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K1270

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X(3), X(15), X(16), X(99), X(11676), X(21444), X(47265), X(47266)

infinite points of K003

vertices of the CircumNormal triangle N1N2N3

imaginary points L1, L2 on (O) and the Lemoine axis, isogonal conjugates of the Steiner infinity points

orthogonal projections R1, R2, R3 of X(11676) on the sidelines of N1N2N3

Geometric properties :

K003 and K736 generate a pencil of stelloids with radial center on the Euler line. See K736 for details and also Table 25.

K1270 is another member of this pencil.

The asymptotes are parallel at X(13586) to those of the McCay cubic K003.

The polar conic of X(11676) is the rectangular hyperbola (H) passing through X(3), X(74), X(99), X(2574), X(2575), hence homothetic to the Jerabek hyperbola.

K1270 is the McCay cubic of the non proper triangle with vertices X(99), L1, L2.

Other points on K1270 :

M1 = a^2*(Sqrt[3]*(a^2 - b^2 - c^2)*(a^4*b^4 - a^4*b^2*c^2 - a^2*b^4*c^2 + a^4*c^4 - a^2*b^2*c^4 + b^4*c^4) - 2*(a^4*b^4 - 7*a^4*b^2*c^2 + 5*a^2*b^4*c^2 + a^4*c^4 + 5*a^2*b^2*c^4 - 5*b^4*c^4)*S) : : , on lines {3, 5106}, {15, 99}, {62, 729}, {9431, 11486}, {33757, 35918}, SEARCH = -14.2771323541213.

M2 = a^2*(Sqrt[3]*(a^2 - b^2 - c^2)*(a^4*b^4 - a^4*b^2*c^2 - a^2*b^4*c^2 + a^4*c^4 - a^2*b^2*c^4 + b^4*c^4) + 2*(a^4*b^4 - 7*a^4*b^2*c^2 + 5*a^2*b^4*c^2 + a^4*c^4 + 5*a^2*b^2*c^4 - 5*b^4*c^4)*S) : : ,on lines {3, 5106}, {16, 99}, {61, 729}, {9431, 11485}, {33757, 35917}, SEARCH = 7.34558120381172.

Note that O, M1, M2 are collinear on the cubic. These two points are X(47265), X(47266) in ETC.