X3586

INTERSECTION OF LINES X1X_4 AND X30X_57

X3586 = insimilicenter of hexyl and 2nd Johnson-Yff circles; the exsimilicenter is X9614
< 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, 4 }: Incenter–Orthocenter Line { 30, 57 }: Central Line L(30, 57) { 9, 80 }: Central Line L(9, 80) { 90, 191 }: Central Line L(90, 191) { 10, 452 }: Central Line L(10, 452) { 329, 519 }: Central Line L(329, 519) 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_4 = ORTHOCENTER (0.6122, 0.3958) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_9 = MITTENPUNKT (0.5110, 0.1615) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_90 = -CROSS CONJUGATE OF X(1) (0.1022, 0.0201) X_191 = -CEVA CONJUGATE OF X(1) (0.4859, 0.1209) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_452 = -HIRST INVERSE OF X(449) (0.5268, 0.1722) X_329 = ISOTOMIC CONJUGATE OF X(189) (0.3847, -0.0222) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9566, 0.6107) X_3586 = INTERSECTION OF LINES X(1)X(4) AND X(30)X(57) (0.5472, 0.1370) A B C X1 X4 X30 X57 X9 X80 X90 X191 X10 X452 X329 X519 X3586 (a, b, c) = (5, 6, 7) | AB = 1

X3586 (INTERSECTION OF LINES X1X_4 AND X30X_57) 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 \(f(a,b,c) = bc(2b^{4} + 2c^{4} - 3a^{4} + a^{3}b + a^{3}c - ab^{3} - ac^{3} + a^{2}b^{2} + a^{2}c^{2} + abc(b + c) + 2a^{2}bc - 4b^{2}c^{2})\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(3.585338199588\)
▸ Incident Central Lines23 lines

{1, 4}, {7, 3543}, {9, 80}, {10, 452}, {11, 3576}, {20, 1210}, {30, 57}, {40, 1728}, {46, 2955}, {90, 191}, {165, 1737}, {329, 519}, {355, 1697}, {380, 1826}, {405, 1376}, {496, 1420}, {516, 2093}, {517, 1864}, {936, 2478}, {938, 3146}, {993, 1005}, {1449, 1901}, {1453, 1834}

▸ Functional Relations3 relations

  • X3586 = reflection of Xi in Xj for these (i,j): (1, 497), (1750, 4)

  • X3586 = insimilicenter of hexyl and 2nd Johnson-Yff circles; the exsimilicenter is X9614

  • X3586 = {X1, X1479}-harmonic conjugate of X9614