too complicated to be written here. Click on the link to download a text file. 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).