X₈

NAGEL POINT

Let N_aN_bN_c be the points in which the A-excircle meets BC, the B-excircle meets CA, and the C-excircle meets AB, respectively. The lines AN_a, BN_b, CN_c concur in X₈. Another construction of N_a is to start at A and trace around ABC half its perimeter, and similarly for N_b and N_c. Also, X₈ is the incenter of the anticomplementary triangle.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₈ (NAGEL POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Extouch Triangle △N_aN_bN_c — vertices N_a, N_b, N_c (Contact triangle of Excircle touchpoints on BC, CA, AB)

• Concurrence Lines: A–N_a, B–N_b, C–N_c concurring at X₈

Trilinears: \((b + c - a)/a : (c + a - b)/b : (a + b - c)/c\)

Barycentrics: \(b + c - a : c + a - b : a + b - c\)

Tripolars: \(\sqrt{a (a^{2}-a b-2 b^{2}-a c+4 b c-2 c^{2})} : :\)

Search Value (6, 9, 13): \(4.507489358552\)

▸ Alternate Coordinates7 forms

• Trilinears: \(\csc^{2}(A/2) : \csc^{2}(B/2) : \csc^{2}(C/2)\)

• Trilinears: \((bc + S_{A})/a : (ca + S_{B})/b : (ab + S_{C})/c\)

• Trilinears: \((1 + \cos A)a^{-2} : :\)

• Trilinears: \((r/R) - \sin B \sin C : :\)

• Barycentrics: \(\cot A/2 : \cot B/2 : \cot C/2\)

• Barycentrics: square of semi-minor axis of A-Soddy ellipse : :

• Barycentrics: \(bc + S_{A} : ca + S_{B} : ab + S_{C}\)

▸ Linear Combinations2 combinations

• X₈ = \(2\) X₁ - \(3\) X₂ = \(3\) X₂ - \(4\) X₁₀ = \(4\) X₁₀ - X₁₄₅

• X₈ = A\(R_{a} +\) B\(R_{b} +\) C\(R_{c}\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) = radii of Soddy circles

X₈ lies on these curves: Feuerbach hyperbola, anticomplementary Feuerbach hyperbola, Mandart hyperbola, Fuhrmann circle, K007, K013, K028, K033, K034, K154, K157, K199, K200, K201, K259, K308, K311, K338, K366, K386, K387, K455, K461, K506, K521, K623, K651, K654, K680, K692, K696, K697, K702, K744, K767, Q045

X₈ lies on the Lucas cubic.

Related Collection 1 Topic Guides

T_2: Isotomic Conjugate • T_3: Complement & Anticomplement • T_7: The Nagel Line • T_12: The Fuhrmann & Spieker Central Lines • T_14: The Incircle & Excircles • T_17: The Spieker Circle • T_20: Medial & Anticomplementary Triangles • T_22: Intouch (Contact) & Extouch Triangles

▸ Incident Central Lines456 lines

{1, 2}, {3, 100}, {4, 72}, {5, 1389}, {6, 594}, {7, 65}, {9, 346}, {11, 1320}, {12, 2099}, {19, 1891}, {20, 40}, {21, 55}, {22, 8193}, {23, 8185}, {25, 7718}, {29, 219}, {30, 3578}, {31, 987}, {32, 5291}, {33, 1039}, {34, 1041}, {35, 993}, {36, 4188}, {37, 941}, {38, 986}, {39, 7976}, {41, 2329}, {44, 4217}, {45, 3943}, {46, 3218}, {56, 404}, {57, 1219}, {58, 996}, {60, 7058}, {76, 668}, {79, 758}, {80, 149}, {81, 1010}, {86, 2334}, {90, 2994}, {101, 1311}, {109, 2370}, {113, 7978}, {114, 7970}, {115, 7983}, {119, 6941}, {125, 7984}, {140, 1483}, {141, 3242}, {142, 3243}, {144, 516}, {147, 9864}, {150, 1930}, {151, 2817}, {153, 2800}, {165, 3522}, {171, 1468}, {172, 4386}, {175, 1270}, {176, 1271}, {177, 556}, {178, 236}, {181, 959}, {188, 2090}, {190, 528}, {191, 3065}, {192, 256}, {193, 894}, {194, 730}, {197, 1603}, {201, 1937}, {210, 312}, {213, 981}, {214, 5445}, {215, 9701}, {220, 294}, {221, 651}, {224, 914}, {226, 3340}, {238, 983}, {244, 3976}, {253, 307}, {274, 1002}, {277, 1280}, {278, 1257}, {279, 7273}, {291, 330}, {304, 3263}, {313, 2997}, {314, 1264}, {315, 760}, {326, 1442}, {344, 480}, {348, 664}, {350, 3789}, {354, 3698}, {376, 3579}, {381, 8148}, {392, 1000}, {394, 3562}, {405, 943}, {406, 1061}, {411, 3428}, {442, 495}, {443, 942}, {474, 999}, {475, 1063}, {479, 7182}, {484, 4299}, {491, 1267}, {492, 5391}, {496, 3820}, {514, 4546}, {521, 4397}, {522, 4474}, {523, 4774}, {524, 4363}, {527, 4454}, {529, 3474}, {535, 3245}, {536, 4419}, {537, 4440}, {573, 3588}, {595, 1724}, {596, 4674}, {599, 1086}, {618, 7975}, {619, 7974}, {631, 1385}, {637, 7595}, {641, 7981}, {642, 7980}, {643, 1098}, {645, 4092}, {646, 3271}, {663, 4147}, {672, 3501}, {673, 4437}, {704, 8264}, {712, 4805}, {726, 1278}, {860, 1068}, {885, 3900}, {901, 2757}, {908, 946}, {912, 5553}, {961, 1460}, {965, 2256}, {971, 9961}, {982, 4457}, {1015, 1574}, {1016, 1083}, {1018, 3730}, {1019, 4807}, {1034, 1895}, {1036, 1183}, {1046, 4418}, {1054, 9457}, {1071, 6916}, {1104, 3744}, {1106, 9363}, {1107, 2276}, {1120, 3445}, {1124, 1377}, {1126, 6539}, {1147, 9933}, {1159, 6147}, {1191, 4383}, {1209, 7979}, {1211, 1834}, {1212, 3693}, {1215, 4865}, {1229, 6601}, {1237, 4485}, {1251, 5239}, {1266, 4346}, {1279, 6687}, {1281, 1282}, {1309, 2745}, {1312, 2103}, {1313, 2102}, {1317, 1388}, {1319, 6049}, {1331, 2988}, {1332, 8759}, {1335, 1378}, {1361, 3042}, {1364, 3040}, {1386, 3618}, {1392, 5048}, {1397, 2985}, {1420, 3911}, {1449, 4982}, {1453, 5294}, {1467, 8732}, {1500, 1573}, {1512, 5720}, {1575, 2275}, {1656, 5901}, {1672, 1680}, {1673, 1681}, {1674, 1678}, {1675, 1679}, {1682, 9564}, {1699, 3832}, {1738, 3620}, {1739, 3953}, {1743, 4058}, {1748, 6197}, {1757, 3923}, {1759, 5011}, {1783, 8743}, {1812, 3193}, {1836, 3962}, {1857, 1896}, {1869, 5307}, {1897, 7358}, {1914, 4426}, {1943, 4296}, {1953, 3949}, {1959, 7379}, {1992, 3758}, {1997, 3816}, {2007, 2013}, {2008, 2014}, {2053, 8851}, {2077, 5450}, {2093, 4001}, {2122, 2123}, {2170, 3061}, {2175, 4157}, {2176, 2238}, {2242, 5277}, {2310, 9365}, {2318, 2654}, {2320, 2646}, {2335, 3694}, {2363, 6043}, {2399, 8058}, {2463, 2467}, {2464, 2468}, {2477, 9702}, {2482, 9884}, {2564, 2568}, {2565, 2569}, {2647, 4332}, {2650, 4938}, {2785, 4088}, {2787, 4730}, {2796, 8596}, {2801, 5696}, {2810, 3888}, {2883, 7973}, {2891, 3754}, {2893, 2897}, {2894, 6839}, {2896, 9857}, {2901, 3995}, {2943, 9355}, {3021, 3039}, {3022, 3041}, {3038, 6018}, {3056, 4110}, {3058, 3715}, {3068, 7969}, {3069, 7968}, {3090, 5886}, {3096, 9997}, {3152, 6360}, {3158, 3601}, {3174, 7675}, {3177, 4712}, {3247, 5257}, {3254, 4858}, {3304, 4413}, {3305, 5129}, {3306, 3333}, {3309, 4462}, {3336, 4317}, {3339, 4298}, {3361, 4315}, {3427, 6836}, {3452, 3680}, {3467, 4309}, {3475, 3925}, {3496, 5282}, {3523, 3576}, {3524, 3655}, {3545, 3656}, {3583, 3899}, {3585, 4067}, {3619, 3844}, {3629, 7227}, {3631, 7232}, {3647, 5441}, {3649, 6175}, {3663, 4452}, {3664, 4924}, {3666, 4646}, {3670, 3987}, {3671, 5290}, {3672, 3755}, {3683, 5302}, {3716, 4895}, {3721, 3959}, {3731, 3950}, {3735, 3954}, {3738, 4768}, {3739, 4648}, {3740, 3983}, {3742, 4731}, {3746, 5248}, {3760, 6381}, {3762, 3887}, {3772, 4952}, {3775, 4085}, {3814, 5154}, {3817, 5068}, {3823, 4864}, {3826, 4966}, {3829, 7173}, {3841, 5425}, {3879, 3945}, {3881, 3918}, {3884, 3992}, {3891, 4972}, {3892, 3968}, {3894, 3919}, {3896, 3931}, {3898, 3956}, {3901, 4084}, {3904, 4528}, {3907, 4041}, {3928, 5128}, {3929, 7285}, {3947, 5726}, {3963, 9052}, {3967, 4005}, {3977, 4304}, {3978, 6382}, {3986, 4898}, {3993, 4704}, {4002, 5045}, {4003, 4706}, {4004, 5551}, {4018, 4980}, {4026, 4360}, {4036, 8702}, {4054, 9612}, {4080, 4792}, {4082, 4866}, {4086, 7253}, {4087, 4531}, {4125, 4857}, {4160, 4761}, {4163, 6332}, {4181, 4182}, {4208, 5249}, {4234, 4921}, {4312, 5850}, {4314, 4512}, {4342, 4900}, {4364, 4748}, {4373, 4862}, {4404, 6003}, {4407, 4743}, {4421, 5217}, {4424, 7226}, {4432, 4473}, {4439, 4527}, {4470, 4670}, {4534, 6558}, {4542, 4582}, {4595, 8299}, {4657, 4852}, {4658, 8025}, {4667, 4747}, {4672, 4753}, {4675, 4688}, {4694, 9335}, {4699, 4732}, {4729, 6002}, {4736, 6758}, {4756, 9668}, {4767, 9669}, {4867, 5141}, {4922, 9508}, {4999, 5432}, {5010, 5267}, {5056, 8227}, {5059, 5493}, {5187, 10043}, {5221, 5434}, {5284, 6767}, {5285, 7520}, {5286, 9620}, {5429, 8258}, {5534, 6908}, {5584, 7411}, {5590, 5604}, {5591, 5605}, {5592, 6546}, {5597, 5600}, {5598, 5599}, {5714, 9654}, {5791, 6857}, {5985, 10053}, {6001, 6223}, {6062, 7068}, {6144, 7277}, {6154, 9963}, {6174, 10031}, {6193, 9928}, {6245, 6282}, {6260, 7971}, {6261, 6838}, {6265, 6949}, {6292, 7977}, {6326, 6960}, {6462, 8214}, {6463, 8215}, {6553, 8056}, {6653, 6655}, {6691, 7231}, {6739, 6742}, {6828, 7680}, {6835, 7686}, {6856, 8164}, {6945, 7681}, {6995, 7713}, {7003, 7020}, {7018, 7033}, {7028, 8242}, {7043, 7126}, {7048, 8422}, {7090, 7133}, {7161, 7206}, {7279, 9723}, {7373, 9342}, {7486, 9624}, {7987, 9588}, {8092, 8125}, {8094, 9795}, {8126, 8351}, {8162, 8167}, {8163, 8169}, {8210, 8222}, {8211, 8223}, {8372, 9787}, {8591, 9881}, {8972, 8983}, {9317, 9451}, {9783, 9805}, {9859, 9943}, {10087, 10093}, {10090, 10094}

▸ Functional Relations73 relations

• X₈ = perspector of ABC and the intouch triangle of the medial triangle of ABC. (Randy Hutson, 9/23/2011)

• X₈ = midpoint of X_i and X_j for these {i,j}: {1, 3632}, {10, 3625}, {40, 5881}, {145, 3621}, {3057, 3893}, {3626, 4701}, {3679, 4677}, {4474, 4814}, {4668, 4816}, {4900, 8275}, {5541, 9897}, {5691, 7991}, {5903, 5904}

• X₈ = reflection of X_i in X_j for these (i,j): (1, 10), (2, 3679), (3, 5690), (4, 355), (7, 2550), (8, 8), (10, 3626), (11, 3036), (20, 40), (56, 8256), (65, 5836), (69, 3416), (75, 3696), (78, 6736), (86, 4733), (100, 1145), (144, 5223), (145, 1), (147, 9864), (149, 80), (192, 984), (193, 3751), (210, 4711), (315, 4769), (329, 3421), (346, 4901), (376, 3654), (388, 5794), (390, 9), (551, 4745), (663, 4147), (944, 3), (950, 5795), (960, 4662), (962, 4), (1019, 4807), (1043, 3704), (1120, 3756), (1125, 4691), (1280, 4904), (1317, 3035), (1320, 11), (1361, 3042), (1364, 3040), (1392, 7705), (1482, 5), (1483, 140), (1697, 5837), (2098, 1329), (2099, 2886), (2102, 1313), (2103, 1312), (3021, 3039), (3022, 3041), (3057, 960), (3146, 5691), (3161, 10005), (3189, 3913), (3241, 2), (3242, 141), (3243, 142), (3244, 1125), (3434, 3419), (3476, 1376), (3486, 958), (3488, 9708), (3555, 942), (3600, 1706), (3616, 3617), (3617, 4668), (3621, 3632), (3623, 1698), (3625, 4701), (3626, 4746), (3632, 3625), (3633, 3244), (3635, 3634), (3679, 4669), (3685, 3717), (3786, 4111), (3868, 65), (3869, 72), (3872, 4847), (3873, 3753), (3874, 3754), (3875, 3755), (3877, 210), (3878, 3678), (3881, 3918), (3883, 3686), (3884, 4015), (3885, 3057), (3886, 2321), (3889, 3698), (3890, 3697), (3892, 3968), (3894, 3919), (3898, 3956), (3899, 4134), (3901, 4084), (3902, 3706), (3952, 4738), (4318, 1861), (4344, 2345), (4360, 4026), (4363, 4665), (4419, 4643), (4430, 5902), (4454, 4659), (4511, 6735), (4560, 4041), (4643, 4690), (4644, 4363), (4673, 3714), (4693, 4439), (4720, 4046), (4861, 6734), (4864, 3823), (4895, 3716), (4922, 9508), (5048, 5123), (5080, 5176), (5180, 5080), (5263, 594), (5441, 3647), (5603, 5790), (5697, 3878), (5698, 5220), (5710, 5835), (5716, 5793), (5731, 5657), (5734, 5818), (5882, 6684), (5905, 1478), (5919, 3740), (5984, 9860), (6018, 3038), (6193, 9928), (6224, 100), (6327, 4680), (6332, 4163), (6737, 6743), (6740, 6741), (6742, 6739), (6758, 4736), (7192, 4761), (7253, 4086), (7962, 3452), (7970, 114), (7971, 6260), (7972, 214), (7973, 2883), (7974, 619), (7975, 618), (7976, 39), (7977, 6292), (7978, 113), (7979, 1209), (7980, 642), (7981, 641), (7982, 946), (7983, 115), (7984, 125), (8241, 2090), (8591, 9881), (8596, 9875), (8834, 6552), (9263, 291), (9780, 4678), (9785, 2551), (9791, 1654), (9797, 938), (9802, 149), (9809, 153), (9856, 9947), (9884, 2482), (9933, 1147), (9957, 5044), (9963, 6154), (9965, 2093), (10031, 6174)

• X₈ = isogonal conjugate of X₅₆

• X₈ = isotomic conjugate of X₇

• X₈ = cyclocevian conjugate of X₁₈₉

• X₈ = complement of X₁₄₅

• X₈ = anticomplement of X₁

• X₈ = complementary conjugate of X₂₈₈₅

• X₈ = anticomplementary conjugate of X₈

• X₈ = circumcircle-inverse of X₁₇₁₀₀

• X₈ = Conway-circle-inverse of X₃₈₄₇₅

• X₈ = X_i-Ceva conjugate of X_j for these (i,j): (2, 3161), (4, 2899), (7, 8055), (69, 329), (75, 2), (190, 3239), (290, 3948), (312, 346), (314, 312), (318, 5552), (319, 2895), (333, 9), (341, 7080), (643, 7253), (645, 3700), (646, 650), (664, 6332), (668, 4391), (765, 3952), (1016, 644), (1043, 78), (1219, 3616), (1222, 1), (1494, 3936), (1909, 1655), (2319, 7155), (2985, 6), (3596, 345), (3699, 522), (4076, 3699), (4102, 2321), (4373, 6557), (4554, 4130), (4555, 3904), (4582, 1639), (4997, 2325), (4998, 190), (6063, 344), (6064, 645), (6079, 900), (7017, 281), (7033, 192), (7155, 4903), (7257, 4560), (8817, 7)

• X₈ = X_i-cross conjugate of X_j for these (i,j): (1, 280), (4, 1034), (9, 2), (10, 318), (11, 522), (40, 7080), (55, 281), (56, 2123), (72, 78), (200, 346), (210, 9), (219, 345), (312, 7155), (346, 6557), (497, 7), (521, 100), (522, 3699), (650, 646), (950, 29), (960, 21), (1145, 6735), (1146, 4391), (1639, 4582), (1837, 4), (1857, 8805), (1864, 282), (2170, 4560), (2321, 312), (2325, 4997), (2968, 4397), (3057, 1), (3059, 200), (3239, 190), (3271, 650), (3680, 6553), (3683, 7110), (3686, 333), (3687, 4451), (3688, 55), (3700, 645), (3703, 3596), (3704, 3701), (3706, 314), (3717, 4518), (3877, 2320), (3880, 1320), (3885, 1392), (3893, 3680), (3900, 644), (3907, 7257), (4012, 5423), (4046, 2321), (4051, 330), (4060, 4102), (4081, 3239), (4086, 3952), (4092, 3700), (4111, 210), (4124, 885), (4130, 4554), (4152, 2325), (4180, 4182), (4531, 41), (4534, 514), (4542, 1639), (4546, 6558), (4847, 75), (4853, 1219), (4863, 6601), (4875, 274), (4965, 7192), (5245, 7026), (5246, 7043), (5795, 1220), (6062, 7359), (6068, 6745), (6555, 3161), (6736, 341), (6737, 1043), (6741, 4086), (7063, 3709), (7067, 3712), (8058, 1897), (8611, 4552), (9785, 5558)

• X₈ = cevapoint of X_i and X_j for these (i,j): {1, 40}, {2, 144}, {4, 3176}, {6, 197}, {9, 200}, {10, 72}, {11, 522}, {34, 8899}, {42, 3588}, {55, 219}, {56, 2122}, {65, 5930}, {175, 176}, {210, 2321}, {312, 4110}, {346, 6555}, {497, 4012}, {513, 3756}, {514, 4904}, {519, 1145}, {521, 2968}, {523, 8286}, {650, 3271}, {758, 6739}, {960, 3704}, {966, 4859}, {1125, 3650}, {1146, 3900}, {1639, 4542}, {2170, 4041}, {2175, 4548}, {2325, 4152}, {3057, 6736}, {3059, 4847}, {3161, 4859}, {3239, 4081}, {3686, 4046}, {3688, 3703}, {3700, 4092}, {3706, 4111}, {3709, 7063}, {3712, 7067}, {3893, 4895}, {4136, 4531}, {4180, 4181}, {4530, 4543}, {4534, 4546}, {6062, 7359}, {6068, 6745}, {7358, 8058}

• X₈ = crosspoint of X_i and X_j for these (i,j): {1, 979}, {2, 4373}, {7, 8051}, {9, 2319}, {75, 312}, {190, 4998}, {314, 333}, {643, 765}, {645, 6064}, {668, 1016}, {3596, 7017}, {3699, 4076}

• X₈ = crosssum of X_i and X_j for these (i,j): {1, 978}, {6, 3052}, {25, 3209}, {31, 604}, {57, 1423}, {244, 4017}, {649, 3271}, {663, 7117}, {667, 1015}, {1042, 1410}, {1284, 8850}, {1400, 1402}

• X₈ = crossdifference of every pair of points on line X₆₄₉X₈₅₄

• X₈ = X₁-aleph conjugate of X₁₀₅₀

• X₈ = X_i-beth conjugate of X_j for these (i,j): (8, 1), (341, 341), (643, 3), (668, 8), (1043, 8)

• X₈ = exsimilicenter of incircle and Spieker circle

• X₈ = exsimilicenter of Conway circle and Spieker radical circle

• X₈ = trilinear product of vertices of Hutson-intouch triangle

• X₈ = trilinear product of vertices of Caelum triangle

• X₈ = orthocenter of X_iX_jX_k for these (i,j,k): (1, 4, 5556), (4, 7, 885)

• X₈ = perspector of ABC and pedal triangle of X₄₀

• X₈ = perspector of ABC and reflection of medial triangle in X₁₀

• X₈ = perspector of ABC and reflection of intouch triangle in X₁

• X₈ = pedal antipodal perspector of X₁

• X₈ = pedal antipodal perspector of X₃₆

• X₈ = X₁₄₉₈-of-intouch-triangle

• X₈ = X₁₈₅-of-excentral-triangle

• X₈ = X₇₄-of-Fuhrmann-triangle

• X₈ = X₅₉₉₂-of-Brocard-triangle

• X₈ = perspector of circumconic with center X₃₁₆₁

• X₈ = center of circumconic that is locus of trilinear poles of lines passing through X₃₁₆₁

• X₈ = X₂-Ceva conjugate of X₃₁₆₁

• X₈ = trilinear pole of line X₅₂₂X₆₅₀ (the radical axis of circumcircle and excircles radical circle)

• X₈ = pole wrt polar circle of trilinear polar of X₂₇₈ (line X₅₁₃X₁₈₃₅)

• X₈ = X₄₈-isoconjugate (polar conjugate) of X₂₇₈

• X₈ = X₆-isoconjugate of X₅₇

• X₈ = X₇₅-isoconjugate of X₁₃₉₇

• X₈ = X₁₁₀₁-isoconjugate of X₁₃₆₅

• X₈ = antigonal image of X₁₃₂₀

• X₈ = {X₁,X₂}-harmonic conjugate of X₃₆₁₆

• X₈ = {X₁,X₁₀}-harmonic conjugate of X₂

• X₈ = {X₂,X₁₀}-harmonic conjugate of X₆₆₇

• X₈ = inverse-in-polar-circle of X₁₈₇₈

• X₈ = inverse-in-Steiner-circumellipse of X₃₉₁₂

• X₈ = inverse-in-Mandart-inellipse of X₂₃₂₅

• X₈ = inverse-in-circumconic-centered-at-X₁ of X₄₅₁₁

• X₈ = X₄ of 2nd Conway triangle (the extraversion triangle of X₈)

• X₈ = trilinear square root of X₃₄₁

• X₈ = perspector of 5th extouch triangle and anticevian triangle of X₇

• X₈ = centroid of cross-triangle of Gemini triangles 20 and 28

• X₈ = X_i-aleph conjugate of X_j for these (i,j): (1, 1050), (188, 2943), (1222, 8)

• X₈ = X_i-beth conjugate of X_j for these (i,j): (8, 1), (99, 3160), (200, 4517), (333, 5222), (341, 341), (346, 4873), (643, 3), (644, 3730), (668, 8), (1043, 8), (2287, 4266), (3699, 8), (6558, 8), (7256, 8), (7257, 76), (7259, 220), (8706, 8)

• X₈ = X_i-anticomplementary conjugate of X_j for these (i,j): (1, 8), (2, 69), (3, 20), (5, 2888), (6, 2), (7, 3434), (8, 3436), (9, 829), (10, 1330), (13, 621), (14, 622), (15, 616), (16, 617), (18, 634), (19, 59015), (20, 6225), (21, 3869), (22, 5596), (24, 6193), (25, 192), (28, 3868), (30, 146), (31, 192), (32, 194), (54, 3), (55, 144), (56, 145), (57, 7), (58, 1), (59, 100), (74, 30), (81, 75), (98, 511), (99, 512), and many others

• X₈ = X_i-complementary conjugate of X_j for these (i,j): (1, 2885), (31, 3161), (513, 5510), (1293, 513), (3445, 10), (3680, 1329), (4373, 2887), (8056, 141)

• X₈ = perspector of ABC and mid-triangle of excentral and 2nd extouch triangles

• X₈ = perspector of 5th extouch triangle and cross-triangle of ABC and 5th extouch triangle

• X₈ = X₁₅₉₃-of-2nd-extouch-triangle

• X₈ = excentral-to-2nd-extouch similarity image of X₁₆₉₇

• X₈ = Cundy-Parry Phi transform of X₁₀₄

• X₈ = Cundy-Parry Psi transform of X₅₁₇

• X₈ = perspector of ABC and cross-triangle of ABC and Gemini triangle 39

• X₈ = barycentric product of vertices of Gemini triangle 39

• X₈ = perspector of Gemini triangle 13 and cross-triangle of Gemini triangles 1 and 13

• X₈ = excentral-to-ABC barycentric image of X₁

• X₈ = X₄-of-Yff-contact-triangle

• X₈ = Yff-contact-isogonal conjugate of X₅₅₉₂

• X₈ = perspector (Brianchon point) of the Mandart inellipse

• X₈ = isoconjugate of X_i and X_j for these {i,j}: {1, 56}, {2, 604}, {3, 34}, {4, 603}, {6, 57}, {7, 31}, {8, 1106}, {9, 1407}, {10, 1408}, {12, 849}, {19, 222}, {21, 1042}, {25, 77}, {27, 1409}, {28, 73}, {29, 1410}, {32, 85}, {33, 7053}, {36, 1411}, {37, 1412}, {40, 1413}, {41, 279}, {42, 1014}, {48, 278}, {54, 1393}, {55, 269}, {58, 65}, {59, 244}, {60, 1254}, {63, 608}, {64, 1394}, {69, 1395}, {71, 1396}, {75, 1397}, {78, 1398}, {79, 1399}, {81, 1400}, {82, 1401}, {84, 221}, {86, 1402}, {87, 1403}, {88, 1404}, {89, 1405}, {90, 1406}, {101, 3669}, {102, 1455}, {103, 1456}, {104, 1457}, {105, 1458}, {106, 1319}, {108, 1459}, {109, 513}, {110, 4017}, {154, 8809}, {158, 7335}, {163, 7178}, {171, 1431}, {172, 1432}, {181, 757}, {184, 273}, {189, 2199}, {198, 1422}, {200, 7023}, {208, 1433}, {212, 1119}, {213, 1434}, {219, 1435}, {220, 738}, {223, 1436}, {225, 1437}, {226, 1333}, {241, 1438}, {255, 1118}, {259, 7370}, {267, 8614}, {270, 1425}, {281, 7099}, {282, 6611}, {283, 1426}, {284, 1427}, {291, 1428}, {292, 1429}, {296, 1430}, {307, 2203}, {326, 7337}, {331, 9247}, {346, 7366}, {347, 2208}, {348, 1973}, {388, 1472}, {393, 7125}, {479, 1253}, {512, 1414}, {514, 1415}, {518, 1416}, {519, 1417}, {552, 872}, {560, 6063}, {593, 2171}, {607, 7177}, {614, 1037}, {643, 7250}, {649, 651}, {650, 1461}, {657, 4617}, {658, 3063}, {661, 4565}, {662, 7180}, {663, 934}, {664, 667}, {669, 4625}, {672, 1462}, {692, 3676}, {727, 1463}, {741, 1284}, {756, 7341}, {759, 1464}, {765, 1357}, {798, 4573}, {893, 7175}, {896, 7316}, {904, 7176}, {909, 1465}, {923, 7181}, {937, 1466}, {939, 1467}, {951, 1104}, {959, 1468}, {961, 1193}, {983, 7248}, {985, 1469}, {998, 1470}, {1002, 1471}, {1015, 4564}, {1019, 4559}, {1020, 7252}, {1027, 2283}, {1035, 3345}, {1036, 4320}, {1041, 1473}, {1073, 3213}, {1086, 2149}, {1088, 2175}, {1089, 7342}, {1096, 1804}, {1098, 7143}, {1101, 1365}, {1110, 1358}, {1149, 8686}, {1170, 1475}, {1174, 1418}, {1191, 7091}, {1201, 1476}, {1214, 1474}, {1245, 5323}, {1262, 2170}, {1279, 1477}, {1420, 3445}, {1421, 3446}, {1423, 2162}, {1424, 3224}, {1439, 2299}, {1440, 2187}, {1441, 2206}, {1442, 6186}, {1443, 6187}, {1447, 1911}, {1453, 2213}, {1576, 4077}, {1616, 2137}, {1617, 2191}, {1631, 7213}, {1769, 2720}, {1790, 1880}, {1795, 1875}, {1813, 6591}, {1919, 4554}, {1922, 10030}, {1974, 7182}, {1980, 4572}, {2003, 2160}, {2006, 7113}, {2099, 2163}, {2114, 9500}, {2150, 6354}, {2159, 6357}, {2176, 7153}, {2194, 3668}, {2207, 7183}, {2210, 7233}, {2212, 7056}, {2218, 4306}, {2221, 2285}, {2260, 2982}, {2263, 3423}, {2275, 7132}, {2279, 5228}, {2291, 6610}, {2306, 2307}, {2310, 7339}, {2324, 6612}, {2334, 3361}, {2353, 7210}, {2362, 6502}, {3121, 4620}, {3212, 7121}, {3248, 4998}, {3271, 7045}, {3451, 3752}, {3709, 4637}, {3733, 4551}, {3777, 8685}, {3900, 6614}, {3911, 9456}, {3937, 7012}, {3942, 7115}, {4252, 5665}, {4296, 8615}, {4557, 7203}, {4626, 8641}, {4822, 5545}, {5018, 8852}, {5546, 7216}, {6129, 8059}, {6180, 9315}, {7011, 7129}, {7013, 7151}, {7051, 7052}, {7054, 7147}, {7084, 7195}, {7104, 7196}, {7117, 7128}, {7122, 7249}, {9309, 9316}, {9363, 9435}, {9364, 9432}

Notes

Let A_b, A_c, B_c, B_a, C_a, C_b be as in the construction of the Conway circle (http://mathworld.wolfram.com/ConwayCircle.html). Let O_a be the circumcircle of AB_cC_b, and define O_b, O_c cyclically. X₈ is the radical center of O_a, O_b, O_c. see also X₂₁ and X₂₇₄. (Randy Hutson, April 9, 2016)

Let A'B'C' be Triangle T(-2,1). Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₅₁₉. The lines A'A″, B'B″, C'C″ concur in X₈. (Randy Hutson, November 18, 2015)

Let I_a be the reflection of X₁ in the perpendicular bisector of BC, and define I_b and I_c cyclically. X₈ = X₁-of-I_aI_bI_c. (Randy Hutson, September 14, 2016)

Let A₂₈B₂₈C₂₈ be the Gemini triangle 28. Let L_A be the line through A₂₈ parallel to BC, and define L_B and L_C cyclically. Let A'₂₈ = L_B∩L_C, and define B'₂₈, C'₂₈ cyclically. Triangle A'₂₈B'₂₈C'₂₈ is homothetic to ABC at X₈. (Randy Hutson, November 30, 2018)

For yet another construction of X₈, see Dasari Naga Vijay Krishna, “On A Simple Construction of Triangle Centers X₈, X₁₉₇, X(K) & X₅₉₄”, Scientific Inquiry and Review, Vol. 2, Issue 3, July 2018.

Another construction is given by Xavier Dussau: Elementary construction of the Nagel point. (NagelPointDussauConstruction.pdf) (April 29, 2020)

Let T be any one of these trianges: {Ascella, 1st circumperp, 2nd circumperp}. Let O_A be the circle centered at the A-vertex of T and passing through A; define O_B and O_C cyclically. X₈ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

X₈ is the {X₆₉,X₇₅}-harmonic conjugate of X₇.