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X(6), X(66), X(69), X(157), X(264), X(393), X(2165), X(6389), X(40697), X(40497)

infinite points of K627

points of pK(X4, P2) on the circumcircle with P2 = X(3767) ÷ X(6)

vertices of the MacBeath triangle = cevian triangle of X(264)

other points below

Geometric properties :

 

P1 = (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (a^4+b^4-2 b^2 c^2+c^4) : : , SEARCH = 8.837048581226908

P2 = b^2 c^2 (a^4+b^4-2 b^2 c^2+c^4) : : , SEARCH = 2.707937036988545

P3 = (a^4-b^4-c^4) (a^4+b^4-2 b^2 c^2+c^4) : : , SEARCH = 10.15773328841999

P4 = (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4+b^4-2 b^2 c^2+c^4) : : , SEARCH = -0.9510035912464061

P5 = (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4+b^4-2 b^2 c^2+c^4) (a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+a^2 c^4-b^2 c^4-c^6) : : , SEARCH = 0.8491200881303797

P6 = b^2 c^2 (a^4+b^4-2 b^2 c^2+c^4) (-3 a^6+3 a^4 b^2-a^2 b^4+b^6+3 a^4 c^2-2 a^2 b^2 c^2-b^4 c^2-a^2 c^4-b^2 c^4+c^6) : : , SEARCH = -4.026050785906658

P7 = b^2 c^2 (a^4+b^4-2 b^2 c^2+c^4) (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-b^4 c^2+a^2 c^4+b^2 c^4-c^6) (-a^6+a^4 b^2-a^2 b^4+b^6+a^4 c^2-b^4 c^2+a^2 c^4+b^2 c^4-c^6) : : , SEARCH = 6.094944411311998

P8 = (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 (a^6+a^4 b^2-a^2 b^4-b^6+a^4 c^2-2 a^2 b^2 c^2+b^4 c^2-a^2 c^4+b^2 c^4-c^6) : : , SEARCH = -0.3960788593395253

P9 = (a^2-b^2-c^2) (a^4+b^4-2 b^2 c^2+c^4) (a^8-2 a^6 b^2+2 a^4 b^4-2 a^2 b^6+b^8-2 a^6 c^2+4 a^4 b^2 c^2-2 b^6 c^2+2 a^4 c^4+2 b^4 c^4-2 a^2 c^6-2 b^2 c^6+c^8) : : , SEARCH = -6.607907042702455

P10 = a^2 (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-2 a^2 b^2 c^2-b^4 c^2+3 a^2 c^4+3 b^2 c^4-3 c^6) (a^6-a^4 b^2+3 a^2 b^4-3 b^6-a^4 c^2-2 a^2 b^2 c^2+3 b^4 c^2-a^2 c^4-b^2 c^4+c^6) : : , SEARCH = 7.403848350132711

P11 = (a^2-b^2-c^2) (a^6+a^4 b^2+a^2 b^4+b^6-a^4 c^2-2 a^2 b^2 c^2-b^4 c^2+a^2 c^4+b^2 c^4-c^6) (a^6-a^4 b^2+a^2 b^4-b^6+a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+a^2 c^4-b^2 c^4+c^6) : : , SEARCH = 2.930270961585107

Triads of collinear points on the curve :

X(6) – X(66) – P5 // X(6) – X(69) – P1 // X(6) – X(157) – X(40947) // X(6) – X(264) – P2 // X(6) – X(393) – P3

X(6) – X(2165) – P9 // X(66) – X(69) – P6 // X(66) – X(157) – X(6389) // X(66) – X(264) – P3 // X(66) – P2 – P8

X(69) – X(264) – P4 // X(157) – X(264) – P7 // X(157) – X(2165) – P4 // X(157) – P3 – P11 // X(264) – X(393) – X(6389)

X(264) – X(2165) – P1 // X(264) – P5 – P11 // X(264) – P6 – P10 // X(393) – X(2165) – P6 // X(393) – X(40947) – P8

X(6389) – X(40947) – P9 // X(6389) – P1 – P2 // X(40947) – P1 – P3 // X(40947) – P2 – P6 // X(40947) – P4 – P5

P1 – P5 – P7 // P1 – P8 – P10 // P2 – P3 – P4 // P3 – P7 – P9

These points are now X(41760) -> X(41770) in ETC.

K1188a