X3599

2nd LIU POINT

Let PAPBPC be as at X3598. Let TA be the point of intersection of the lines BPC and CPB. X3599 is the perspector of triangles PAPBPC and TATBTC.
< 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) { 2, 7 }: Centroid–Gergonne Line { 165, 279 }: Central Line L(165, 279) { 55, 479 }: Central Line L(55, 479) { 651, 1190 }: Central Line L(651, 1190) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_7 = GERGONNE POINT (0.5903, 0.2769) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_279 = ISOGONAL CONJUGATE OF X(220) (0.6236, 0.3540) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_479 = -CROSS CONJUGATE OF X(279) (0.6392, 0.4221) X_651 = TRILINEAR POLE OF LINE X(1)X(3) (0.3048, 0.5599) X_1190 = 2nd SARAGOSSA POINT OF X(55) (0.5884, 0.2303) X_3599 = 2nd LIU POINT (0.5716, 0.2497) A B C X2 X7 X165 X279 X55 X479 X651 X1190 X3599 (a, b, c) = (5, 6, 7) | AB = 1

X3599 (2nd LIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc(a - b + c)(a + b - c)(5a^{4} + b^{4} + c^{4} - 8a^{3}b - 8a^{3}c - 4b^{3}c - 4bc^{3} + 6b^{2}c^{2} + 2a^{2}b^{2} + 2a^{2}c^{2} + 12a^{2}bc) : :\)
Barycentrics: \((a + b - c) \cdot (a - b + c) \cdot (5 a^{4} - 8 a^{3} b + 2 a^{2} b^{2} + b^{4} - 8 a^{3} c + 12 a^{2} b c - 4 b^{3} c + 2 a^{2} c^{2} + 6 b^{2} c^{2} - 4 b c^{3} + c^{4}) : :\)
Search Value (6, 9, 13): \(0.926753218576\)
▸ Incident Central Lines22 lines

{1, 8916}, {2, 7}, {55, 479}, {165, 279}, {354, 5543}, {390, 31527}, {497, 36620}, {651, 1190}, {658, 10580}, {1088, 9778}, {1323, 31508}, {1615, 6180}, {1996, 9812}, {2898, 30332}, {3748, 31721}, {4350, 10857}, {5218, 30623}, {7056, 10578}, {8236, 31526}, {10004, 13405}, {14100, 15913}, {32624, 37578}

▸ Functional Relations1 relation

  • X3599 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {55, 479, 3160}