[Antreas P. Hatzipolakis]:
Denote:
A', B', C' = the the reflections of I in BC, CA, AB, resp.
The orthologic center (A"B"C", A'B'C') is the de Longchamps point of A'B'C'.
Which point is it wrt triangle ABC?
(ie the perpendiculars from A", B", C" to B'C', C'A', A'B', resp. are concurrent.
Which is the point of concurrence on the OI line?)
Not a big deal to be added, but as A’B’C’ and A”B”C” are symmetric triangles with respect to a point, they are homothetic and orthologic, the orthologic centers being the orthocenters of each triangle (also these centers are symmetric w/r to the center of symmetry). Then, if I is replaced for any other point P=u:v:w (trilinears), the orthologic centers are:
Q’ = Q(A’->A”) = -a*b*c*u*(b*w*u + c*u*v-a*v*w) - 2*u*(v^2*SC*c^2 + b^2*SB*w^2) + 2*b*a*(2*SA + SB)*v*w^2 + 2*c*a*(2*SA + SC)*v^2*w : : (trilinears)
Q” = Q(A”->A’) = -a*b*c*u*(a*v*w + b*w*u + c*u*v) + 2*SA*a*v*w*(c*v + b*w) - 2*u*(v^2*SC*c^2 + b^2*SB*w^2) : : (trilinears)
ETC pairs, excluding circumcircle and infinity:
(P,Q’(P) ): (1,5697), (3,382), (4,11412), (15,13), (16,14), (36,9897), (54,2888), (501,13514), (1157,11671)
(P,Q”(P)): (1, 5903), (2,11188), (3,3), (4,5889), (5,13368), (15,13), (16,14), (23,895), (36,1), (54,195), (59,651), (186,110), (187,2482), (249,110), (250,648), (501,1), (1157,3), (1687,1689), (1688,1690), (2065,2987), (2070,195), (2459,13873), (2460,13926), (5961,14889), (10419,2986), (15380,2989), (15381,2990), (15382,2991), (15404,14919)
Some others:
Q’(G) = X(6)X(110) ∩ X(20)X(185)
= ((b^2+c^2)*a^4-7*b^2*c^2*a^2-( b^4-3*b^2*c^2+c^4)*(b^2+c^2))* a^2 :: (barycentrics)
= 4*X(6)-3*X(5640), 4*X(6)-X(12272), 2*X(51)-3*X(5032), 2*X(69)-3*X(7998), X(193)+2*X(6467), 2*X(193)+X(12220), 3*X(373)-2*X(14913), 16*X(575)-13*X(15028), 2*X(1351)+X(12283), 4*X(1353)-X(6403), 2*X(1843)-3*X(11002), 3*X(5640)-2*X(11188), 3*X(5640)-X(12272), X(5889)+2*X(15073), 4*X(6467)-X(12220)
= on lines: {2, 8681}, {6, 110}, {20, 185}, {51, 5032}, {54, 5050}, {68, 6804}, {69, 3266}, {195, 11255}, {323, 11511}, {373, 14913}, {524, 2979}, {542, 15305}, {568, 1353}, {575, 9545}, {1351, 11456}, {1598, 5093}, {1843, 11002}, {1992, 2393}, {1993, 10602}, {1994, 8541}, {3313, 11008}, {3564, 11459}, {3620, 5650}, {3629, 8705}, {3917, 11160}, {5890, 14984}, {5921, 15030}, {8550, 10574}, {8584, 9971}, {9730, 14912}, {9781, 11482}, {11412, 15074}, {11898, 15067}, {15056, 15069}
= reflection of X(i) in X(j) for these (i,j): (568, 1353), (3060, 1992), (5921, 15030), (6403, 568), (9971, 8584), (11160, 3917), (11188, 6), (11898, 15067), (12272, 11188), (15072, 6776)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (6, 11188, 5640), (193, 6467, 12220), (5640, 12272, 11188)
= [ 1.1010140223589690, 4.5168588202248770, 0.0054480575860942 ]
Q’(N) = X(20)X(1154) ∩ X(206)X(576)
= (2*cos(4*A)-1)*cos(B-C)-2*cos( 3*A)*cos(2*(B-C))+4*cos(3*A) (trilinears)
= ((3*R^2+2*SW)*SA^2+(2*R^4+6*R^ 2*SW-4*SW^2)*SA-(23*R^2-6*SW)* S^2)*(SB+SC) :: (barys)
= 4*X(54)-3*X(5946), 2*X(2888)-3*X(15067), 3*X(5946)-2*X(13368), 4*X(6153)-5*X(15026), X(10263)+2*X(12291)
= on lines: {20, 1154}, {54, 5946}, {143, 9706}, {156, 11702}, {195, 1614}, {206, 576}, {2888, 15067}, {3448, 13418}, {6153, 15026}, {10610, 11577}, {10627, 12325}, {12316, 13391}
= midpoint of X(195) and X(12291)
= reflection of X(i) in X(j) for these (i,j): (10263, 195), (10610, 11577), (12325, 10627), (13368, 54), (13423, 143), (13491, 12254)
= {X(54), X(13368)}-Harmonic conjugate of X(5946)
= [ -31.3424041105588000, 37.2729065668630200, -7.6979304748936540 ]
Q’(X(6)) = reflection of X(6) in X(599)
= 5*a^2-4*b^2-4*c^2 : : (barys)
= 4*X(2)-3*X(6), X(2)-3*X(69), 5*X(2)-6*X(141), 7*X(2)-3*X(193), 7*X(2)-6*X(597), 2*X(2)-3*X(599), 5*X(2)-3*X(1992), 13*X(2)-12*X(3589), 17*X(2)-15*X(3618), 11*X(2)-15*X(3620), 11*X(2)-6*X(3629), X(2)+6*X(3630), 7*X(2)-12*X(3631), 14*X(2)-15*X(3763), 13*X(2)-9*X(5032), 10*X(2)-3*X(6144), 13*X(2)-3*X(11008), X(2)+3*X(11160), X(6)-4*X(69), 5*X(6)-8*X(141), 7*X(6)-4*X(193), 7*X(6)-8*X(597), 5*X(6)-4*X(1992), 13*X(6)-16*X(3589), 17*X(6)-20*X(3618), 11*X(6)-20*X(3620), 11*X(6)-8*X(3629), X(6)+8*X(3630), 7*X(6)-16*X(3631), 7*X(6)-10*X(3763), 13*X(6)-12*X(5032), 5*X(6)-2*X(6144), 9*X(6)-8*X(8584), 13*X(6)-4*X(11008), X(6)+4*X(11160), 5*X(69)-2*X(141), 7*X(69)-X(193), 7*X(69)-2*X(597), 5*X(69)-X(1992), 13*X(69)-4*X(3589), 17*X(69)-5*X(3618), 19*X(69)-7*X(3619), 11*X(69)-5*X(3620), 11*X(69)-2*X(3629), X(69)+2*X(3630), 7*X(69)-4*X(3631), 14*X(69)-5*X(3763)
= on lines: {2, 6}, {22, 2930}, {30, 15069}, {76, 11317}, {99, 10488}, {154, 5648}, {315, 8352}, {381, 11477}, {511, 3830}, {518, 4677}, {519, 4655}, {532, 11296}, {533, 11295}, {538, 5077}, {542, 1350}, {549, 10541}, {576, 5055}, {671, 7850}, {732, 11055}, {754, 11159}, {1351, 11178}, {1352, 3845}, {1503, 11001}, {1975, 9855}, {2482, 5210}, {2854, 2979}, {3053, 7801}, {3416, 4669}, {3524, 8550}, {3564, 8703}, {3917, 9027}, {3933, 5023}, {5013, 7810}, {5066, 10516}, {5085, 5965}, {5102, 5476}, {5309, 10542}, {5463, 11480}, {5464, 11481}, {5585, 6390}, {5969, 11161}, {6179, 8366}, {7751, 11318}, {7754, 7883}, {7758, 8359}, {7768, 7841}, {7775, 7882}, {7811, 8716}, {7813, 11165}, {7817, 7896}, {7827, 7879}, {8355, 13881}, {8369, 14023}, {8546, 15246}, {8594, 11128}, {8595, 11129}, {9466, 13330}, {9830, 11057}, {9887, 9983}, {11164, 14712}, {11173, 14537}, {11179, 12100}, {11216, 13857}
= midpoint of X(69) and X(11160)
= reflection of X(i) in X(j) for these (i,j): (6, 599), (193, 597), (597, 3631), (599, 69), (1351, 11178), (1992, 141), (6144, 1992), (10488, 99), (11160, 3630), (11477, 381), (13330, 9466)
= anticomplement of X(8584)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (69, 193, 3631), (141, 6144, 6), (183, 7840, 11184), (193, 3631, 3763), (193, 3763, 6), (591, 1991, 7610), (5858, 5859, 8667), (5860, 5861, 9740), (9770, 20022, 11168), (13846, 13847, 230)
= [ 7.5387496847578630, 2.5435618855077350, -1.5996859087169130 ]
Q”(X(6)) = reflection of X(6) in X(1992)
= 7*a^2-2*b^2-2*c^2 : : (barycentrics)
= 2*X(2)-3*X(6), 5*X(2)-3*X(69), 7*X(2)-6*X(141), X(2)+3*X(193), 5*X(2)-6*X(597), 4*X(2)-3*X(599), X(2)-3*X(1992), 11*X(2)-12*X(3589), 13*X(2)-15*X(3618), 19*X(2)-15*X(3620), X(2)-6*X(3629), 13*X(2)-6*X(3630), 17*X(2)-12*X(3631), 16*X(2)-15*X(3763), 5*X(2)-9*X(5032), 4*X(2)+3*X(6144), 7*X(2)+3*X(11008), 7*X(2)-3*X(11160), 5*X(6)-2*X(69), 7*X(6)-4*X(141), X(6)+2*X(193), 5*X(6)-4*X(597), 11*X(6)-8*X(3589), 13*X(6)-10*X(3618), 23*X(6)-14*X(3619), 19*X(6)-10*X(3620), X(6)-4*X(3629), 13*X(6)-4*X(3630), 17*X(6)-8*X(3631), 8*X(6)-5*X(3763), 5*X(6)-6*X(5032), 2*X(6)+X(6144), 19*X(6)-16*X(6329), 3*X(6)-4*X(8584), 7*X(6)+2*X(11008), 7*X(6)-2*X(11160), 7*X(69)-10*X(141), X(69)+5*X(193), 4*X(69)-5*X(599), X(69)-5*X(1992), 11*X(69)-20*X(3589), X(69)-10*X(3629), 13*X(69)-10*X(3630), 17*X(69)-20*X(3631), X(69)-3*X(5032), 4*X(69)+5*X(6144)
= on lines: {2, 6}, {25, 2930}, {30, 11477}, {32, 12151}, {51, 9027}, {99, 8787}, {187, 11165}, {194, 9855}, {376, 8550}, {381, 576}, {511, 3534}, {518, 3899}, {519, 5695}, {532, 11295}, {533, 11296}, {538, 11159}, {542, 1351}, {543, 10488}, {575, 5054}, {732, 12156}, {754, 5077}, {895, 8877}, {1350, 1353}, {1352, 5066}, {1384, 2482}, {1853, 11216}, {2502, 10554}, {2854, 3060}, {3167, 5648}, {3416, 4745}, {3524, 10541}, {3564, 3845}, {3679, 4663}, {3751, 4677}, {3767, 8355}, {3793, 8182}, {3849, 7798}, {3882, 5043}, {4644, 4969}, {4669, 5847}, {4675, 4700}, {5017, 14645}, {5052, 14711}, {5055, 11482}, {5064, 8541}, {5085, 12100}, {5093, 5476}, {5097, 11178}, {5107, 11648}, {5210, 7618}, {5319, 8360}, {5463, 11485}, {5464, 11486}, {5477, 11173}, {5480, 11180}, {5839, 7277}, {5969, 8593}, {6353, 15471}, {6636, 8546}, {6776, 11001}, {7529, 13431}, {7622, 10485}, {7739, 10542}, {7754, 7812}, {7758, 8369}, {7759, 11318}, {7760, 7841}, {7762, 8352}, {7775, 7805}, {7776, 7817}, {7784, 7827}, {7796, 8366}, {7801, 7890}, {7810, 9605}, {7839, 9939}, {7883, 7894}, {7926, 9166}, {8359, 14023}, {8540, 11238}, {8542, 15004}, {8598, 8716}, {8681, 9971}, {9830, 10754}, {10109, 14561}, {11663, 14449}, {11742, 14712}
= midpoint of X(i) and X(j) for these {i,j}: {193, 1992}, {599, 6144}, {11008, 11160}
= reflection of X(i) in X(j) for these (i,j): (2, 8584), (6, 1992), (69, 597), (99, 8787), (376, 8550), (381, 576), (599, 6), (1350, 11179), (1853, 11216), (1992, 3629), (3679, 4663), (5648, 15303), (10516, 5093), (11160, 141), (11178, 5097), (11179, 1353), (11180, 5480), (11898, 11178), (15069, 381)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 1992, 8584), (2, 8584, 6), (6, 193, 6144), (69, 1992, 5032), (69, 5032, 597), (193, 3629, 6), (385, 11163, 7610), (591, 1991, 11184), (597, 5032, 6), (3180, 3181, 7777), (5858, 5859, 9766), (6189, 6190, 8859), (7736, 9740, 11168), (7837, 14614, 9766), (11168, 15480, 9740)
= [ -2.2800120997804270, 0.9622631711438901, 4.0267955633218840 ]
César Lozada
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