X397

CROSSPOINT OF X4 AND X17

X397 is the crosspoint of X4 and X17.
< 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) { 4, 6 }: Van Aubel Line { 5, 13 }: Central Line L(5, 13) { 16, 17 }: Central Line L(16, 17) { 30, 61 }: Central Line L(30, 61) { 3, 396 }: Central Line L(3, 396) { 51, 462 }: Central Line L(51, 462) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_17 = 1st NAPOLEON POINT (0.5666, 0.2383) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_61 = ISOGONAL CONJUGATE OF X(17) (0.5764, 0.2283) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_396 = MIDPOINT OF X(13) AND X(15) (0.5699, 0.2349) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_462 = EULER LINE INTERCEPT OF LINE X(51)X(397) (0.7090, 0.6489) X_397 = CROSSPOINT OF X(4) AND X(17) (0.6029, 0.2975) A B C X4 X17 X6 X5 X13 X16 X30 X61 X3 X396 X51 X462 X397 (a, b, c) = (5, 6, 7) | AB = 1

X397 (CROSSPOINT OF X4 AND X17) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cos(B - C) - 2 \cos(A + \pi /3) : :\)
Search Value (6, 9, 13): \(-0.077081121117\)
▸ Incident Central Lines13 lines

{3, 396}, {4, 6}, {5, 13}, {14, 546}, {15, 550}, {16, 17}, {30, 61}, {51, 462}, {141, 634}, {184, 463}, {202, 496}, {524, 633}, {532, 635}

▸ Functional Relations2 relations

Notes

X397 is the pole of the Euler line wrt every conic passing through X13, X16, X17 and X62. (Randy Hutson, January 17, 2020)

X397 is the {X4,X6}-harmonic conjugate of X398.