Interesting locus: Conic through 34 ETC centers

What is the locus of P such that the anticevian triangle of P w/r to ABC is perspective to the intangents triangle of ABC?

The answer is:

1) The antiorthic axis of ABC, line (X(44)X(513)), with trilinear equation u+v+w = 0, or

2) The circumhyperbola U with trilinear equation

U: ∑ [ a*(b-c)*(b+c-a)*v*w ] = 0

The perspector of U is X(663).

The center of U is:

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

and lies on lines (X(I)X(J)) for these (I,J): (41,4557), (649,4014), (657,3271), (1200,2246), (3124,5075)      

U pasess through A, B, C and these 34 ETC centers: X(6), X(9), X(19), X(55), X(57), X(284), X(333), X(673), X(893), X(909), X(1024), X(1174), X(1436), X(1751), X(1945), X(2160), X(2161), X(2164), (2195), X(2258), X(2259), X(2291), X(2299), X(2316), X(2319), X(2337), X(2339), X(2343), X(2364), X(2432), X(2590), X(2591), X(3451), X(3512) 

Mappings:

1) If P is on the antiorthic axis of ABC then the perspector Z(P) lies on line (X(650)X(663)) with trilinear equation: u/(s-a)+v/(s-b)+w/(s-c)=0. This line passes through this 15 ETC-centers: X(650), X(663), X(861), X(2340), X(3689), X(3900), X(4041), X(4105), X(4162), (4433), X(4435), X(4477), X(4814), X(4895), X(4959)

There are 92 known ETC centers on the antiorthic axis (up to X(5546)) and for only two of them the correspondant perspectors are defined in ETC. These [ P, Z(P) ] are [X(650), X(4162)] and [X(652), X(663)].

You can see/download the mapping from this part of the locus in table 1 and table 1.2 at end of this page, [Note: most of pairs (P, Z(P)) are not easy to handle and only the most simple ones has been calculated.]

2) If P lies on conic U, then the perspectors Z(P) lie on the hyperbola V with trilinear equation:

V: ∑ [ a*(b-c)*(b+c-a)^2*v*w ] = 0

The perspector of V is X(657).

The center of V has trilinears:

a*(b-c)^2*(-a+b+c)^2*(a^2*(a^2-b^2+b*c-c^2)-a*(b+c)*(-b^2-c^2+a^2)-b*c*(b-c)^2) : :

or, equivalently:

cos(A/2)^2*(cos(2*A)+r/R+3)*sin(B/2-C/2)^2 : :

and lies on lines (X(I)X(J)) for these (I,J): (1,4566), (11,3138), (663,3022), (997,4319), (1015,3269), (1064,2293), (1984,2310)

The hyperbola V passes through A, B, C and these 14 ETC centers: X(1), X(33), X(55), X(64), X(103), X(200), X(220),X(963), X(1043), X(2192),X(2328), X(2332), X(2342), X(4845).

For P on cubic U, (P, Z(P)) = (X(I),X(J)) for these [I,J]:   [6, 55], [9, 200], [19, 33], [55, 220], [57, 1], [284, 2328], [333, 1043], [909, 2342], [1436, 2192], [2291, 4845], [2299, 2332] .

Please see/download tables 2.1 and 2.2  for complete lists of points and perspectors.

  Direct mapping for P on antiorthic axis   Download

  Inverse mapping for Z(P) on line (X(650)X(663))   Download

  Direct mapping for P on conic U   Download

  Inverse mapping for Z(P) on conic V   Download

Note: Show tables seems to work well only with Google Chrome.

Nice trilinear coordinates

The sum and difference of two simple terms were applied to the ETC search triangle (6,9,13). The resulting numerical coordinates were sought in the ETC tables and then they were checked algebraically.

SA, SB, SC, SW are the standard variables for Conway notation.

S=2*area(ABC), R=circumradius, r=inradius and s=semiperimeter of triangle.

X(7)	(b*c-SA)/a	X(610)	SB*SC-a^2*SA	X(2223)	a^2*(a*s-SW)
X(8)	(b*c+SA)/a	X(612)	b*c+SW	X(2285)	SB*SC+a^2*b*c
X(9)	a-s	X(614)	b*c-SW	X(2300)	a^2*(a*s+SA)
X(19)	a^2*SA-S^2	X(615)	a*R-b*c	X(2308)	a*(a*r+S)
X(37)	a*r-S	X(661)	SB-SC	X(2548)	a^3*R+b*c*S
X(38)	SA+SW	X(672)	a*(a*s-SW)	X(2964)	a^2-R^2
X(41)	a^2*(a-s)	X(748)	a^2*r*s-b*c*S	X(2965)	a*(a^2-R^2)
X(42)	a*(a*r-S)	X(750)	a^2*r*s+b*c*S	X(2999)	b*c-s^2
X(47)	a^2*r*s-S*R^2	X(798)	SB^2-SC^2	X(3053)	a*(a^2-SA)
X(48)	SB*SC-S^2	X(894)	(b*c+a^2)/a	X(3068)	(S+a^2)/a
X(55)	a*(a-s)	X(940)	a*s+b*c	X(3069)	(S-a^2)/a
X(57)	SA-b*c	X(968)	a^2*r-S*s	X(3083)	b*c+S
X(58)	a^2*s+S*R	X(1100)	a*r+S	X(3084)	b*c-S
X(63)	SA	X(1124)	a*(S+b*c)	X(3299)	a+R
X(75)	SA^2+S^2	X(1193)	a*(a*s+SA)	X(3301)	a-R
X(77)	SA*(SA-b*c)	X(1203)	a*s-R*r	X(3553)	a*R+r*s
X(78)	SA*(SA+b*c)	X(1267)	(b*c+S)/a	X(3553)	a^2*b*c+S^2
X(171)	a^2+b*c	X(1335)	a*(S-b*c)	X(3554)	a*R-r*s
X(191)	SA+R*r	X(1386)	a*s+SW	X(3554)	a^2*b*c-S^2
X(193)	(SA-a^2)/a	X(1449)	a+s	X(3624)	b*c+R*r
X(213)	a^2*(a*r-S)	X(1468)	a*(a*s+b*c)	X(3666)	SA+a*s
X(238)	a^2-b*c	X(1572)	a^3*s+S^2	X(3751)	a*S-SW*r
X(239)	(b*c-a^2)/a	X(1580)	a^4-b^2*c^2	X(3915)	a*(a*s-b*c)
X(326)	b^2*c^2-S^2	X(1582)	a^4+b^2*c^2	X(4258)	a*(a^2-s^2)
X(405)	b*c*S+SA*r*a	X(1698)	b*c-R*r	X(4383)	a*s-b*c
X(474)	b*c*S-SA*r*a	X(1707)	a^2-SA	X(4512)	a^2-s^2
X(491)	(SA-S)/a	X(1743)	a*S-s^2*r	X(4641)	SA-a*s
X(492)	(SA+S)/a	X(1953)	SB*SC+S^2	X(5019)	a^2*(a*s+b*c)
X(518)	a*s-SW	X(1958)	SA^2+SB*SC	X(5058)	a^2*(a*R-b*c)
X(560)	a^2*(SA-SW)	X(1958)	a^2*SA-b^2*c^2	X(5062)	a^2*(a*R+b*c)
X(590)	a*R+b*c	X(1959)	SA^2-SB*SC	X(5266)	SA*r*a-S*SW
X(595)	a^2*s-S*R	X(1964)	SA^2-SW^2	X(5277)	a^3*r+b*c*S
X(604)	a^2*(SA-b*c)	X(1964)	a^2*(SA+SW)	X(5280)	a*SW+S*R
X(605)	a^2*(S+b*c)	X(2082)	SB*SC-a^2*b*c	X(5299)	a*SW-S*R
X(606)	a^2*(S-b*c)	X(2175)	a^3*(a-s)	X(5336)	SB*SC-a^3*s
X(609)	a^3+S*R	X(2210)	a^2*(a^2-b*c)	X(5391)	(b*c-S)/a

Parallel and perpendicular central lines

Central lines determined by 8-first ETC centers have many parallel and perpendicular central lines. 

These results have been found using ETC centers X(1)-X(5543) as published by August 25th, 2013.

Note: The line at infinity L­_∞ is both parallel and perpendicular to any line and has been removed from all lists.


NAGEL LINE (X(1), X(2))=(IG)

• Parallel lines intersecting L­_∞ at X(519):  
(3, 3654), (4, 3680), (5, 3813), (6, 996), (7, 4900), (9, 1000), (11, 3814), (12, 4870), (20, 5493), (21, 3746), (35, 2975), (36, 100), (37, 1573), (38, 3896), (40, 376), (44, 2325), (45, 3707), (46, 4311), (55, 956), (56, 4848), (57, 3476), (58, 1043), (63, 1727), (65, 553), (69, 3663), (72, 950), (74, 2692), (75, 3664), (76, 4479), (80, 908), (81, 4720), (86, 4803), (98, 2705), (99, 2712), (101, 2726), (102, 2731), (103, 2737), (104, 2077), (105, 2748), (106, 1120), (107, 2755), (108, 2756), (109, 2757), (110, 2758), (111, 2759), (112, 2760), (141, 3946), (142, 4361), (149, 3583), (171, 3996), (188, 1128), (190, 4693), (192, 4416), (193, 3729), (210, 392), (213, 3780), (214, 1145), (218, 4513), (226, 2099), (238, 765), (244, 1739), (291, 3227), (312, 4090), (319, 4021), (320, 679), (321, 3902), (329, 3586), (333, 3750), (345, 3749), (346, 1743), (350, 668), (354, 3753), (355, 381), (388, 3340), (390, 4929), (391, 3731), (405, 3303), (428, 1829), (447, 648), (474, 3304), (484, 3218), (495, 2886), (496, 1329), (497, 3421), (549, 1385), (573, 3169), (594, 1100), (595, 2985), (597, 1386), (599, 3242), (643, 5127), (644, 5526), (664, 1323), (666, 1121), (672, 1018), (751, 984), (894, 4431), (902, 3977), (942, 3754), (958, 3295), (960, 3678), (962, 3543), (966, 3247), (999, 1376), (1001, 4078), (1010, 4658), (1015, 1575), (1023, 4530), (1056, 2550), (1058, 2551), (1089, 3702), (1104, 3695), (1107, 1500), (1108, 3694), (1126, 1220), (1146, 5199), (1150, 2177), (1155, 4973), (1211, 4914), (1212, 3991), (1213, 3723), (1215, 3706), (1279, 3932), (1377, 3297), (1378, 3298), (1387, 3036), (1392, 5154), (1420, 1788), (1449, 2345), (1457, 4551), (1468, 5264), (1478, 3434), (1479, 3436), (1621, 5251), (1697, 1776), (1699, 3839), (1706, 3333), (1723, 3692), (1724, 3710), (1738, 4684), (1757, 3685), (1770, 3901), (1785, 1897), (1834, 3454), (1837, 2098), (1838, 5174), (1861, 1870), (1862, 1878), (1914, 5291), (1962, 4981), (1992, 3751), (2087, 4169), (2093, 2094), (2170, 3930), (2223, 4433), (2238, 3230), (2241, 4426), (2242, 4386), (2280, 4390), (2292, 3578), (2329, 4251), (2348, 4534), (2475, 5270), (2478, 3984), (2646, 4995), (2650, 4647), (2654, 3191), (2891, 5484), (3125, 3726), (3158, 3524), (3208, 3730), (3245, 4880), (3246, 4422), (3251, 4448), (3263, 4986), (3264, 4783), (3290, 4541), (3294, 3691), (3339, 3600), (3361, 4308), (3452, 3940), (3485, 3947), (3501, 4050), (3509, 5011), (3545, 3817), (3550, 4257), (3612, 4917), (3647, 4918), (3653, 5054), (3666, 4030), (3670, 4642), (3672, 4460), (3681, 3877), (3696, 4688), (3698, 5439), (3701, 4935), (3703, 3744), (3704, 5266), (3711, 5316), (3722, 4933), (3727, 3954), (3739, 4399), (3740, 3956), (3742, 3833), (3745, 4046), (3761, 4441), (3762, 4895), (3765, 4044), (3772, 4035), (3775, 4026), (3812, 3918), (3816, 3820), (3834, 4395), (3836, 4864), (3842, 4755), (3868, 4084), (3869, 3885), (3873, 3919), (3876, 3890), (3891, 3914), (3894, 4430), (3899, 4525), (3951, 4309), (3952, 4937), (3953, 3987), (3966, 4104), (3969, 5294), (3992, 4358), (3999, 4706), (4011, 4082), (4015, 4662), (4037, 4115), (4066, 4385), (4101, 4894), (4129, 4879), (4136, 5306), (4153, 5309), (4190, 4317), (4323, 5261), (4345, 5274), (4359, 4714), (4363, 4667), (4364, 4407), (4367, 4807), (4371, 4648), (4373, 4902), (4442, 4938), (4445, 4657), (4452, 4862), (4457, 4883), (4472, 4758), (4499, 4665), (4527, 4663), (4640, 4884), (4644, 4659), (4649, 5263), (4730, 4922), (4857, 5046), (4866, 5129), (4919, 5179), (5064, 5090), (5239, 5246), (5240, 5245), (5249, 5425), (5259, 5260), (5525, 5540)
• Perpendicular lines intersecting L­_∞ at X(3667):
(2, 4786), (3, 4057), (4, 2457), (74, 2758), (98, 2712), (99, 2705), (100, 2743), (101, 2737), (102, 2757), (103, 2726), (104, 2718), (109, 2731), (110, 2692), (145, 2403), (190, 4076), (572, 1919), (649, 3239), (650, 4773), (656, 4129), (661, 4765), (764, 4301), (885, 3062), (1027, 1721), (1292, 2748), (1293, 3699), (1294, 2755), (1295, 2756), (1296, 2759), (1297, 2760), (1519, 1769), (1577, 4811), (1768, 2957), (2254, 3798), (2394, 3429), (2487, 2496), (2517, 4791), (2976, 4394), (3676, 3999), (3700, 4790), (3737, 4794), (3756, 5510), (3762, 4397), (4017, 4170), (4025, 5211), (4040, 5529), (4086, 4807), (4162, 4504), (4380, 4468), (4404, 4462), (4724, 5524), (4893, 4984), (4931, 4979)

EULER LINE (X(2), X(3))=(GO)

• Parallel lines intersecting L­_∞ at X(30):
(1, 79), (6, 2549), (7, 3488), (8, 3578), (9, 3587), (10, 3579), (11, 36), (12, 35), (13, 15), (14, 16), (17, 5238), (18, 5237), (32, 5254), (33, 1060), (34, 1062), (40, 191), (46, 1837), (49, 1614), (50, 1989), (52, 185), (53, 577), (54, 3521), (55, 495), (56, 496), (57, 3586), (58, 1834), (61, 397), (62, 398), (63, 3419), (64, 68), (65, 1770), (69, 3426), (74, 265), (80, 484), (84, 3928), (98, 671), (99, 316), (100, 2687), (101, 2688), (102, 2689), (103, 2690), (104, 1290), (105, 2691), (106, 2692), (107, 2693), (108, 2694), (109, 2695), (110, 477), (111, 2696), (112, 2697), (113, 1495), (114, 2482), (115, 187), (119, 2077), (133, 3184), (141, 3098), (143, 389), (146, 323), (148, 385), (155, 1498), (156, 1147), (182, 597), (226, 4304), (262, 598), (284, 1901), (298, 616), (299, 617), (315, 1975), (329, 3940), (340, 1494), (371, 3070), (372, 3071), (388, 3295), (390, 1056), (485, 1151), (486, 1152), (489, 638), (490, 637), (497, 999), (498, 5217), (499, 5204), (551, 946), (553, 942), (567, 5012), (568, 3060), (574, 3815), (582, 1724), (599, 1350), (618, 623), (619, 624), (620, 625), (664, 5195), (841, 1302), (908, 5440), (910, 5179), (935, 1297), (944, 962), (956, 3434), (993, 2886), (1043, 1330), (1058, 3600), (1125, 3824), (1141, 1157), (1145, 5176), (1146, 5011), (1155, 1737), (1213, 4877), (1292, 2752), (1293, 2758), (1294, 1304), (1295, 2766), (1296, 2770), (1319, 1387), (1337, 3479), (1338, 3480), (1351, 1353), (1376, 3820), (1465, 1877), (1565, 4872), (1587, 3311), (1588, 3312), (1625, 3289), (1699, 3576), (1754, 5398), (1768, 5535), (1807, 3465), (1838, 1852), (1865, 2193), (1870, 3100), (1990, 3163), (2021, 2023), (2094, 2095), (2132, 2133), (2292, 5492), (2456, 5182), (2548, 5013), (2646, 4870), (2654, 4303), (2895, 4720), (2931, 2935), (2968, 5081), (3035, 3814), (3053, 3767), (3085, 5229), (3086, 5225), (3255, 3577), (3303, 4309), (3304, 4317), (3424, 5485), (3429, 4052), (3481, 3482), (3485, 4305), (3486, 4295), (3487, 4313), (3565, 5203), (3589, 4045), (3665, 4056), (3703, 4680), (3746, 4330), (3829, 5450), (3911, 5122), (3925, 5251), (4030, 4692), (4252, 5292), (4298, 5045), (4325, 4857), (4511, 5057), (4669, 5493), (4999, 5267), (5008, 5355), (5032, 5093), (5103, 5149), (5107, 5477), (5119, 5252), (5459, 5478), (5463, 5473), (5464, 5474)
• Perpendicular lines intersecting L­_∞ at X(523):
(1, 2605), (2, 1649), (4, 1552), (6, 879), (8, 4774), (10, 4049), (11, 1090), (12, 2599), (23, 385), (37, 3709), (59, 655), (66, 2435), (72, 3657), (74, 477), (75, 876), (98, 842), (99, 691), (100, 1290), (101, 2690), (102, 2695), (103, 2688), (104, 2687), (105, 2752), (106, 2758), (107, 1304), (108, 2766), (109, 2689), (110, 476), (111, 2770), (112, 935), (115, 5099), (125, 2677), (140, 1116), (141, 882), (160, 3164), (230, 231), (250, 648), (253, 2419), (325, 684), (351, 4108), (649, 4773), (656, 2457), (661, 3700), (764, 4647), (827, 1287), (878, 3425), (885, 2346), (930, 1291), (1086, 2643), (1101, 2612), (1145, 4736), (1146, 3708), (1222, 2403), (1292, 2691), (1293, 2692), (1294, 2693), (1295, 2694), (1296, 2696), (1297, 2697), (1325, 3733), (1459, 4449), (1577, 3932), (1635, 4809), (1638, 4379), (1639, 4893), (1962, 4448), (2407, 4226), (2486, 4516), (2487, 4369), (2525, 2526), (2527, 4394), (2528, 5189), (2530, 4768), (2549, 5486), (2594, 2616), (3569, 5112), (3726, 5098), (3762, 4985), (3776, 4818), (3777, 4801), (3835, 4500), (4092, 4934), (4170, 4983), (4240, 5502), (4391, 4490), (4444, 4665), (4462, 4811), (4467, 4608), (4552, 4557), (4707, 4730), (4724, 5160), (4776, 4951), (4833, 4879), (4885, 5159)

SODDY LINE (X(1), X(7))=(IG_e)

• Parallel lines intersecting L­_∞ at X(516):
(2, 165), (3, 142), (4, 9), (5, 3579), (6, 3755), (8, 144), (11, 1155), (27, 2328), (31, 1754), (35, 411), (46, 1210), (55, 226), (57, 497), (63, 1709), (65, 950), (69, 3886), (72, 3059), (74, 2690), (75, 3883), (79, 2346), (80, 655), (86, 4229), (98, 2702), (99, 2700), (100, 908), (101, 2724), (102, 929), (103, 927), (104, 1308), (105, 2736), (106, 2737), (107, 2738), (108, 2739), (109, 1936), (110, 2688), (111, 2740), (112, 2741), (118, 910), (145, 5059), (146, 2948), (149, 1768), (152, 1282), (153, 5541), (190, 3717), (200, 329), (214, 1537), (238, 673), (255, 1777), (284, 5327), (320, 4684), (321, 4450), (354, 553), (355, 382), (376, 551), (377, 5250), (381, 3828), (388, 1697), (412, 1838), (484, 1737), (550, 1385), (580, 3073), (612, 4656), (899, 5400), (902, 3011), (938, 3339), (944, 3243), (972, 1543), (993, 1012), (1054, 5121), (1058, 3333), (1076, 1771), (1086, 1279), (1158, 3358), (1284, 2223), (1292, 2725), (1293, 2726), (1294, 2727), (1295, 2728), (1296, 2729), (1317, 3328), (1350, 4655), (1376, 3452), (1386, 3946), (1387, 5126), (1389, 3255), (1428, 5091), (1463, 4014), (1478, 5119), (1482, 1657), (1490, 3174), (1519, 2077), (1532, 3814), (1538, 3035), (1571, 2548), (1572, 2549), (1587, 1702), (1588, 1703), (1621, 5249), (1633, 3220), (1698, 3091), (1700, 2546), (1701, 2547), (1736, 2310), (1764, 3741), (1788, 5128), (1829, 1885), (1837, 4848), (1848, 4219), (1852, 1888), (1860, 4055), (1902, 3575), (2017, 2544), (2018, 2545), (2051, 4192), (2093, 3586), (2321, 3416), (2325, 3932), (2635, 4551), (2886, 4640), (2941, 4418), (2947, 3190), (3006, 3977), (3021, 3323), (3052, 3772), (3085, 3947), (3336, 4857), (3340, 3486), (3424, 4052), (3475, 4654), (3485, 3601), (3522, 3616), (3523, 3624), (3534, 3656), (3543, 3679), (3550, 3944), (3582, 5131), (3627, 4691), (3654, 3830), (3683, 3925), (3685, 3912), (3686, 3696), (3687, 4388), (3710, 5300), (3744, 3782), (3745, 4854), (3748, 3982), (3771, 4138), (3823, 4422), (3836, 4432), (3935, 5531), (4031, 4860), (4085, 4672), (4104, 4703), (4220, 4425), (4263, 4646), (4357, 5263), (4413, 4679), (4429, 4676), (4511, 5180), (4701, 5073), (4702, 4966), (4779, 4869), (4780, 4856), (4972, 5294), (5183, 5532), (5218, 5219), (5226, 5281), (5274, 5435)
• Perpendicular lines intersecting L­_∞ at X(514):
(1,663), (2,1022), (8,4546), (10,764), (11,3328), (57,2401), (74,2688), (75,4406), (85,2140), (98,2700), (99,2702), (100,1308), (101,664), (102,2723), (103,2724), (104,2717), (105,2725), (106,2726), (107,2727), (108,2728), (109,929), (110,2690), (111,2729), (116,1146), (189,2399), (190,1016), (239,649), (241,650), (242,1459), (330,3249), (551,4448), (651,655), (653,1461), (659,667), (661,693), (668,3807), (996,4363), (1000,4419), (1024,2402), (1086,4403), (1111,2170), (1125,4874), (1292,2736), (1293,2737), (1294,2738), (1295,2739), (1296,2740), (1297,2741), (1317,3322), (1358,3323), (1635,4453), (1639,4927), (1729,3188), (1734,2254), (1768,2958), (1921,3261), (2487,2516), (2517,4086), (2526,4163), (3064,4077), (3177,3730), (3212,4253), (3675,4124), (3700,4106), (3803,4504), (3942,4858), (4010,4983), (4024,4382), (4088,4474), (4089,4530), (4105,5527), (4122,4963), (4170,4804), (4380,4467), (4397,4404), (4521,4885), (4543,4677), (4581,5018), (4774,4808), (4775,4879), (4784,4834), (4790,4897), (4806,4992), (4815,4985), (4818,4913)

LINE (X(1), X(3))=(IO)

• Parallel lines intersecting L­_∞ at X(517):
(2,392), (4,8), (5,10), (6,998), (7,1000), (9,374), (11,1737), (19,219), (20,145), (21,1389), (31,5398), (33,1905), (34,1753), (37,573), (42,1064), (44,1168), (52,1858), (59,1870), (63,956), (71,1243), (73,5399), (74,1290), (78,945), (79,5270), (80,3583), (81,4221), (84,3680), (88,1318), (98,2703), (99,2699), (100,953), (101,910), (102,1807), (103,1308), (104,901), (105,2742), (106,2743), (107,2744), (108,2745), (109,1455), (110,1325), (111,2746), (112,2747), (113,5520), (115,5164), (116,5074), (119,908), (140,1125), (150,4872), (151,1535), (169,220), (182,1386), (191,5258), (200,3940), (201,2599), (210,381), (218,2082), (221,3157), (226,495), (238,1052), (244,1149), (347,1439), (376,2094), (382,3632), (386,4646), (388,4295), (389,950), (390,3488), (399,2948), (404,5330), (405,5250), (411,3871), (496,1210), (500,2650), (546,3626), (547,3828), (548,3635), (549,551), (550,1483), (572,1100), (576,4663), (579,1108), (580,595), (582,602), (601,1468), (631,3616), (664,5088), (668,4087), (672,2170), (759,5127), (840,1292), (899,4695), (906,1951), (936,1706), (938,1058), (958,3560), (990,1350), (993,4640), (995,3752), (997,1376), (1006,1621), (1018,3693), (1042,1066), (1046,2943), (1051,2944), (1068,1426), (1072,3914), (1113,2103), (1114,2102), (1122,4862), (1124,2362), (1146,5179), (1148,1895), (1160,3640), (1161,3641), (1193,4642), (1212,3730), (1276,5239), (1277,5240), (1293,2718), (1294,2719), (1295,2720), (1296,2721), (1297,2722), (1317,3025), (1351,3751), (1352,3416), (1361,1785), (1362,3328), (1364,3319), (1387,3911), (1391,1443), (1411,2361), (1437,3193), (1451,1497), (1457,1465), (1478,1836), (1479,1837), (1490,2136), (1571,5013), (1656,1698), (1657,3633), (1682,5530), (1702,3311), (1703,3312), (1730,4245), (1739,4674), (1768,4880), (1788,3086), (1830,1877), (1838,1888), (1841,3990), (1864,3586), (2080,5184), (2171,2269), (2176,3959), (2182,2323), (2270,2324), (2292,3989), (2295,3727), (2329,3496), (2348,5526), (2550,3781), (2975,3916), (3022,3322), (3061,3501), (3062,4900), (3073,5247), (3085,3485), (3090,4002), (3091,3617), (3125,3230), (3146,3621), (3158,4930), (3190,3198), (3197,3211), (3208,3991), (3212,3673), (3216,3987), (3294,4520), (3474,3476), (3486,4294), (3509,4919), (3522,3623), (3523,3622), (3524,3653), (3526,3624), (3530,3636), (3534,3894), (3543,4661), (3545,3921), (3553,4254), (3554,5120), (3562,4296), (3584,4870), (3625,3627), (3628,3634), (3683,5251), (3689,4867), (3755,4260), (3794,4234), (3811,3913), (3822,3838), (3830,4677), (3832,4533), (3839,4539), (3843,4005), (3845,4134), (3850,4015), (3851,3983), (3853,4127), (3856,4547), (3861,3988), (3870,3895), (3892,4744), (3897,4189), (3920,4220), (3927,4853), (3952,4723), (3956,4745), (3992,4009), (4109,4167), (4330,5441), (4345,5435), (4390,5282), (4731,5055), (4816,5076), (4915,5223)
• Perpendicular lines intersecting L­_∞ at X(513):
(1,764), (2,4448), (3,3657), (6,1024), (7,885), (9,3126), (11,3025), (36,238), (37,876), (44,649), (59,651), (74,2687), (98,2699), (99,2703), (100,765), (101,1308), (102,2716), (103,2717), (104,953), (105,840), (106,2718), (107,2719), (108,2720), (109,2222), (110,1290), (111,2721), (112,2722), (125,5520), (190,660), (269,2424), (320,350), (484,1734), (644,3908), (663,855), (665,3709), (668,889), (676,2488), (751,4850), (875,4375), (884,3423), (927,1275), (957,2401), (1002,4644), (1037,1486), (1052,1054), (1086,3271), (1292,2742), (1293,2743), (1294,2744), (1295,2745), (1296,2746), (1297,2747), (1357,3756), (1361,3319), (1362,3322), (1364,3326), (1406,4186), (1423,3941), (1430,2201), (1577,4985), (1835,1874), (1960,3246), (2310,3942), (2473,2487), (2490,2505), (2500,2532), (2517,2533), (2529,3239), (2977,4925), (3004,4025), (3022,3328), (3123,3248), (3245,4730), (3261,3766), (3290,5098), (3510,5143), (3573,4585), (3700,4949), (3716,3834), (3762,4086), (3776,4458), (3789,4643), (3814,3836), (3823,5123), (3882,4436), (3909,4427), (4024,4820), (4041,4490), (4120,4944), (4162,4449), (4170,4815), (4379,4728), (4380,4824), (4382,4804), (4397,4462), (4401,5122), (4404,4768), (4453,4809), (4474,4774), (4507,4830), (4589,4601), (4594,4623), (4801,5180), (4841,4976), (4931,4958), (4960,5214)

BROCARD AXIS (X(3), X(6))=(OK)

• Parallel lines intersecting L­_∞ at X(511):
(1, 256), (2, 51), (4, 69), (5, 141), (9, 3781), (20, 185), (22, 184), (23, 110), (24, 1092), (25, 394), (26, 206), (35, 2330), (36, 1428), (40, 1045), (49, 2937), (54, 1176), (55, 611), (56, 613), (57, 3784), (66, 68), (67, 265), (72, 4416), (74, 691), (81, 4220), (83, 3399), (98, 385), (99, 2698), (100, 2699), (101, 2700), (102, 2701), (103, 2702), (104, 2703), (105, 2704), (106, 2705), (107, 450), (108, 2707), (109, 2708), (111, 352), (112, 2710), (113, 5099), (114, 325), (125, 858), (140, 143), (154, 3167), (155, 159), (165, 3097), (171, 181), (186, 249), (195, 2916), (199, 1790), (209, 4641), (230, 2023), (232, 2211), (238, 3271), (242, 1944), (283, 3145), (287, 401), (291, 3510), (295, 3509), (298, 1080), (299, 383), (343, 427), (355, 3416), (375, 3740), (376, 1992), (381, 599), (399, 2930), (403, 1568), (468, 1112), (546, 3631), (549, 597), (550, 1353), (631, 3567), (694, 3229), (843, 1296), (852, 2972), (942, 3664), (982, 1401), (984, 3688), (1113, 2105), (1114, 2104), (1177, 5504), (1194, 3051), (1196, 1613), (1292, 2711), (1293, 2712), (1294, 2713), (1295, 2714), (1297, 2715), (1364, 1936), (1370, 1899), (1385, 1386), (1397, 5329), (1425, 4296), (1437, 2915), (1482, 3242), (1569, 5477), (1595, 3867), (1654, 3786), (1656, 3763), (1733, 4459), (1757, 3507), (1764, 4192), (1812, 4231), (1818, 2183), (1959, 5360), (1976, 2065), (1986, 5095), (1994, 5012), (2003, 3955), (2070, 3447), (2095, 2097), (2274, 3764), (2323, 3220), (2653, 2670), (3033, 3684), (3090, 3619), (3091, 3620), (3100, 3270), (3101, 3611), (3124, 3231), (3155, 5408), (3156, 5409), (3218, 3937), (3219, 3690), (3266, 4576), (3448, 5189), (3508, 4876), (3576, 5429), (3578, 3681), (3579, 4663), (3627, 3630), (3888, 4645), (3909, 3936), (3932, 4553), (4846, 5486), (5054, 5215))
• Perpendicular lines intersecting L­_∞ at X(512):
(1, 875), (2, 3111), (4, 879), (6, 2444), (10, 4129), (25, 2433), (31, 5170), (32, 878), (39, 881), (51, 1640), (64, 2435), (74, 842), (98, 2698), (99, 805), (100, 2703), (101, 2702), (102, 2708), (103, 2700), (104, 2699), (105, 2711), (106, 2712), (107, 2713), (108, 2714), (109, 2701), (110, 249), (111, 843), (112, 2715), (115, 2679), (125, 5099), (187, 237), (263, 2395), (316, 850), (460, 2501), (598, 5466), (625, 3835), (650, 2499), (659, 4040), (660, 1016), (661, 4041), (670, 886), (764, 2650), (798, 3709), (884, 2440), (1015, 4128), (1292, 2704), (1293, 2705), (1294, 2706), (1295, 2707), (1296, 2709), (1297, 2710), (1326, 2605), (1491, 1734), (1500, 2084), (1570, 2451), (1577, 2533), (1691, 2483), (1692, 2211), (1962, 3251), (1968, 2909), (2021, 2491), (2024, 2507), (2030, 2492), (2031, 2510), (2032, 2508), (2142, 2143), (2254, 2530), (2378, 2379), (2421, 5118), (2611, 3937), (2643, 3271), (2959, 3737), (3022, 3708), (3267, 5207), (3669, 5194), (3777, 4905), (3801, 4707), (3837, 4992), (3919, 4049), (4014, 4403), (4088, 4808), (4162, 4790), (4378, 4449), (4401, 4782), (4498, 4724), (4813, 4814), (4895, 4979)

LINE (X(1), X(4))=(IH)

• Parallel lines intersecting L­_∞ at X(515):
(2, 3576), (3, 10), (5, 1125), (7, 3577), (8, 20), (9, 3427), (11, 1319), (12, 2646), (29, 947), (36, 80), (46, 4299), (48, 1826), (55, 1012), (56, 1210), (57, 4293), (58, 3072), (71, 1765), (74, 2689), (78, 3436), (79, 1389), (98, 2701), (99, 2708), (100, 2077), (101, 2723), (102, 1309), (103, 929), (105, 2730), (106, 2731), (107, 2732), (108, 2733), (109, 2734), (110, 2695), (111, 2735), (117, 1455), (119, 214), (140, 3634), (145, 962), (150, 5088), (153, 908), (165, 376), (200, 3421), (281, 610), (284, 1065), (354, 5434), (381, 551), (382, 1482), (411, 2975), (412, 5174), (484, 1768), (498, 3612), (546, 3636), (548, 4691), (549, 3828), (550, 3579), (573, 3686), (580, 5247), (595, 3073), (601, 5264), (602, 1724), (603, 1771), (631, 1698), (664, 4872), (910, 1146), (936, 2551), (938, 3333), (942, 4298), (956, 3419), (990, 3755), (997, 3452), (999, 4315), (1000, 3062), (1006, 1746), (1276, 5246), (1277, 5245), (1292, 2751), (1293, 2757), (1294, 2762), (1295, 2765), (1296, 2768), (1297, 2769), (1317, 1537), (1323, 1565), (1350, 3416), (1386, 5480), (1387, 1538), (1420, 3086), (1483, 3627), (1498, 3173), (1593, 5090), (1657, 3625), (1697, 4294), (1709, 4302), (1766, 2321), (1829, 3575), (1836, 2099), (1839, 1953), (1885, 1902), (2048, 5393), (2051, 5396), (2093, 2096), (2183, 2250), (2476, 3897), (2950, 5541), (2951, 4915), (3035, 5123), (3085, 3601), (3090, 3624), (3091, 3616), (3218, 5535), (3241, 3543), (3295, 4314), (3336, 4325), (3338, 4317), (3340, 4295), (3358, 3587), (3522, 3617), (3529, 3632), (3534, 3654), (3560, 5248), (3621, 5059), (3622, 3832), (3646, 5129), (3651, 5258), (3653, 5055), (3656, 3830), (3674, 4911), (3741, 4192), (3746, 5441), (3811, 5534), (3982, 5425), (4853, 5082), (5531, 5538)
• Perpendicular lines intersecting L­_∞ at X(522):
(1,1459), (7,2400), (8,4474), (9,657), (10,4036), (11,3326), (74,2695), (75,3261), (98,2708), (99,2701), (100,655), (101,929), (102,2734), (103,2723), (104,2716), (105,2751), (106,2757), (107,2762), (108,2765), (109,1309), (110,2689), (111,2768), (112,2769), (124,2968), (142,3126), (190,666), (240,656), (243,652), (244,4939), (596,3813), (649,3509), (650,1639), (659,4122), (661,4502), (663,1944), (664,1275), (676,4885), (693,1266), (1019,5214), (1026,2397), (1027,2402), (1086,4953), (1090,2310), (1292,2730), (1293,2731), (1294,2732), (1295,2733), (1296,2735), (1317,3319), (1324,4057), (1491,3835), (1635,4931), (2321,4140), (2490,2496), (2526,3004), (3063,3287), (3465,4040), (3699,4076), (3717,4041), (3762,4404), (3798,4369), (3904,4895), (3935,4088), (3971,4448), (4120,4893), (4318,4449), (4379,4750), (4459,4516), (4784,4932), (4786,4789), (4813,4988), (4838,4979), (4905,4978)

LINE (X(1), X(5))=(IN)

• Parallel lines intersecting L­_∞ at X(952):
(3, 8), (4, 145), (7, 1159), (10, 140), (20, 3621), (40, 550), (49, 3045), (84, 2136), (101, 1146), (106, 3756), (110, 3109), (150, 664), (165, 3654), (182, 996), (381, 3241), (382, 962), (390, 1000), (499, 1388), (546, 946), (547, 551), (548, 3579), (549, 3576), (572, 594), (631, 3617), (632, 1698), (942, 5083), (954, 3488), (970, 3032), (997, 3820), (999, 3476), (1125, 3628), (1319, 1737), (1352, 3242), (1353, 3751), (1478, 2099), (1479, 2098), (1595, 5090), (1656, 3616), (1699, 3656), (1871, 1891), (1898, 3057), (2246, 4530), (3065, 3652), (3090, 3622), (3091, 3623), (3243, 3254), (3245, 4316), (3295, 3486), (3421, 3940), (3523, 4678), (3530, 3626), (3627, 3633), (3635, 3850), (3649, 5270), (3817, 5066), (3853, 4301), (3911, 5126), (4221, 4720), (4311, 4848), (4317, 5221), (4437, 4482), (4511, 5176), (4534, 5540), (4861, 5086), (5046, 5330), (5258, 5428), (5326, 5444)
• Perpendicular lines intersecting L­_∞ at X(900):
(2,4800), (6,4435), (11,244), (37,665), (75,3766), (80,4674), (100,190), (149,4440), (214,1960), (335,876), (649,2527), (654,2161), (661,4976), (667,4990), (673,885), (693,4406), (889,4583), (903,4453), (1120,1320), (1145,3762), (1213,2642), (1317,4895), (1387,3960), (1491,4776), (1635,1639), (1830,1862), (1846,5151), (2092,2511), (2487,3798), (2490,3239), (2516,4521), (2517,4811), (2530,4170), (3004,4467), (3035,3716), (4024,4979), (4025,4106), (4036,4985), (4364,4486), (4374,4828), (4500,4932), (4522,4782), (4560,4833), (4765,4949), (4784,4789), (4786,4874), (4790,4820), (4813,4841)

LINE (X(1), X(6))=(IK)

• Parallel lines intersecting L­_∞ at X(518):
(2,210), (3,3433), (7,8), (10,141), (11,908), (20,3189), (21,2346), (31,3744), (35,3916), (36,3446), (38,42), (39,3774), (43,982), (55,63), (56,78), (57,200), (58,5266), (59,765), (74,2691), (80,3254), (81,1390), (86,3786), (92,1859), (98,2704), (99,2711), (100,840), (101,2725), (102,2730), (103,2736), (104,2742), (105,1280), (106,2748), (107,2749), (108,2750), (109,2751), (110,2752), (111,2753), (112,2754), (120,5519), (144,145), (149,5057), (165,3158), (171,3961), (181,3687), (182,1385), (190,3685), (191,3746), (209,306), (214,5126), (226,2886), (239,335), (241,1458), (243,1897), (244,899), (291,1575), (312,3967), (318,1887), (321,3706), (329,497), (333,3757), (355,1352), (386,4719), (404,4420), (474,3338), (484,4880), (551,597), (579,3694), (583,1009), (599,3679), (612,940), (614,4383), (643,2651), (650,5098), (651,1456), (668,1921), (672,3693), (677,1814), (756,3720), (846,3750), (869,2274), (872,1193), (894,5263), (896,902), (910,1282), (936,3333), (938,2551), (946,3813), (959,1219), (961,1257), (976,1468), (978,3976), (986,4646), (988,4255), (993,5138), (997,999), (1026,3675), (1046,5255), (1054,5524), (1066,3682), (1086,1738), (1125,3589), (1156,1320), (1210,1329), (1214,3190), (1215,3741), (1222,1431), (1260,1617), (1278,3621), (1330,5015), (1331,2361), (1351,1482), (1353,1483), (1362,3323), (1400,3965), (1418,4334), (1423,4073), (1465,4551), (1478,3419), (1621,3219), (1697,4326), (1698,3697), (1706,3339), (1707,3052), (1737,5123), (1768,5537), (1824,1889), (1829,1843), (1836,3434), (1837,3436), (1861,1876), (1875,5081), (1888,5174), (1961,4038), (1962,3989), (1992,3241), (2076,3099), (2093,2097), (2102,2104), (2103,2105), (2113,3797), (2136,2951), (2177,4414), (2223,3286), (2238,3290), (2260,3949), (2264,5279), (2280,5282), (2285,3713), (2292,2667), (2308,4722), (2330,2646), (2352,3998), (2475,5178), (2650,3728), (2895,4914), (2930,2948), (2999,3677), (3006,3936), (3030,5212), (3035,3660), (3062,3680), (3097,3795), (3098,3579), (3187,3891), (3216,3953), (3240,4003), (3244,3629), (3245,5528), (3293,3670), (3295,3927), (3303,3951), (3304,3984), (3305,3715), (3306,3711), (3340,4853), (3361,5438), (3452,3816), (3501,4515), (3550,4650), (3576,5085), (3616,3618), (3617,3620), (3619,3983), (3623,3890), (3624,4533), (3625,3630), (3626,3631), (3632,3901), (3633,4718), (3634,4015), (3635,3884), (3636,3988), (3644,3885), (3662,4429), (3663,3755), (3688,3879), (3699,5205), (3702,4043), (3705,4417), (3712,3977), (3714,4385), (3717,3912), (3721,3780), (3729,3886), (3730,3991), (3756,5121), (3771,4438), (3781,4851), (3782,3914), (3784,4952), (3792,4553), (3817,3829), (3821,4085), (3824,3841), (3828,3833), (3840,4090), (3883,4416), (3898,4525), (3918,4691), (3919,4669), (3922,4678), (3925,5249), (3929,4428), (3946,4353), (3952,4009), (3968,4745), (3971,4891), (4000,4310), (4001,4030), (4004,4668), (4026,4357), (4028,4884), (4042,5271), (4071,4119), (4111,4733), (4113,4359), (4141,4933), (4295,5082), (4307,4644), (4327,5228), (4349,4667), (4388,4514), (4407,4708), (4519,4671), (4523,4852), (4541,4987), (4652,5217), (4655,4660), (4701,4757), (4753,4974), (4812,5016), (4855,5204), (4930,5093), (4973,5122), (5104,5184), (5175,5229), (5219,5231), (5322,5347), (5531,5536) [Total: 259 lines]
• Perpendicular lines intersecting L­_∞ at X(3309):
(1,3669), (3,667), (4,885), (8,4462), (40,4063), (74,2752), (98,2711), (99,2704), (100,2742), (101,2736), (102,2751), (103,2725), (104,840), (109,2730), (110,2691), (218,2440), (644,1292), (649,3803), (650,1734), (663,905), (764,1482), (1293,2748), (1294,2749), (1295,2750), (1296,2753), (1297,2754), (1538,3835), (1565,3022), (2488,3798), (2496,3812), (2530,4775), (3579,4782), (3657,4846), (3777,4879), (4041,4724), (4077,4106), (4394,4401), (4449,4895), (4498,4729), (4553,4752), (4790,5525), (4904,5511)

LINE (X(2), X(6))=(GK)

• Parallel lines intersecting L­_∞ at X(524):
(1,4364), (3,5486), (4,5485), (5,576), (7,4361), (8,4363), (9,4851), (10,4472), (11,4396), (12,4400), (15,5463), (16,5464), (23,2930), (32,3933), (37,3879), (44,3912), (50,4558), (53,317), (67,858), (74,2696), (75,4399), (76,598), (98,2709), (99,843), (100,2721), (101,2729), (102,2735), (103,2740), (104,2746), (105,2753), (106,2759), (107,2763), (108,2767), (109,2768), (110,2770), (115,5107), (140,575), (145,4419), (182,549), (187,2482), (190,3943), (237,1634), (238,4966), (239,320), (249,1691), (265,5505), (287,1494), (297,340), (306,4641), (314,3770), (315,5254), (316,671), (319,594), (332,2305), (338,3260), (376,1350), (381,1351), (397,633), (398,634), (428,1843), (441,3284), (468,3292), (487,1152), (488,1151), (500,3811), (547,5097), (551,1386), (620,2030), (621,5318), (622,5321), (623,5459), (625,5461), (626,5305), (637,3070), (638,3071), (670,3978), (694,3228), (750,4023), (896,3712), (980,4277), (1003,4048), (1030,1444), (1046,3704), (1078,5038), (1084,3229), (1100,4357), (1125,4708), (1146,1944), (1238,2965), (1279,4684), (1330,1834), (1428,5298), (1431,4496), (1449,4657), (1469,5434), (1570,5031), (1609,3964), (1692,5215), (1698,4798), (1757,3932), (1901,2893), (1959,4053), (1989,2987), (1999,4415), (2094,2097), (2245,3882), (2330,4995), (2345,4445), (2549,5077), (3008,3834), (3053,3926), (3056,3058), (3187,3782), (3241,3242), (3363,5475), (3416,3679), (3524,5085), (3545,5102), (3616,4748), (3617,4470), (3621,4454), (3626,4796), (3632,4659), (3634,4758), (3661,3758), (3662,3759), (3663,4852), (3664,3686), (3665,4372), (3666,4001), (3703,4376), (3785,5013), (3818,3845), (3828,3844), (3867,5064), (3929,5227), (3946,4856), (4026,4649), (4028,4640), (4046,4418), (4104,4682), (4205,4658), (4370,4437), (4384,4675), (4389,4393), (4433,4436), (4447,4557), (4480,4727), (4681,4889), (4683,4854), (4771,4987), (4956,5057), (5028,5309), (5035,5337), (5050,5054), (5055,5093), (5103,5111)
• Perpendicular lines intersecting L­_∞ at X(1499):
(1,4897), (3,669), (4,1550), (74,2770), (98,843), (99,2709), (100,2746), (101,2740), (102,2768), (103,2729), (104,2721), (109,2735), (110,2696), (125,2682), (352,3288), (879,3426), (1125,2487), (1292,2753), (1293,2759), (1294,2763), (1295,2767), (1513,3569), (1514,1637), (1992,2408), (2394,3424), (2686,5512), (3049,5166), (3265,3804), (3700,4761)

LINE (X(2), X(7))=(GG_e)

• Parallel lines intersecting L­_∞ at X(527):
(1, 4419), (6, 3663), (8, 4454), (10, 3927), (37, 3664), (44, 1086), (45, 4675), (69, 2321), (72, 1242), (75, 3686), (86, 4877), (190, 320), (192, 3879), (193, 3875), (200, 3474), (219, 3668), (239, 1266), (269, 2324), (319, 4060), (321, 4001), (345, 4035), (346, 4488), (347, 1419), (376, 2096), (377, 3951), (381, 2095), (390, 3241), (551, 993), (599, 2097), (651, 2323), (666, 673), (896, 3011), (940, 4656), (950, 3868), (958, 3671), (960, 4298), (1023, 4089), (1100, 4021), (1125, 4364), (1150, 4054), (1156, 3254), (1275, 4564), (1386, 4353), (1429, 5053), (1449, 3672), (1478, 2093), (1654, 4967), (1698, 4470), (1737, 4880), (1738, 1757), (1743, 4000), (1836, 4847), (2295, 4503), (2329, 3674), (2340, 3000), (2346, 3255), (2551, 3339), (2951, 3174), (3161, 4869), (3191, 4303), (3247, 3945), (3263, 4987), (3436, 4848), (3475, 4512), (3578, 4980), (3616, 4747), (3626, 4665), (3629, 4852), (3634, 3824), (3636, 4796), (3638, 5240), (3639, 5239), (3655, 4930), (3661, 4741), (3685, 4684), (3707, 4384), (3717, 4645), (3723, 4909), (3731, 4648), (3751, 3755), (3758, 4389), (3759, 4398), (3782, 4641), (3820, 3822), (3831, 4721), (3834, 4422), (3840, 4713), (3913, 5493), (3935, 5528), (3936, 3977), (3950, 4851), (3973, 4859), (3980, 4104), (3984, 4190), (4007, 4461), (4058, 4445), (4138, 4438), (4315, 5289), (4346, 5222), (4373, 4402), (4393, 4982), (4399, 4726), (4409, 4969), (4465, 4871), (4679, 4860), (4898, 4916)
• Perpendicular lines intersecting L­_∞ at X(?) (see [1] below):
(1,3676), (4,3064), (8,4468), (10,4521), (40,649), (885,3577), (946,3835)

LINE (X(3), X(7))=(OG_e)

• Parallel lines intersecting L­_∞ at X(?) (see [2] below):
(4, 144), (5, 9), (11, 5536), (40, 495), (140, 142), (165, 4654), (355, 5223), (390, 1482), (956, 962), (1385, 5542), (1483, 3243), (1484, 3254), (1699, 3929), (1754, 3782), (3332, 4419), (3817, 5325)
• Perpendicular lines intersecting L­_∞ at X(?) (see [3] below):
(663,676), (1577,4990)

LINE (X(3), X(8))=(ON_a)

• This line is parallel to line (X(1), X(5) ) = (IN) 
(1,5), (4,145), (7,1159), (10,140), (20,3621), (40,550), (49,3045), (84,2136), (101,1146), (106,3756), (110,3109), (150,664), (165,3654), (182,996), (381,3241), (382,962), (390,1000), (499,1388), (546,946), (547,551), (548,3579), (549,3576), (572,594), (631,3617), (632,1698), (942,5083), (954,3488), (970,3032), (997,3820), (999,3476), (1125,3628), (1319,1737), (1352,3242), (1353,3751), (1478,2099), (1479,2098), (1595,5090), (1656,3616), (1699,3656), (1871,1891), (1898,3057), (2246,4530), (3065,3652), (3090,3622), (3091,3623), (3243,3254), (3245,4316), (3295,3486), (3421,3940), (3523,4678), (3530,3626), (3627,3633), (3635,3850), (3649,5270), (3817,5066), (3853,4301), (3911,5126), (4221,4720), (4311,4848), (4317,5221), (4437,4482), (4511,5176), (4534,5540), (4861,5086), (5046,5330), (5258,5428), (5326,5444)

LINE (X(4), X(6))=(HK)

• Parallel lines intersecting L­_∞ at X(1503):
(2,154), (3,66), (5,182), (10,3429), (11,1428), (12,2330), (20,64), (22,161), (23,3448), (25,1619), (40,1761), (51,428), (65,1891), (67,74), (98,230), (99,2710), (100,2747), (101,2741), (102,2769), (104,2722), (110,858), (114,5026), (115,1692), (125,468), (132,1529), (140,5092), (147,325), (184,427), (185,1843), (193,3146), (221,388), (235,1974), (242,1146), (265,1177), (287,297), (323,5189), (376,599), (381,597), (382,1351), (383,395), (394,1370), (396,1080), (440,2328), (497,2192), (546,575), (550,3098), (576,1353), (578,1595), (611,1478), (613,1479), (631,3763), (852,1624), (944,3242), (946,1386), (958,3556), (1211,4220), (1292,2754), (1293,2760), (1294,2764), (1297,2867), (1368,1660), (1456,5236), (1570,5477), (1594,1614), (1650,5502), (1657,3630), (1836,5307), (1854,3486), (1861,2182), (1890,2262), (1976,2450), (1992,3543), (2023,2024), (2096,2097), (2456,5103), (2550,3197), (2892,2930), (2916,2917), (3091,3618), (3426,5486), (3522,3620), (3523,3619), (3830,5093), (3845,5476), (3853,5097), (5012,5133), (5034,5475)
• Perpendicular lines intersecting L­_∞ at X(525):
(2,1640), (3,878), (4,2435), (74,2697), (98,2710), (99,249), (100,2722), (103,2741), (104,2747), (105,2754), (106,2760), (107,2764), (109,2769), (110,935), (112,2867), (127,1562), (297,850), (323,401), (339,3269), (441,647), (448,4467), (449,4468), (656,4064), (669,2528), (676,4990), (1019,4897), (1073,2416), (1565,3708), (1577,3700), (1636,3268), (1975,2422), (2420,4235), (2474,2514), (2485,2506), (2513,2531), (2533,4122), (2632,2968), (3049,4580), (3267,4143), (3716,4142), (3801,4010), (4041,4088), (4049,4052), (4730,4808), (5466,5485)

LINE (X(4), X(7))=(HG_e)

• Parallel lines intersecting L­_∞ at X(971):
(1,1419), (3,9), (5,142), (6,990), (11,1538), (20,72), (33,222), (37,991), (40,2951), (55,1709), (56,1898), (57,1750), (65,4312), (103,910), (104,1156), (152,1530), (165,210), (241,1736), (354,1699), (355,2550), (390,944), (411,3916), (651,3100), (946,5045), (954,1012), (960,4297), (962,3555), (984,1742), (999,4321), (1001,1385), (1155,1768), (1158,3579), (1159,3577), (1214,2947), (1350,1766), (1445,3149), (1458,2310), (1465,2635), (1482,3243), (1498,3157), (1721,3751), (1754,4641), (1836,5173), (2950,5528), (3091,5439), (3146,3868), (3174,5534), (3295,4326), (3332,4644), (3522,3876), (3689,5531), (3740,5325), (3742,3817), (3786,4229), (3931,4335)
• Perpendicular lines intersecting L­_∞ at X(3900):
(1,905), (8,885), (100,4564), (210,4543), (649,4729), (650,663), (657,4130), (667,4394), (764,3680), (884,4513), (960,4522), (1146,3022), (1491,4879), (1960,2516), (2192,2431), (2254,3669), (2488,4765), (3064,3700), (3158,3251), (3239,4148), (3270,4081), (3271,4542), (3716,4147), (3803,4063), (3939,4752), (4131,4467), (4705,4775)

LINE (X(4), X(8))=(HN_a)

• This line is parallel to line (X(1), X(3))=(IO)

LINE (X(5), X(6))=(NK)

• Parallel lines intersecting L­_∞ at X(3564):
(2,3167), (3,69), (4,193), (26,159), (52,1843), (67,5504), (98,325), (110,468), (113,5095), (114,230), (115,1570), (125,3292), (140,141), (147,385), (156,206), (184,343), (265,895), (287,441), (323,858), (355,3751), (381,1992), (383,3181), (394,1368), (427,1993), (428,3060), (429,3193), (495,611), (496,613), (546,576), (547,597), (548,3098), (549,599), (550,1350), (575,3589), (631,3620), (632,3763), (1080,3180), (1176,3519), (1483,3242), (1511,5181), (1625,2211), (1656,3618), (1994,3410), (2023,2025), (2080,3793), (2895,4220), (2930,2931), (3526,3619), (3530,3631), (3545,5032), (3815,5034), (3845,5102), (3850,5097), (5028,5254), (5066,5476)
• Perpendicular lines intersecting L­_∞ at X(3566):
(64,879), (669,3265), (1640,1853), (2088,3143), (2450,3569), (2489,2506), (2491,2524), (2501,5203), (2525,3804), (2533,3700), (4088,4729), (4170,4707), (4367,4897), (4874,4990)

LINE (X(5), X(7))=(NG_e)

• Parallel lines intersecting L­_∞ at X(?) (see [4] below):
(3, 144), (9, 140), (142, 3628), (355, 4312), (390, 1483), (1156, 1484), (2096, 3940), (2951, 5534)
• Perpendicular lines intersecting L­_∞ at X(?) (see [5] below):
(676,4162)

LINE (X(5), X(8))=(NN_a)

• Parallel lines intersecting L­_∞ at X(?) (see [6] below):
(1, 140), (3, 145), (4, 3621), (10, 3628), (36, 1317), (40, 548), (119, 4867), (165, 3655), (355, 546), (495, 2099), (496, 2098), (547, 3679), (549, 3241), (550, 944), (631, 3623), (632, 3616), (946, 3625), (962, 3627), (1000, 2346), (1056, 1159), (1145, 4511), (1320, 1484), (1385, 3244), (1387, 1737), (1656, 3617), (3036, 3814), (3090, 4678), (3526, 3622), (3576, 3654), (3656, 4677), (3820, 5289), (3861, 4301), (4534, 5526), (5535, 5541)
• Perpendicular lines intersecting L­_∞ at X(?) (see [7] below):
(21,3733), (649,1639), (661,4984), (2254,4806), (2487,3835), (2527,3239), (3649,4017), (3700,4958), (3798,4940), (4453,4897), (4773,4893), (4782,4925), (4790,4944), (4813,4976), (4905,4992)

LINE (X(6), X(7))=(KG_e)

• Parallel lines intersecting L­_∞ at X(?) (see [8] below):
(9,141), (41,3665), (69,144), (101,1565), (142,3589), (150,1146), (193,4440), (348,3207), (390,3242), (597,4795), (599,4370), (903,1992), (1001,4364), (1386,4667), (2550,4363), (3416,4901), (3618,4747), (3620,4473), (3751,4312), (3763,4748), (3826,4472), (4904,5540)

LINE (X(6)X(8))=(KN_a)

• Parallel lines intersecting L­_∞ at X(?) (see [9] below):
(1,141), (10,1386), (31,3703), (37,3883), (42,4030), (44,3717), (63,4884), (69,145), (100,5078), (193,3621), (238,3932), (306,3744), (345,3052), (355,5480), (595,3695), (597,3679), (599,3241), (612,3966), (902,3712), (944,1350), (1086,4645), (1125,3844), (1211,3920), (1279,3912), (1352,1482), (1697,5227), (1738,4395), (1743,4901), (1834,5015), (1999,4514), (2550,4361), (2886,4362), (2975,4265), (3008,3823), (3035,4434), (3187,5014), (3244,3631), (3616,3763), (3617,3618), (3619,3622), (3620,3623), (3625,4663), (3629,3632), (3630,3633), (3685,3943), (3704,5255), (3705,3769), (3722,4062), (3745,4914), (3755,4852), (3756,5211), (3782,3891), (3790,4676), (3811,5396), (3867,5090), (4153,5305), (4307,4363), (4349,4670), (4366,4437), (4388,4415), (4684,4864), (5241,5297)
• Perpendicular lines intersecting L­_∞ at X(?) (see [10] below):
(4897, 4905)

LINE (X(7), X(8))=(G_eN_a)

• This line is parallel to line (X(1), X(6))=(IK)

NOTES Non-ETC intersections:  (All barycentrics)

[1]: (b-c)*(3*a^3-(5*(b+c))*a^2+(b+c)^2*a+(b+c)*(b-c)^2) : :

[2]: 2*a^6-(2*(b+c))*a^5-(3*(b^2+c^2))*a^4+(2*(b+c))*(b^2+c^2)*a^3+(2*(b^3-c^3))*(b-c)*a^2-(b^2-c^2)^2*(b-c)^2 : :

[3]: (b-c)*(2*a^3-(3*(b+c))*a^2+(b^2-c^2)*(b-c)) : :

[4]: 2*a^6+(-7*c^2-7*b^2+4*b*c)*a^4+(2*(b+c))*(b^2+c^2)*a^3+(6*(b^3-c^3))*(b-c)*a^2-2*(b-c)^2*(b+c)^3*a-(b^2-c^2)^2*(b-c)^2 : :

[5]: (b-c)*(4*a^3-(7*(b+c))*a^2+2*(b+c)^2*a+(b^2-c^2)*(b-c)) : :

[6]: 2*a^4-(4*(b+c))*a^3-(c^2+b^2-8*b*c)*a^2+(4*(b^2-c^2))*(b-c)*a-(b^2-c^2)^2 : :

[7]: (b-c)*(b+c-4*a) : :

[8]: 2*a^4-(2*(b+c))*a^3+a^2*(b^2+c^2)-(b^2+c^2)*(b-c)^2 : :

[9]: 2*a^3+(b^2+c^2)*a-(b+c)*(b^2+c^2) : :

[10]: (b-c)*(2*a^3+(b+c)*a^2+(2*(b^2+c^2))*a-(b+c)*(b^2+c^2)) : :