X1249

ISOGONAL CONJUGATE OF X1073

X1249 is the isogonal conjugate of X1073.
< 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 { 19, 57 }: Central Line L(19, 57) { 2, 253 }: Central Line L(2, 253) { 1, 281 }: Central Line L(1, 281) { 108, 198 }: Central Line L(108, 198) { 112, 376 }: Central Line L(112, 376) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1073 = -CROSS CONJUGATE OF X(3) (0.4792, -0.4343) X_4 = ORTHOCENTER (0.6122, 0.3958) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_19 = CLAWSON POINT (0.6246, 0.4244) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_2 = CENTROID (0.5374, 0.2000) X_253 = -CROSS CONJUGATE OF X(2) (3.5087, 32.5688) X_1 = INCENTER (0.5714, 0.2333) X_281 = -CROSS CONJUGATE OF X(9) (0.5796, 0.3202) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.5320, 0.6114) X_198 = -CEVA CONJUGATE OF X(6) (0.4741, -0.0381) X_112 = Ψ(ORTHOCENTER, SYMMEDIAN POINT) (0.5162, 0.6121) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4626, 0.0042) X_1249 = ISOGONAL CONJUGATE OF X(1073) (-0.9482, -15.9845) A B C X1073 X4 X6 X19 X57 X2 X253 X1 X281 X108 X198 X112 X376 X1249 (a, b, c) = (5, 6, 7) | AB = 1

X1249 (ISOGONAL CONJUGATE OF X1073) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((\tan A)(\cos A - \cos B \cos C) : :\)
Barycentrics: \([3a^{4} - 2a^{2}b^{2} - 2a^{2}c^{2} - (b^{2} - c^{2})^{2}]/(b^{2} + c^{2} - a^{2}) : :\)
Search Value (6, 9, 13): \(0.670131448662\)

X1249 lies on hyperbola {{X2,X6,X216,X233,X1249,X1560,X3162}}. This hyperbola is a circumconic of the medial triangle, and the locus of perspectors of circumconics centered at a point on the Euler line. The hyperbola is tangent to Euler line at X2.

X1249 lies on the Thomson cubic.

▸ Incident Central Lines47 lines

{1, 281}, {2, 253}, {3, 1033}, {4, 6}, {9, 1712}, {19, 57}, {20, 3172}, {25, 5304}, {69, 648}, {108, 198}, {112, 376}, {154, 3079}, {186, 1609}, {193, 297}, {208, 2270}, {216, 631}, {219, 1783}, {223, 3352}, {232, 800}, {233, 3090}, {264, 3618}, {273, 5222}, {317, 1992}, {346, 1897}, {347, 653}, {461, 3192}, {579, 1715}, {604, 2202}, {608, 2122}, {610, 3213}, {920, 1720}, {1108, 1148}, {1118, 2264}, {1743, 1785}, {1838, 2956}, {1870, 3554}, {1941, 2138}, {1968, 5065}, {2165, 3018}, {3003, 3147}, {3068, 3535}, {3069, 3536}, {3089, 5305}, {3199, 5319}, {3284, 3529}, {3343, 3356}, {5200, 5411}

▸ Functional Relations16 relations

Notes

X1249 is the perspector of triangle ABC and the tangential triangle of the circumconic centered at X4. X1249 is also the perspector of the medial triangle and the triangle formed by the trilinear poles of the sidelines of the orthic triangle. (Randy Hutson, 9/23/2011)

X1249 is the unique point whose anticomplement is also its polar conjugate, namely X253. (Randy Hutson, March 14, 2018)

Let Ea be the ellipse with B and C as foci and passing through X4, and define Eb and Ec cyclically.

Let La be the line tangent to Ea at X4, and define Lb and Lc cyclically.

Let A' be the trilinear pole of line La, and define B' and C' cyclically.

Then A', B', C' lie on the circumconic centered at X1249. (Randy Hutson, March 14, 2018)

Let MaMbMc be the medial triangle. Let A″ be the pole, wrt the polar circle, of line MbMc, and define B″ and C″ cyclically. Also, A″B″C″ is the triangle formed by trilinear poles of sides of the orthic triangle. The lines MaA″, MbB″, McC″ concur in X1249. (Randy Hutson, March 14, 2018)