X2981

DC(X13)

X2981 = perspector of ABC and unary cofactor triangle of outer Napoleon triangle
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 61, 110 }: Central Line L(61, 110) { 15, 1337 }: Central Line L(15, 1337) { 396, 1989 }: Central Line L(396, 1989) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_396 = MIDPOINT OF X(13) AND X(15) (0.5699, 0.2349) X_61 = ISOGONAL CONJUGATE OF X(17) (0.5764, 0.2283) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_1337 = 1st WERNAU POINT (2.6910, 4.6498) X_1989 = ISOGONAL CONJUGATE OF X(323) (0.9916, -0.2095) X_2981 = DC(X(13)) (0.5730, 0.2317) A B C X396 X61 X110 X15 X1337 X1989 X2981 (a, b, c) = (5, 6, 7) | AB = 1

X2981 (DC(X13)) 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) is as given just before X2979, using U = X13
Search Value (6, 9, 13): \(1.790165020057\)
▸ Alternate Coordinates1 form

  • Trilinears: \(1/(\cos(B - C) + 2 \cos(A - \pi /3)) : :\)

X2981 lies on the hyperbola {{A, B, C,X2,X6}}.

▸ Incident Central Lines3 lines

{15, 1337}, {61, 110}, {396, 1989}

▸ Functional Relations3 relations

Notes

Let U be the line tangent to the Neuberg cubic (K001) at X13, and let V be the line tangent to K001 at X15. Then X2981 = U∩V. Let U' be the line tangent to the Napoleon-Feuerbach cubic (K005) at X17, and let V' be the line tangent to K005 are X61. Then X2981 = U'∩V'; see X6151. Also, X2981 = trilinear pole of the line X16X512. (Randy Hutson, January 5, 2015)