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K1267

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X(3), X(3357), X(5450), X(6759), X(6796)

infinite points of K004

vertices of the CircumNormal triangle N1N2N3

vertices of the CircumTangential triangle T1T2T3

Geometric properties :

K1267 is the CircumNormal-isogonal transform of the McCay cubic K003. See K735, a similar cubic where a generalization is given, K1268, K1269, and also Table 25. See CL076 for another generalization and other analogous cubics.

K1267 and K003 generate a pencil which contains the Neuberg strophoid K725. All these cubics pass through O (twice), N1, N2, N3 and four points on two perpendicular lines, secant at X(5), parallel to the asymptotes of the rectangular circum-hyperbola with center X(11792) and perspector X(55280), passing through {4, 140, 252, 1232}.

These four points also lie on the cubic K800 and on the rectangular hyperbola passing through {3, 4, 195, 576, 1147, 2574, 2575, 2888, 2904}.

K1267 is a central cubic with center O, point of inflexion with inflexional tangent the Euler line.

K1267 and the Darboux cubic K004 share the same asymptotes and meet again at O and two points on the line {3,54}, obviously symmetric about O. These two points lie on the circum-conic with perspector X(14346).

The tangentials of X(5450), X(6796) are X(6759), X(3357) respectively and the reflections of X(3357), X(5450) in O are X(6759), X(6796) respectively.

Note that the CircumTangential-isogonal transform of K1267 is the reflection of the McCay cubic about O.