X617

ANTICOMPLEMENT OF X14

X617 is the anticomplement of X14.
< 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, 14 }: Central Line L(2, 14) { 69, 74 }: Central Line L(69, 74) { 13, 148 }: Central Line L(13, 148) { 3, 298 }: Central Line L(3, 298) { 30, 299 }: Central Line L(30, 299) 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_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_148 = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE (1.3904, -0.2646) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4724, 0.1301) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-1.1465, -1.6097) X_617 = ANTICOMPLEMENT OF X(14) (-0.2146, 0.8285) A B C Ka Kb Kc X2 X14 X69 X74 X13 X148 X3 X298 X30 X299 X617 (a, b, c) = (5, 6, 7) | AB = 1

X617 (ANTICOMPLEMENT OF X14) 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 \(f(a,b,c) = [S_{B}S_{C} - 2S_{A}(a^{2} - \sqrt{3} \operatorname{area})]/a\)
Search Value (6, 9, 13): \(-0.858757166561\)
▸ Alternate Coordinates1 form

  • Trilinears: \(F(14)/a - 2 \csc(A - \pi /3) : F(14)/b - 2 \csc(B - \pi /3) : F(14)/c - 2 \csc(C - \pi /3)\)

X617 lies on the Neuberg cubic.

▸ Incident Central Lines10 lines

{2, 14}, {3, 298}, {4, 628}, {13, 148}, {16, 533}, {20, 634}, {30, 299}, {69, 74}, {302, 549}, {303, 381}

▸ Functional Relations7 relations

Notes

Let OA be the circle centered at the A-vertex of the inner Napoleon triangle and passing through A; define OB and OC cyclically. X617 is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)