X377

EULER LINE INTERCEPT OF LINE X7X_8

X377 is the {X3,X20}-harmonic conjugate of X21.
< A B C >: Reference Triangle △ABC < K_a K_b K_c >: Anticomplementary Triangle △K_a K_b K_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 405 }: Euler Line { 2, 3 }: Euler Line { 7, 8 }: Gergonne–Nagel Line { 10, 46 }: Central Line L(10, 46) { 1, 224 }: Central Line L(1, 224) { 78, 226 }: Central Line L(78, 226) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) K_a = Anticomplementary Vertex K_a (1.6122, 0.5999) K_b = Anticomplementary Vertex K_b (-0.3878, 0.5999) K_c = Anticomplementary Vertex K_c (0.3878, -0.5999) X_2 = CENTROID (0.5374, 0.2000) X_405 = EULER LINE INTERCEPT OF LINE X(1)X(6) (0.5325, 0.1870) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_7 = GERGONNE POINT (0.5903, 0.2769) X_8 = NAGEL POINT (0.4694, 0.1333) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_1 = INCENTER (0.5714, 0.2333) X_224 = -CEVA CONJUGATE OF X(63) (1.2479, 0.4419) X_78 = ISOGONAL CONJUGATE OF X(34) (0.3835, 0.0491) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_377 = EULER LINE INTERCEPT OF LINE X(7)X(8) (0.5473, 0.2258) A B C Ka Kb Kc X2 X405 X3 X7 X8 X10 X46 X1 X224 X78 X226 X377 (a, b, c) = (5, 6, 7) | AB = 1

X377 (EULER LINE INTERCEPT OF LINE X7X_8) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b^{4} + c^{4} - a^{4} - 2b^{2}c^{2} - 2abc(a + b + c))\)
Barycentrics: \(h(a,b,c) : h(b,c,a) : h(c,a,b)\), where \(h(a,b,c) = b^{4} + c^{4} - a^{4} - 2b^{2}c^{2} - 2abc(a + b + c)\)
Search Value (6, 9, 13): \(2.042891249441\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \cos A +(\cos A + \cos B + \cos C) \cos B \cos C\)

▸ Incident Central Lines105 lines

{1, 224}, {2, 3}, {7, 8}, {10, 46}, {12, 1259}, {56, 2886}, {58, 1714}, {72, 5905}, {77, 5930}, {78, 226}, {79, 5692}, {81, 387}, {84, 5587}, {100, 3085}, {142, 950}, {145, 1056}, {149, 1058}, {171, 5230}, {200, 5290}, {225, 1038}, {274, 315}, {278, 4296}, {283, 1754}, {318, 1947}, {329, 3876}, {348, 3188}, {355, 1071}, {394, 3193}, {495, 5687}, {497, 2646}, {498, 3822}, {516, 5250}, {527, 3951}, {528, 3303}, {667, 1155}, {908, 936}, {938, 5175}, {940, 1834}, {942, 3419}, {958, 3925}, {960, 1836}, {962, 3877}, {965, 1901}, {966, 2245}, {993, 3841}, {1001, 6284}, {1060, 1068}, {1125, 1479}, {1155, 2551}, {1159, 3621}, {1210, 3306}, {1220, 4429}, {1329, 4413}, {1330, 5739}, {1454, 1788}, {1621, 4294}, {1698, 3585}, {1765, 5816}, {1771, 3561}, {1837, 3812}, {1935, 3215}, {2096, 5818}, {2182, 5749}, {2287, 5746}, {2327, 5747}, {2345, 5279}, {2549, 5283}, {2893, 5738}, {2975, 4293}, {3086, 5253}, {3189, 3475}, {3304, 3813}, {3361, 5231}, {3421, 3617}, {3476, 4861}, {3485, 4511}, {3583, 3624}, {3618, 5135}, {3679, 5270}, {3710, 3729}, {3767, 5277}, {3869, 4295}, {3873, 5178}, {3897, 5731}, {3916, 5791}, {4000, 5262}, {4255, 5718}, {4298, 4847}, {4302, 5248}, {4312, 5785}, {4359, 5016}, {4652, 5745}, {4999, 5204}, {5217, 6690}, {5219, 5438}, {5225, 5550}, {5254, 5275}, {5256, 5717}, {5276, 5286}, {5334, 5367}, {5335, 5362}, {5439, 5722}, {5691, 5732}, {5715, 6282}, {5927, 6259}

▸ Functional Relations2 relations

Notes

As a point on the Euler line, X377 has Shinagawa coefficients \((abc \$a, 2S^{2})\).