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K814

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X(3), X(4), X(20),

isogonal conjugate of X(3529) and its reflection about O

X3-OAP points, see Table 53

their reflections about O

antipodes of A, B, C on (O)

K814 is a cubic passing through the X3-OAP points, see Table 53.

It is a central cubic with center X(3), a member of the pencil generated by K004 and the cubic which is the union of the circumcircle with the Euler line. This pencil also contains K047, K080, K426, K443, K566, K706, K850. All these cubics are spK(P, Q) with P, Q on the Euler line such that Q is the midpoint of P, X(20). See Table 54, column P = [X20].

K814 is also a member of the pencil generated by K006 and K026, also containing K405, K762, K810, K811, K812, K813. All these cubics pass through the X3-OAP points. See Table 54, column P = S.

At last, K814 is a member of the (less interesting) pencil generated by K187 and pK(X6, X1657). All these cubics pass through the four foci of the inconic with center X(1657), a point on the Euler line. This latter pencil contains the stelloid spK(X3, X1657).