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K1075

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X(1), X(3), X(4), X(218), X(277), X(1010), X(1245), X(1780), X(3494), X(3502), X(8481), X(8482), X(8926), X(8947), X(8949), X(23604), X(23605)

X(23604), X(23605) = isogonal conjugates of X(1780), X(8926)

Geometric properties :

K1075 is an isogonal conico-pivotal nodal cubic with node X(1). The nodal tangents pass through X(7) and X(8).

The pivotal conic (P) is tangent to the Euler line at X and to the nodal tangents at X(2951) and E respectively where E = a (a^2+2 a b+b^2+2 a c-6 b c+c^2) : : , with SEARCH = -1.40947355334035, now X(23511) in ETC (2018-09-22).

The contact X of (P) and the Euler line is : a^6+a^5 b-a^2 b^4-a b^5+a^5 c+3 a^4 b c-2 a^3 b^2 c-2 a^2 b^3 c+a b^4 c-b^5 c-2 a^3 b c^2+2 a^2 b^2 c^2-2 a^2 b c^3+2 b^3 c^3-a^2 c^4+a b c^4-a c^5-b c^5 : : , with SEARCH = 20.0010392423145, also on the line {171, 516}, now X(23512) in ETC (2018-09-22).

K1075 is the image of the circum-conic (C) with perspector X(1) under the transformation 𝛕P with P = X(3732) which is described in page K1065.

See the analogous K1073 and also Table 69.