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X(2), X(3), X(6), X(99), X(385), X(5104), X(5106), X(11676)

infinite points of K002

vertices of the Thomson triangle

imaginary common points of (O) and the Lemoine axis

other points below

Geometric properties :

K1115 is a member of the pencil of cubics generated by the Thomson cubic K002 and the union of the line at infinity with the Thomson-Jerabek hyperbola. The central cubic K764 and K1114 are two other members.

K1115 meets the circumcircle at the vertices of the Thomson triangle and three other points – apart A, B, C – on K017 = nK0(X6, X385).

P1 = {X6,X694} /\ {X99,X511} = a^2 (a^6 b^6-a^4 b^8+a^8 b^2 c^2+a^6 b^4 c^2-a^4 b^6 c^2+a^6 b^2 c^4-5 a^4 b^4 c^4+a^2 b^6 c^4+a^6 c^6-a^4 b^2 c^6+a^2 b^4 c^6+2 b^6 c^6-a^4 c^8) : : , SEARCH = 2.20602620610655

P2 = {X39,X729} /\ {X99,X187} = a^2 (2 a^6 b^4-a^4 b^6-5 a^6 b^2 c^2-2 a^4 b^4 c^2+4 a^2 b^6 c^2+2 a^6 c^4-2 a^4 b^2 c^4+7 a^2 b^4 c^4-4 b^6 c^4-a^4 c^6+4 a^2 b^2 c^6-4 b^4 c^6) : : , SEARCH = 7.50358555290657

P3 on {X6, X99} = 2 a^8 b^4-5 a^8 b^2 c^2-2 a^6 b^4 c^2+4 a^4 b^6 c^2+2 a^8 c^4-2 a^6 b^2 c^4+4 a^4 b^4 c^4-4 a^2 b^6 c^4+4 a^4 b^2 c^6-4 a^2 b^4 c^6+b^6 c^6 : : , SEARCH = -2.73527391512195

P4 on {X3, X99} = 2 a^10 b^6-2 a^8 b^8+a^12 b^2 c^2+2 a^10 b^4 c^2-a^8 b^6 c^2-4 a^6 b^8 c^2+2 a^10 b^2 c^4-6 a^8 b^4 c^4+2 a^6 b^6 c^4-2 a^4 b^8 c^4+2 a^10 c^6-a^8 b^2 c^6+2 a^6 b^4 c^6+11 a^4 b^6 c^6+b^10 c^6-2 a^8 c^8-4 a^6 b^2 c^8-2 a^4 b^4 c^8-2 b^8 c^8+b^6 c^10 : : , SEARCH = 1.97366540166108

P5 on {X3, X385} = -2 a^12 b^4+a^10 b^6-2 a^8 b^8+3 a^6 b^10+4 a^12 b^2 c^2-a^10 b^4 c^2+7 a^8 b^6 c^2+15 a^6 b^8 c^2-7 a^4 b^10 c^2-2 a^12 c^4-a^10 b^2 c^4+8 a^8 b^4 c^4+45 a^6 b^6 c^4-10 a^4 b^8 c^4+5 a^2 b^10 c^4+a^10 c^6+7 a^8 b^2 c^6+45 a^6 b^4 c^6-29 a^4 b^6 c^6-5 a^2 b^8 c^6-b^10 c^6-2 a^8 c^8+15 a^6 b^2 c^8-10 a^4 b^4 c^8-5 a^2 b^6 c^8+2 b^8 c^8+3 a^6 c^10-7 a^4 b^2 c^10+5 a^2 b^4 c^10-b^6 c^10 : : , SEARCH = -0.822158986921623

These points are X(33755), X(33756), X(33757), X(33758), X(33759) in ETC (2019-08-04).