X167

NAGEL POINT OF EXCENTRAL TRIANGLE

X167 = X8-of-excentral triangle
< A B C >: Reference Triangle △ABC < J_a J_b J_c >: Excentral Triangle △J_a J_b J_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ J_a X_167 ]: Line [ J_a X_167 ] [ J_b X_167 ]: Line [ J_b X_167 ] [ J_c X_167 ]: Line [ J_c X_167 ] { 164, 165 }: Central Line L(164, 165) { 1, 174 }: Central Line L(1, 174) { 166, 188 }: Central Line L(166, 188) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) J_a = A-Excenter J_a (1.2857, 0.5249) J_b = B-Excenter J_b (-0.2857, 0.6999) J_c = C-Excenter J_c (0.4286, -1.0498) X_164 = INCENTER OF EXCENTRAL TRIANGLE (0.4930, 0.1115) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_1 = INCENTER (0.5714, 0.2333) X_174 = YFF CENTER OF CONGRUENCE (0.5813, 0.2549) X_166 = GERGONNE POINT OF EXCENTRAL TRIANGLE (0.5156, 0.1554) X_188 = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE (0.5245, 0.1802) X_167 = NAGEL POINT OF EXCENTRAL TRIANGLE (0.4425, -0.0480) A B C Ja Jb Jc X164 X165 X1 X174 X166 X188 X167 (a, b, c) = (5, 6, 7) | AB = 1

X167 (NAGEL POINT OF EXCENTRAL TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Derived Triangle: Excentral Triangle △JaJbJc — vertices Ja, Jb, Jc (Anticevian triangle of Incenter X1 (the three excenters))

  • Concurrence Lines: Ja–X167, Jb–X167, Jc–X167 concurring at X167

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = s(B,C,A)t(B,C,A) + s(C,A,B)t(C,A,B) - s(A,B,C)t(A,B,C)\), where \(s(A,B,C) = \sin(A/2)\) and \(t(A,B,C) = (\cos B/2 + \cos C/2 - \cos A/2) \sec A/2\)
Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin A)f(A,B,C)\)
Search Value (6, 9, 13): \(14.236467851794\)
▸ Incident Central Lines3 lines

{1, 174}, {164, 165}, {166, 188}

▸ Functional Relations2 relations