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K569

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X(3), X(4), X(54), X(140), X(1493)

midpoints of ABC

infinite points of the Napoleon cubic K005

points of pK(X6, X140) on the circumcircle

K569 is a cubic related to circular psK cubics with pseudo-pivot G. See Pseudo-Pivotal Cubics and Poristic Triangles.

Its isogonal transform is K849 = spK(X140, X3628).

It is a member of the pencil generated by K002 and K009. See Table 50.

It is also a member of the pencil generated by K005 and the union of the line at infinity with the Jerabek hyperbola. All these cubics are spK(X5, Q) with Q on the Euler line as in CL055. Apart K005, this pencil contains two psKs which are K569 = spK(X5, X3628) and K361 = psK(X5, X140). The central cubic K566 = spK(X5, X548) is another remarkable member.