596}, {184, 3541}, {206, 13336}, {221, 495}, {343, 11414}, {355, 5836}, {381, 2883}, {382, 13093}, {427, 1181}, {485, 12964}, {486, 12970}, {496, 2192}, {499, 10535}, {511, 11411}, {542, 2892}, {548, 8567}, {550, 10606}, {569, 11179}, {578, 3088}, {631, 10282}, {858, 11441}, {1204, 13399}, {1370, 5562}, {1478, 7355}, {1479, 6285}, {1495, 3147}, {1593, 6146}, {1594, 11456}, {1597, 12241}, {1598, 13567}, {1619, 6642}, {1657, 5894}, {1872, 5928}, {1907, 10982}, {2393, 10625}, {2777, 3146}, {2781, 6243}, {2917, 7525}, {3091, 5643}, {3410, 3522}, {3523, 11202}, {3526, 10192}, {3546, 9306}, {3548, 10539}, {3575, 10605}, {3583, 12950}, {3585, 12940}, {3627, 5895}, {3818, 7401}, {3819, 11487}, {3843, 5893}, {4846, 6145}, {5418, 10533}, {5420, 10534}, {5480, 11432}, {5654, 13371}, {5810, 6907}, {5889, 7391}, {5907, 6643}, {6102, 6293}, {6221, 8991}, {6284, 10060}, {6398, 13980}, {6640, 10540}, {6756, 9786}, {7354, 10076}, {7386, 11793}, {7387, 12359}, {7392, 11695}, {7487, 11438}, {7507, 12174}, {7528, 9730}, {7544, 10574}, {8550, 11426}, {10113, 11744}, {10117, 10264}, {10182, 10303}, {10201, 13561}, {10274, 11003}, {10628, 12284}, {11598, 12121}, {11818, 13630}, {12084, 12118}, {12383, 13293}, {12586, 12675}

= midpoint of X(i) and X(j) for these {i,j}: {4, 12324}, {382, 13093}, {3146, 12250}, {12317, 13203}

= reflection of X(i) in X(j) for these (i,j): (3, 6247), (20, 3357), (1352, 66), (1498, 5), (1657, 5894), (5596, 182), (5878, 4), (5895, 3627), (6193, 13346), (6293, 6102), (7387, 12359), (9833, 3), (9934, 125), (10117, 10264), (11744, 10113), (12118, 12084), (12121, 11598), (12315, 2883), (12383, 13293)

= anticomplement of X(6759)

= anticomplementary circle-inverse-of-X(6761)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (4, 11433, 10110), (4, 11457, 1899), (185, 11550, 4), (381, 12315, 2883), (631, 11206, 10282), (1498, 1853, 5), (1907, 11245, 10982), (3818, 9729, 7401), (7528, 9730, 9815), (11438, 13419, 7487), (12278, 13445, 20)

= [ 19.168341760916870, 20.78986994641083, -19.599249370646290 ]

 

Q( X(80) ) = Reflection of X(40) in X(11)

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

= 2*sin(A/2)*(6*cos((B-C)/2)-cos (3*(B-C)/2))+3*cos(A)+cos(2*A) -6 : : (trilinears)

= 2*X(119)-3*X(1699) = 3*X(165)-4*X(6713) = 2*X(214)-3*X(5603) = 2*X(1145)-3*X(5587) = 4*X(1387)-3*X(3576) = 4*X(1484)-3*X(11219) = 3*X(1699)-X(5541) = 4*X(3035)-5*X(8227) = 3*X(5603)-X(13199) = 3*X(11219)-2*X(12515)

= On lines:

On lines:

{1, 5840}, {4, 2802}, {11, 40}, {20, 11715}, {30, 12737}, {46, 5533}, {65, 13274}, {80, 517}, {100, 946}, {104, 516}, {119, 1699}, {149, 151}, {153, 9802}, {165, 6713}, {214, 5603}, {497, 12736}, {515, 1320}, {528, 1537}, {529, 11256}, {952, 3627}, {1145, 5587}, {1317, 1836}, {1387, 3576}, {1482, 7972}, {1484, 5535}, {1768, 9589}, {1770, 10074}, {2077, 10090}, {2093, 12832}, {2099, 12743}, {2809, 10772}, {2817, 10777}, {2829, 6264}, {3035, 8227}, {3057, 13273}, {3149, 13205}, {3585, 12749}, {3898, 6951}, {4295, 5083}, {4301, 10698}, {5057, 12531}, {5119, 8068}, {5180, 12532}, {5443, 11849}, {5657, 6702}, {5660, 12331}, {5697, 10057}, {5805, 9945}, {5854, 5881}, {5856, 11372}, {6284, 11014}, {7743, 13528}, {8148, 12747}, {9612, 10956}, {9616, 13913}, {9897, 11531}, {10058, 11012}, {10087, 12047}, {10265, 10707}, {10624, 10902}, {10993, 11522}, {11571, 12750}, {12608, 13278}, {12619, 12702}, {12672, 13271}, {12763, 13600}

= midpoint of X(i) and X(j) for these {i,j}: {149, 962}, {153, 9802}, {1320, 10724}, {1768, 9589}, {5691, 12653}, {8148, 12747}, {9897, 11531}

= reflection of X(i) in X(j) for these (i,j): (20, 11715), (40, 11), (80, 10738), (100, 946), (5541, 119), (6326, 1537), (7972, 1482), (10698, 4301), (10993, 11729), (12119, 1), (12331, 12611), (12515, 1484), (12702, 12619), (12751, 4), (13199, 214), (13528, 7743)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1484, 12515, 11219), (1699, 5541, 119), (5603, 13199, 214), (9802, 9812, 153), (12331, 12611, 5660), (12700, 12701, 40)

= [ -6.588147036445251, -3.35409225332791, 9.003411597186124 ]

 

 

César Lozada