X3591

2nd DIXIT POINT

This point is obtained in the manner of X3590 using the inward semicircles instead of outward.
< 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, 1328 }: Central Line L(20, 1328) { 615, 1132 }: Central Line L(615, 1132) { 140, 3317 }: Central Line L(140, 3317) { 6, 3590 }: Central Line L(6, 3590) { 486, 3523 }: Central Line L(486, 3523) { 1131, 3069 }: Central Line L(1131, 3069) 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_1328 = ARCTAN(-3) KIEPERT POINT (0.4997, 1.5024) X_615 = ISOGONAL CONJUGATE OF X(589) (0.4337, 0.0885) X_1132 = ARCTAN(-2) KIEPERT POINT (0.8864, -0.3774) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_3317 = ISOGONAL CONJUGATE OF X(3312) (0.4661, 0.1442) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_3590 = 1st DIXIT POINT (0.5693, 0.2429) X_486 = INNER VECTEN POINT (-0.0810, -0.0170) X_3523 = EULER LINE INTERCEPT OF LINE X(35)X(390) (0.5160, 0.1440) X_1131 = ARCTAN(2) KIEPERT POINT (0.5904, 0.2888) X_3069 = INTERSECTION OF X(2)X(6) AND X(4)X(372) (0.8529, 0.5391) X_3591 = 2nd DIXIT POINT (0.4080, 0.1149) A B C X20 X1328 X615 X1132 X140 X3317 X6 X3590 X486 X3523 X1131 X3069 X3591 (a, b, c) = (5, 6, 7) | AB = 1

X3591 (2nd 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): \(-2.337113387402\)
▸ Alternate Coordinates1 form

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

X3591 lies on the Kiepert hyperbola.

▸ Incident Central Lines10 lines

{2, 3592}, {6, 3590}, {20, 1328}, {76, 3593}, {140, 3317}, {486, 3523}, {615, 1132}, {1131, 3069}, {1327, 3091}, {1656, 3316}