X3590

1st DIXIT POINT

The radius of circle TA is |BC|/6, and the Kiepert angle (e.g., B-to-C-to-X3890) is arctan(2/3).
< 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) { 20, 1327 }: Central Line L(20, 1327) { 590, 1131 }: Central Line L(590, 1131) { 140, 3316 }: Central Line L(140, 3316) { 485, 3523 }: Central Line L(485, 3523) { 1132, 3068 }: Central Line L(1132, 3068) { 1328, 3091 }: Central Line L(1328, 3091) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_1327 = ARCTAN(3) KIEPERT POINT (0.5967, 0.3083) X_590 = ISOGONAL CONJUGATE OF X(588) (0.5615, 0.2258) X_1131 = ARCTAN(2) KIEPERT POINT (0.5904, 0.2888) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_3316 = ISOGONAL CONJUGATE OF X(3311) (0.5639, 0.2341) X_485 = VECTEN POINT (0.5773, 0.2578) X_3523 = EULER LINE INTERCEPT OF LINE X(35)X(390) (0.5160, 0.1440) X_1132 = ARCTAN(-2) KIEPERT POINT (0.8864, -0.3774) X_3068 = INTERSECTION OF X(2)X(6) AND X(4)X(371) (0.5722, 0.2373) X_1328 = ARCTAN(-3) KIEPERT POINT (0.4997, 1.5024) X_3091 = INTERSECTION OF LINES X(2)X(3) AND X(11)X(153) (0.5673, 0.2783) X_3590 = 1st DIXIT POINT (0.5693, 0.2429) A B C X20 X1327 X590 X1131 X140 X3316 X485 X3523 X1132 X3068 X1328 X3091 X3590 (a, b, c) = (5, 6, 7) | AB = 1

X3590 (1st DIXIT POINT) 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 \(f(a,b,c) = bc/(b^{2} + c^{2} - a^{2} + 6 \operatorname{area}(ABC))\)
Barycentrics: 1/(\(2S_{A}\) + 3S) : 1/(\(2S_{B}\) + 3S) : 1/(\(2S_{C}\) + 3S) [Conway notation]
Search Value (6, 9, 13): \(1.253197919441\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = 1/(2 \cos A + 3 \sin A)\)

X3590 lies on the Kiepert hyperbola.

▸ Incident Central Lines14 lines

{2, 3594}, {4, 6221}, {6, 3591}, {20, 1327}, {76, 3595}, {140, 3316}, {485, 3523}, {486, 5056}, {590, 1131}, {1132, 3068}, {1271, 5490}, {1328, 3091}, {1656, 3317}, {3543, 6482}

Notes

On Dec. 22, 2000, Atul Dixit constructed this point in Hyacinthos message #2183, as follows. Let ABC be a triangle with medians AD, BE, FC and centroid G. Construct semicircles with diameters BD, DC, BC outwardly. Let TA be the circle tangent to the three semicircles BD, DC, BC, and let A' be the center of TA. Define B' and C' cyclically. Then the lines AA', BB', CC' concur in X3590. Barycentric coordinates were found by Paul Yiu (2000), and further properties and related points, by Peter Moses (2010).