X3642

FERMAT POINT OF 1st BROCARD TRIANGLE

X3642 and X3644 lie on the line through X141 parallel to the Euler line.
< 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) { 2, 14 }: Central Line L(2, 14) { 13, 76 }: Central Line L(13, 76) { 30, 141 }: Central Line L(30, 141) { 16, 298 }: Central Line L(16, 298) { 32, 395 }: Central Line L(32, 395) { 69, 532 }: Central Line L(69, 532) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_141 = COMPLEMENT OF SYMMEDIAN POINT (0.5061, 0.1663) X_16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X_298 = ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER (0.4724, 0.1301) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_395 = MIDPOINT OF X(14) AND X(16) (1.3794, 1.1048) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_532 = DIRECTION OF VECTOR AX + BX + CX, where X = X(17) (0.8927, 0.6671) X_3642 = FERMAT POINT OF 1st BROCARD TRIANGLE (0.5234, 0.2116) A B C X2 X14 X13 X76 X30 X141 X16 X298 X32 X395 X69 X532 X3642 (a, b, c) = (5, 6, 7) | AB = 1

X3642 (FERMAT POINT OF 1st BROCARD TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \(2 \sqrt{3} (a^{4}+2 b^{2} c^{2}) S+a^{6}-(b^{2}+c^{2}) a^{4}+2 (b^{4}+3 b^{2} c^{2}+c^{4}) a^{2}-2 (b^{4}-c^{4}) \cdot (b^{2}-c^{2}) : :\)
Search Value (6, 9, 13): \(2.257234575004\)
▸ Incident Central Lines16 lines

{2, 14}, {3, 618}, {4, 636}, {13, 76}, {16, 298}, {17, 628}, {30, 141}, {32, 395}, {62, 633}, {69, 532}, {381, 624}, {530, 599}, {616, 2896}, {629, 631}, {630, 1656}, {2926, 3131}

▸ Functional Relations1 relation