X616

ANTICOMPLEMENT OF X13

The midpoint of X616 and X617 is the Steiner point, X99.
< 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, 13 }: Central Line L(2, 13) { 69, 74 }: Central Line L(69, 74) { 14, 148 }: Central Line L(14, 148) { 3, 299 }: Central Line L(3, 299) { 30, 298 }: Central Line L(30, 298) 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_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) 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_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_148 = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE (1.3904, -0.2646) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_299 = ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER (-1.1465, -1.6097) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4724, 0.1301) X_616 = ANTICOMPLEMENT OF X(13) (0.4364, 0.0360) A B C Ka Kb Kc X2 X13 X69 X74 X14 X148 X3 X299 X30 X298 X616 (a, b, c) = (5, 6, 7) | AB = 1

X616 (ANTICOMPLEMENT OF 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 \(f(a,b,c) = [S_{B}S_{C} - 2S_{A}(a^{2} + \sqrt{3} \operatorname{area})]/a\)
Search Value (6, 9, 13): \(7.736216073857\)
▸ Alternate Coordinates1 form

  • Trilinears: \(F(13)/a - 2 \csc(A + \pi /3) : F(13)/b - 2 \csc(B + \pi /3) : F(13)/c - 2 \csc(C + \pi /3)\)

X616 lies on the Neuberg cubic.

▸ Incident Central Lines12 lines

{2, 13}, {3, 299}, {4, 627}, {14, 148}, {15, 532}, {20, 633}, {30, 298}, {69, 74}, {302, 381}, {303, 549}, {489, 2043}, {490, 2044}

▸ Functional Relations8 relations

Notes

Let OA be the circle centered at the A-vertex of the outer Napoleon triangle and passing through A; define OB and OC cyclically. X616 is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)