I guess this is well-known theorem, but I do not have references
Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and L a line.
Denote:
A", B", C" = the intersections of L and BC, CA, AB, resp.
A* = the intersection of BB", CC"
B* = the intersection of CC", AA"
C* = the intersection of AA", BB"
>>The lines A'A*, B'B*, C'C* are concurrent (The triangles A'B'C', A*B*C* are perspective).
Let L be the trilinear polar of X = x:y:z (trilinears) and P=u:v:w (trilinears). The perspector Z(X,P) of the given triangles is:
Z(X,P) = CevaConjugate( P, X)
>>If A'B'C' = the cevian triangle of G, then the parallels to A'A*, B'B*, C'C* through A, B, C, resp. are concurrent.
Let L be the trilinear polar of X = x:y:z (trilinears)
If P=G then the given parallel lines concur at Q(G, X) = IsotomicConjugate( Anticomplement(X))
In general, the given parallel lines concur if any of the following conditions is satisfied:
- P lies on the circumconic(*) with center X, in this case the lines are parallel and Q(X,P) lies in the infinity. The general expression for Q(X,P) is rather complicated.
(*) The trilinear pole of the isogonal conjugate of this conic is: Isogonal(Complement(Isotomic( Anticomplement(X)))).
The appearance of (i, j, k) in the following lists means that Q(X(i),X(j))=X(k):
(1, 100, 513), (1, 643, 6003), (1, 644, 3309), (1, 664, 514), (1, 1120, 519), (1, 1280, 518), (1, 1320, 517), (1, 1897, 522), (1, 3699, 3667), (1, 3903, 512), (1, 4614, 0), (1, 6740, 30), (1, 6742, 523), (1, 7257, 6002), (1, 8851, 0), (1, 13138, 521), (1, 14942, 516)
(2, 99, 523), (2, 190, 514), (2, 290, 511), (2, 648, 525), (2, 664, 522), (2, 666, 918), (2, 668, 513), (2, 670, 512), (2, 671, 524), (2, 886, 888)
(4, 107, 523), (4, 10152, 0), (4, 14944, 0)
(6, 110, 512), (6, 287, 1503), (6, 648, 523), (6, 651, 513), (6, 677, 926), (6, 895, 2393), (6, 1331, 8676), (6, 1332, 0), (6, 1797, 2390), (6, 1813, 0)
Missing:
Q( X(1), X(4614) ) = Infinity point of line X(1)X(4822)
= a*(b-c)*(a^2+2*(b+c)*a+3*b*c+ c^2+b^2) : : (barys)
= On lines: {1, 4822}, {30, 511}, {661, 1019}, {1577, 7192}, {3960, 4813}, {4063, 4979}, {4129, 4369} et als.
= [ 1.836289928538072, -3.07440994139809, 1.280919223181108 ]
Q( X(1), X(8851) ) = Infinity point of line X(44)X(573)
= a*((b-c)^2*a^3+b*c*(b+c)*a^2-( b^4+c^4)*a+(b^2-c^2)*(b-c)*b* c): : (barys)
= On lines: {1, 1463}, {3, 238}, {4, 4645}, {5, 3836}, {30, 511}, {40, 1757}, {44, 573}, {320, 10446}, {497, 3784}, {991, 1279}, {1633, 7193}, {1738, 3271} et als.
= [ 1.632664435563391, 0.89677809575561, -1.374383805783139 ]
Q( X(4), X(10152) ) = isogonal conjugate of X(5897)
= (8*R^2-SA-SW)*S^2-4*(6*R^2-SW) *SB*SC: : (barys)
= On lines: {2, 10606}, {3, 1661}, {4, 64}, {5, 3357}, {20, 394}, {30, 511}, {40, 12779}, {55, 12940}, {56, 12950}, {66, 3426}, {74, 403} et als.
= isogonal conjugate of X(5897)
= [ 1.632664435563391, 0.89677809575561, -1.374383805783139 ]
Q( X(4), X(14944) ) = Infinity point of line X(4)X(253)
= 2*S^4-(SA-SW)*(16*R^2-3*SA-4* SW)*S^2+4*(4*R^2-SW)*(SA-SW)* SA*SW : : (barys)
= On lines: {3, 1033}, {4, 253}, {5, 6523}, {30, 511}, {64, 15238}, {140, 15274}, {1073, 6525}, {1853, 13157} et als.
= [ 1.279528964954112, 1.00351881897122, -1.285295627728128 ]
Q( X(6), X(1332) ) = isogonal conjugate of X(13397)
= a*(b-c)*(a^3-(b+c)*a^2-(b+c)^ 2*a+(b+c)*(b^2+c^2)) : : (barys)
= On lines: {4, 3657}, {30, 511}, {66, 10099}, {100, 1618}, {649, 8611}, {656, 663}, {905, 2605} et als.
= complementary conjugate of X(5521)
= isogonal conjugate of X(13397)
= [ 6.252297464240273, 0.88832147140443, -3.500667540621658 ]
Isogonal conjugates of these points not-in-ETC will be calculated later.
- P lies on the line GX. In this case, if GP=p*GX then
Q(X, p) = 1/(a*((x^2*a^2-y^2*b^2+z*y*b* c-z^2*c^2)*p-(x*a+y*b+z*c)*(x* a-y*b-z*c))) : : (trilinears)
which lies on the circumconic with center O(X) = (b*y-c*z)^2*(a*x-b*y-c*z)/a :: (trilinears)
ETC pairs (X, O(X)): (1,11), (3,125), (4,122), (5,2972), (6,125), (7,13609), (8,3756), (9,11), (10,244), (11,3126), (20,13611), (30,1650), (37,244), (39,3124), (57,5514), (63,6506), (69,6388), (75,6377), (86,6627), (114,868), (115,1649), (120,3675), (141,3124), (142,3119), (216,2972), (223,5514), (230,868), (282,13612), (323,10413), (440,4466), (513,14434), (514,6544), (519,1647), (523,1649), (524,1648), (525,14401), (536,1646), (538,1645), (597,8288), (599,6791), (650,3126), (1015,14434), (1073,13613), (1086,6544), (1125,3120), (1210,7004), (1211,3125), (1212,3119), (1213,3120), (1249,122), (1645,888), (1646,891), (1647,900), (1648,690), (1649,690), (1650,9033), (2482,1648), (3117,6784), (3150,684), (3160,13609), (3161,3756), (3163,1650), (3229,2086), (3290,3675), (3341,13612), (3343,13613), (3452,2170), (3580,2088), (3666,3125), (3720,2486), (3739,3121), (3741,3122), (3752,2170), (3772,7117), (3840,3123), (4000,14936), (4370,1647), (5664,1637), (6337,6388), (6376,6377), (6505,6506), (6509,3269), (6544,900), (6554,14936), (6626,6627), (10190,14443), (10196,14442), (11019,2310), (11165,6791), (13466,1646), (13567,3269), (13636,523), (14434,891)
Some particular cases for part b):
- For X=X(1)=I, ETC pairs(P, Q(X, P)) (all P on the line IG and all Q(X, P) on the Feuerbach hyperbola):
(1,1), (2,7), (8,4), (10,79), (42,256), (43,3551), (78,84), (145,8), (200,3062), (239,2481), (976,987), (997,7284), (1125,5557), (1999,314), (2340,9442), (2398,885), (3187,2997), (3241,1000), (3244,5559), (3616,3296), (3617,5556), (3621,7319), (3622,5558), (3623,7320), (3632,5560), (3635,13606), (3679,5561), (3811,90), (3870,9), (3920,2298), (3935,1156), (3938,983), (3957,2346), (4420,10308), (4511,104), (4666,10390), (4861,1389), (5550,5551), (5552,5553), (5554,5555), (6542,7261), (7080,10309), (11679,10435)
Q( I, X(306) ) = isogonal conjugate of X(5285)
= (a^4+b*a^3+(b^2-c^2)*b*a+b^4- c^4)*(a^4+c*a^3-(b^2-c^2)*c*a- b^4+c^4) : : (barys)
= on Feuerbach hyperbola and lines: {1, 1503}, {8, 2893}, {9, 440}, {21, 3220}, {80, 2831}, {90, 1756}, {226, 2298},
{307,5285}, {314, 4872}, {1172, 1848}, {1474, 4466}, {1891, 3668}
= isogonal conjugate of X(5285)
= isotomic conjugate of X(7270)
= [ 27.980311713830330, 33.30569175610508, -32.331112140240830 ]
Q( I, X(386) ) = isogonal conjugate of X(5264)
= (b*a^2+(b^2+c^2)*a+c^2*(b+c))* (c*a^2+(b^2+c^2)*a+b^2*(b+c))* a : : (barys)
= on Feuerbach hyperbola and lines: {1, 7186}, {4, 3670}, {7, 3953}, {8, 4424}, {9, 3216}, {21, 995}, {35, 983}, {36, 987}, {38, 4894}, {46, 989}, {79, 982}, {80, 986}, {90, 988}, {314, 4389}, {1476, 4306}, {2344, 5299}, {2997, 3663}, {3296, 4694}, {3976, 5557}, {4443, 6763}
= isogonal conjugate of X(5285)
= [ 2.057840085462697, 1.96564884963397, 1.330058315793464 ]
- For X=X(3)=O, ETC pairs(P, Q(X, P)) (all P on the Euler line and all Q(X, P) on the Jerabek hyperbola):
(2,69), (3,3), (4,68), (5,3519), (20,4), (21,72), (22,6), (23,895), (25,6391), (186,5504), (376,4846), (401,290), (548,14861), (858,67), (1113,2574), (1114,2575), (1370,66), (1817,1439), (2071,74), (2937,15002), (3146,15077), (4184,71), (4225,73), (4226,879), (4230,2435), (4236,10099), (6636,1176), (7471,14220), (7488,54), (7493,5486), (7560,1246), (8613,8795), (8703,13623), (10296,11564), (10298,3431), (11413,64), (11414,3527), (11634,10097), (12225,6145)
Q( O, X(24) ) = isogonal conjugate of X(3542)
= ((SB-SC)^2*S^4-(SB+SC)^2*SA^4) *(S^2-SB*SC) : : (barys)
= 3*X(3167)-X(12309)
= on Jerabek hyperbola and lines: {4, 155}, {5, 14457}, {6, 1147}, {26, 1177}, {54, 9932}, {64, 12085}, {65, 921}, {66, 3564}, {68, 394}, {69, 3546}, {70, 858}, {74, 9938}, {140, 5486}, {265, 12429}, {511, 9908}, {539, 6145}, {1069, 9931}, {1173, 1995}, {1181, 4846}, {1498, 11744}, {1503, 12420}, {1657, 10293}, {1899, 12421}, {2931, 8907}, {3167, 3527}, {3426, 12164}, {3532, 7689}, {5654, 10982}, {7464, 13452} , {8057, 10279} , {9820, 11427}, {12038, 14528}, {12161, 14542}, {12423, 15232}
= isogonal conjugate of X(3542)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (1147, 12235, 6642), (1993, 6193, 155)
= [ 7.467418731673147, 0.40576151165678, -0.086748286934843 ]
Q( O, X(26) ) = isogonal conjugate of X(7505)
= ((SB-3*SC)*S^2+(SB+SC)*SA^2)*( (SC-3*SB)*S^2+(SB+SC)*SA^2) )*(S^2-SB*SC) : : (barys)
= 3*X(3167)-X(12309)
= on Jerabek hyperbola and lines: {4, 1994}, {6, 49}, {24, 12236}, {54, 6644}, {68, 2072}, {69, 3548}, {70, 1993}, {74, 5889}, {155, 265}, {195, 6145}, {382, 11744}, {394, 3519}, {895, 9925}, {1173, 11422}, {1176, 15073}, {1181, 3521}, {2071, 11270}, {3526, 13622}, {3527, 14627}, {5504, 9932}, {9703, 15002}, {12086, 13452}
= isogonal conjugate of X(7505)
= [ 0.690863916517471, 4.46088425096225, 0.233499731310141 ]
- For X=X(4)=H, ETC pairs(P, Q(X, P)) (all P on the Euler line and all Q(X, P) on the rectangular circum-hyperbola { 4, 20, 253, 1249, 1294, 3346, 3668, 5930, 6188, 8804, 8806, 9307, 10152, 14249, 14615, 14863}):
(2,253), (4,4), (20,3346), (25,9307), (27,3668), (140,14863), (3146,20), (13619,6188)
Q( H, X(3) ) = isogonal conjugate of X(6759)
= (S^4-(SB+2*SC)*SA*S^2+(SB+SC)* SA^2*SB)*(S^4-(2*SB+SC)*SA*S^ 2+(SB+SC)*SA^2*SC) : : (barys)
= 2*X(5)-3*X(14059) = 4*X(5)-3*X(14249) = 5*X(631)-3*X(1075)
= on cubics K071, K080, K268, K671, K827 and lines: {2, 14363}, {3, 14371}, {4, 8798}, {5, 14057}, {20, 2979}, {64, 1294}, {216, 631}, {382, 10152}, {1093, 2972}, {5930, 11362}, {7796, 14615} , {13409, 13599}
= isogonal conjugate of X(6759)
= reflection of X(i) in X(j) for these (i,j): (4, 8798), (14249, 14059)
= [ 7.600894940587217, 5.52051724622546, -3.689337430365811 ]
Q( H, X(5) ) = isogonal conjugate of X(10282)
= (S^4-(SB+2*SC)*SA*S^2+(SB+SC)* SA^2*SB)*(S^4-(2*SB+SC)*SA*S^ 2+(SB+SC)*SA^2*SC) : : (barys)
= 7*X(3090)-5*X(3462)
= on cubic K566 and lines: {20, 2888}, {140, 6760}, {233, 1249}, {381, 14249}, {1294, 6247}, {3627, 10152}, {7917, 14615}
= isogonal conjugate of X(10282)
= [ 20.146886480662310, -12.30220077497829, 2.859009719663665 ]
- For X=X(5)=N, ETC pairs(P, Q(X, P)) (all P on the Euler line and all Q(X, P) on the circum-conic 3, 5, 216, 264, 3463, 5562, 6662, 8439, 8798, 13599}):
(2,264), (3,6662), (4,3), (5,5), (3091,13599)
Q( N, X(20) ) = Q( H, X(3) )
- For X=X(6)=K, ETC pairs(P, Q(X, P)) (all P on the line GK and all Q(X, P) on the Jerabek hyperbola):
(2,4), (6,6), (69,66), (81,65), (193,69), (323,74), (333,15232), (343,6145), (385,290), (394,64), (395,11138), (396,11139), (1654,8044), (1992,5486), (1993,3), (1994,54), (2287,1903), (3051,695), (3181,2993), (3289,1987), (3580,265), (3629,13622), (5422,3527), (5590,10262), (5591,10261), (6515,68), (8115,2574), (8116,2575), (11004,3431), (11064,11744), (11427,14542), (11433,14457), (15018,14483), (15066,3426)
Q( K, X(86) ) = isogonal conjugate of X(4184)
= (a^2+(b-c)*(a+b))*(a^2-(b-c)*( a+c))*(b+c) : : (barys)
= 7*X(3090)-5*X(3462)
= on Jerabek hyperbola and lines: {3, 142}, {6, 1836}, {65, 1893}, {68, 916}, {69, 674}, {71, 1213}, {72, 3696}, {73, 3649}, {79, 4649}, {265, 2772}, {1175, 5327}, {1243, 6001}, {1400, 2486}, {1770, 3286}, {1918, 3120}, {2293, 3058}, {4675, 10013}, {5132, 12047}, {6391, 9028}
= isogonal conjugate of X(4184)
= trilinear pole of the line {647, 4988}
= [ -2.041413098842202, -2.46615516405044, 6.290193333408009 ]
Q( K, X(141) ) = isogonal conjugate of X(6636)
= (3*S^2-4*SA*SB+3*SC^2)*(3*S^2- 4*SA*SC+3*SB^2) : : (barys)
= 3*X(1176)-4*X(3589) = 3*X(2916)-5*X(3763)
= on Jerabek hyperbola and lines: {3, 2916}, {6, 5064}, {54, 1503}, {66, 9971}, {67, 1843}, {68, 10263}, {69, 1369}, {74, 7576}, {248, 2965}, {265, 11807}, {511, 3519}, {879, 1510}, {895, 3629}, {1173, 5480}, {1176, 3589}, {1352, 10627}, {2393, 13622}, {2435, 6368}, {3521, 13474} , {4846, 11818} , {5504, 14982}, {6144, 6391}, {6776, 13472}, {8718, 14788} , {10575, 14861}
= isogonal conjugate of X(6636)
= isotomic conjugate of X(7768)
= trilinear pole of the line {647, 7950}
= [ -146.276001396641700, -157.71055607809010, 180.336896257496800 ]
César Lozada
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