X3312

INTERSECTION OF LINES X3X_6 AND X4X_1132

X3312 = inverse-in-Brocard-circle of X3311
< 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) { 3, 6 }: Brocard Axis { 25, 589 }: Central Line L(25, 589) { 381, 486 }: Central Line L(381, 486) { 4, 1132 }: Central Line L(4, 1132) { 591, 639 }: Central Line L(591, 639) { 5, 1587 }: Central Line L(5, 1587) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3317 = ISOGONAL CONJUGATE OF X(3312) (0.4661, 0.1442) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_589 = 2nd KENMOTU-VAN LAMOEN POINT (0.6340, 0.4099) X_381 = MIDPOINT OF X(2) AND X(4) (0.5748, 0.2979) X_486 = INNER VECTEN POINT (-0.0810, -0.0170) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1132 = ARCTAN(-2) KIEPERT POINT (0.8864, -0.3774) X_591 = 1st VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.4203, 0.0741) X_639 = COMPLEMENT OF X(371) (0.5235, 0.1951) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_1587 = POINT CASTOR I (0.6026, 0.2943) X_3312 = INTERSECTION OF LINES X(3)X(6) AND X(4)X(1132) (0.6365, 0.3274) A B C X3317 X3 X6 X25 X589 X381 X486 X4 X1132 X591 X639 X5 X1587 X3312 (a, b, c) = (5, 6, 7) | AB = 1

X3312 (INTERSECTION OF LINES X3X_6 AND X4X_1132) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cos A - 2 \sin A : \cos B - 2 \sin B : \cos C - 2 \sin C\)
Barycentrics: \((\cos A - 2 \sin A)(\sin A) : (\cos B - 2 \sin B)(\sin B) : (\cos C - 2 \sin C)(\sin C)\)
Search Value (6, 9, 13): \(-0.155142913427\)
▸ Alternate Coordinates4 forms

  • Trilinears: \(\cos(A + t) : \cos(B + t) : \cos(C + t)\), where t = arctan(2)

  • Trilinears: \(a - R \cos A : :\)

  • Trilinears: \(a (S_{A} - 2 S) : :\)

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

▸ Incident Central Lines12 lines

{3, 6}, {4, 1132}, {5, 1587}, {25, 589}, {30, 1588}, {381, 486}, {487, 1992}, {517, 1703}, {591, 639}, {615, 1656}, {642, 1991}, {999, 1335}

▸ Functional Relations7 relations

Notes

X3312 is the perspector of each of the following pairs of triangles:

  • Lucas(-1:1) central triangle and the symmedial triangle

  • Lucas(-1:1) tangents triangle and the Lucas central triangle

  • Lucas(-2:3) central triangle and the circumsymmedial triangle.

Moreover, X3311, is the radical center of the Lucas(-4:1) circles. See X371 and X3311. (Randy Hutson, 9/23/2011)

X3312 is the perspector of each of the pair of the following four triangles:

  • symmedial triangle, Lucas(-1) central triangle, Lucas secondary central triangle, 1st Lucas secondary tangents triangle (Randy Hutson, September 14, 2016)