X999

MIDPOINT OF X1 AND X57

X999 is the radical center of the mixtilinear incircles.
< 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) { 1, 57 }: OI Line (Incenter–Circumcenter Line) { 1, 3 }: OI Line (Incenter–Circumcenter Line) { 6, 101 }: Central Line L(6, 101) { 7, 104 }: Central Line L(7, 104) { 11, 381 }: Central Line L(11, 381) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_7 = GERGONNE POINT (0.5903, 0.2769) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_999 = MIDPOINT OF X(1) AND X(57) (0.5926, 0.2722) A B C X1 X57 X3 X6 X101 X7 X104 X11 X381 X999 (a, b, c) = (5, 6, 7) | AB = 1

X999 (MIDPOINT OF X1 AND X57) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(2 - \cos A : 2 - \cos B : 2 - \cos C\) (Joe Goggins, 2002)
Search Value (6, 9, 13): \(1.035381900487\)
▸ Alternate Coordinates1 form

  • Trilinears: \(a(a^{2} + 4bc - b^{2} - c^{2}) : :\)

▸ Incident Central Lines23 lines

{1, 3}, {2, 495}, {4, 496}, {5, 388}, {6, 101}, {7, 104}, {8, 474}, {11, 381}, {12, 499}, {20, 1058}, {30, 497}, {63, 392}, {77, 1057}, {78, 1059}, {81, 859}, {145, 404}, {329, 405}, {376, 390}, {518, 997}, {527, 551}, {601, 1106}, {938, 944}, {954, 1006}

▸ Functional Relations13 relations

Notes

X999 is the {X1,X56}-harmonic conjugate of X3.