Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Ha, Hb, Hc = the orthocenters of AB'C', BC'A', CA'B', resp.
Which is the locus of P such that
1. A'B'C', HaHbHc are perspective?
I lies on the locus (Thanos Kalogerakis)
2. A'B'C', HaHbHc are orthologic?
[César Lozada]:
Let P=u:v:w (trilinears)
1) The entire plane.
Perspector: Q(P) = u+cos(C)*v+cos(B)*w : :
ETC pairs (P,Q(P)): (1,942), (2,5943), (3,5), (4,389), (5,5462), (6,6), (15,11542), (16,11543), (19,9119), (20,5907), (22,21243), (25,13567), (26,5449), (30,13754), (32,5305), (40,5777), (50,16310), (52,13292), (54,12242), (55,226), (56,1210), (64,4), (66,9969), (68,12235), (69,14913), (74,7687), (84,5908), (109,15252), (110,5972), (113,9826),….
2) The entire plane:
Zah(P) = A’->Ha = (c*v+b*w)*(-cos(A)*v*w+u*(u+v*cos(C)+w*cos(B))) : :
ETC pairs(P, Zah(P)): (1,65), (3,3), (4,52), (5,6153), (6,1992), (15,396), (16,395), (30,13754), (36,1319), (40,18239), (54,1493), (55,8545), (64,3146), (74,30), (98,511), (99,512), (100,513), (101,514), (102,515), (103,516), (104,517), (105,518), (106,519), (107,520), (108,521), (109,522), (110,523), (111,524), (112,525), (187,27088),….
Zha(P) = Ha->A’ = isogonal-conjugate-of-P
Particular cases:
Q(X(7)) = X(1)X(3688) ∩ X(7)X(2808)
= a^2*((b^2+c^2)*a^4-2*(b^3+c^3)*a^3-2*b^2*c^2*a^2+2*(b^3-c^3)*(b^2-c^2)*a-(b^4+c^4+2*(b^2+c^2)*b*c)*(b-c)^2) : : (barys)
= on lines: {1, 3688}, {7, 2808}, {77, 14520}, {511, 5728}, {674, 5572}, {916, 942}, {938, 5933}, {2389, 15587}, {2810, 18412}, {3211, 9306}, {3819, 11018}, {3917, 11020}, {4253, 20793}, {6738, 29311}, {7146, 21746}, {8680, 13563}, {9440, 20683}, {13754, 15939}, {17092, 22440}
= [ 1.2730105116285230, 1.4741497929728930, 2.0325559275592110 ]
Q(X(8)) = COMPLEMENT OF X(23154)
= a^2*((b^2+c^2)*a^3+(b^2-c^2)*(b-c)*a^2-(b^4+c^4)*a-(b+c)*(b^4+c^4-2*(b^2+c^2)*b*c)): : (barys)
= 3*X(51)-X(3868), 3*X(51)-2*X(12109), 3*X(375)-2*X(3812), 2*X(942)-3*X(5943), 3*X(3819)-4*X(5044), 3*X(3819)-2*X(11573), X(3874)-3*X(15049), 5*X(3876)-3*X(3917), 5*X(5439)-6*X(6688), 3*X(10167)-4*X(17704), 3*X(10176)-X(23156), 3*X(10202)-4*X(11695), 2*X(13369)-3*X(16836)
= on lines: {1, 2810}, {2, 23154}, {51, 3868}, {63, 970}, {65, 23841}, {72, 511}, {78, 26892}, {181, 1046}, {185, 2808}, {329, 10441}, {355, 2818}, {375, 3812}, {386, 20805}, {389, 912}, {404, 3937}, {517, 12527}, {581, 20760}, {651, 1425}, {936, 3784}, {942, 5943}, {960, 8679}, {978, 1401}, {986, 23638}, {1331, 3145}, {1463, 24178}, {1757, 10822}, {1762, 7066}, {2390, 5836}, {2392, 3678}, {2842, 3754}, {3061, 23630}, {3157, 9306}, {3219, 22076}, {3732, 17499}, {3819, 5044}, {3869, 16980}, {3874, 15049}, {3876, 3917}, {3916, 15489}, {3927, 5752}, {3951, 26893}, {4339, 9309}, {4415, 18178}, {5396, 22458}, {5439, 6688}, {5462, 24475}, {5777, 5907}, {5904, 9052}, {6743, 29353}, {7078, 24320}, {7248, 11512}, {9021, 9969}, {9822, 24476}, {10110, 24474}, {10167, 17704}, {10176, 23156}, {10202, 11695}, {13369, 16836}, {13731, 21361}, {17114, 24174}, {17572, 26910}, {17768, 22300}
= midpoint of X(i) and X(j) for these {i,j}: {185, 12528}, {3869, 16980}
= reflection of X(i) in X(j) for these (i,j): (65, 23841), (3868, 12109), (5907, 5777), (11573, 5044), (24474, 10110), (24475, 5462), (24476, 9822)
= complement of X(23154)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 3868, 12109), (5044, 11573, 3819)
= [ 1.9049567398322360, -0.0271654519746181, 2.7802605302750070 ]
Zah( X(2) ) = X(49)X(575) ∩ X(51)X(524)
= a^2*(b^2+c^2)*(a^4-b^4+4*b^2*c^2-c^4) : : (barys)
= X(6)-4*X(9822), X(6)+2*X(14913), X(69)+2*X(9969), 2*X(141)+X(1843), 4*X(141)-X(3313), 3*X(373)-2*X(597), 2*X(389)+X(15069), X(1205)-4*X(6698), 2*X(1352)+X(19161), 2*X(1843)+X(3313), X(1992)-3*X(5640), X(2979)-3*X(21356), 7*X(3090)-X(15073), 4*X(3589)-X(6467), 5*X(3618)+X(12272), 7*X(3619)-X(12220), 4*X(3628)-X(15074), 2*X(9822)+X(14913), X(9967)-4*X(24206), X(12162)-4*X(18553)
= on lines: {2, 2393}, {5, 5181}, {6, 1196}, {39, 1634}, {49, 575}, {51, 524}, {67, 3521}, {69, 3060}, {110, 12039}, {141, 427}, {159, 3796}, {338, 6248}, {373, 597}, {381, 511}, {384, 1632}, {389, 15069}, {542, 9730}, {895, 16042}, {1205, 6698}, {1235, 27373}, {1350, 1597}, {1352, 7706}, {1495, 19127}, {1568, 5480}, {1992, 5640}, {1995, 8542}, {2072, 9967}, {2386, 11286}, {2781, 15030}, {2882, 8370}, {2979, 21356}, {3003, 11328}, {3090, 15073}, {3564, 5946}, {3589, 6467}, {3618, 12272}, {3619, 12220}, {3628, 15074}, {3763, 9973}, {3818, 16194}, {3819, 21358}, {5085, 11202}, {5092, 12367}, {5169, 19510}, {5421, 20794}, {5476, 14845}, {5650, 8705}, {5651, 8541}, {5890, 11180}, {5892, 11179}, {6697, 26156}, {6776, 15045}, {7669, 13335}, {8547, 22112}, {10110, 11477}, {10151, 12294}, {10602, 11284}, {10984, 15581}, {11002, 11160}, {11451, 15531}, {11645, 14855}, {11898, 13321}, {12093, 17430}, {14810, 18859}, {15060, 18358}, {15533, 21849}, {16072, 23049}, {19126, 20987}, {21513, 22143}, {21969, 22165}, {22087, 23635}
= midpoint of X(i) and X(j) for these {i,j}: {2, 11188}, {69, 3060}, {599, 9971}, {1843, 3917}, {5890, 11180}, {5943, 14913}
= reflection of X(i) in X(j) for these (i,j): (6, 5943), (51, 16776), (3060, 9969), (3313, 3917), (3917, 141), (5891, 11178), (5943, 9822), (11179, 5892), (15060, 18358), (16194, 3818)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9306, 9813, 6), (9822, 14913, 6)
= [ 1.7570858801305440, 0.1073895966489724, 2.7553551241671470 ]
César Lozada
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