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A cubic is a member of the Steiner net when it is a circum-cubic passing through the (four) foci of the Steiner inscribed ellipse, the in-conic with center the centroid G of ABC. These foci are X{39162, 39163, 39164, 39165} in ETC.

Any such cubic is a spK(P, G) for some P. See CL055 for general properties of these cubics and their construction.

The following table gives a selection of spK(P, G) according to P.

Notes :

• spK(P, G) is circular if and only if P lies at infinity, giving a pencil of focal nKs with root G, focus F on (O), see yellow lines.

• spK(P, G) is also a nK when P lies on the Steiner (circum) ellipse in which case its root lies on this same ellipse and the pole on K229. Two examples are given in the pink lines.

• spK(P, G) is a K0 if and only if P lies on the line GK in which case it contains G and K, see red points P in the table. These cubics are in a same pencil of K0s, stable under isogonal conjugation, generated by K002 and K018 which are the only two self-isogonal cubics.

• if P1 and P2 are symmetric about G, then the cubics spK(P1, G) and spK(P2, G) are swapped under isogonal conjugation. They meet again at two isogonal conjugate points on K002 hence collinear with G.

Other cubics passing through the foci of the Steiner inscribed ellipse

These cubics are not circum-cubics of ABC but, in some cases, they are circum-cubics of some other triangle.

Those in the orange cells are circum-cubics of the Thomson triangle (T) and pass through G and K. They form a pencil which contains the Thomson cubic K002. The remaining points on (O) lie on a nK0(X6, R on GK) and the tangents at these points concur on a hyperbola passing through {2, 6, 110}.

Those in the blue cells are focal cubics with singular focus F on the circle passing through {2, 3, 110, 842, 8724, 14649, 14685, 34291, 35911}, with center X(44814), which is the (O)-inverse of the line {23,110} ans the Psi-image of the line {3,74}. These cubics are in a same pencil of circular cubics passing through X(2), X(3), X(110). They are Psi-invariant, see at the bottom of Table 60. Every such focal cubic meets the circumcircle (O) again at three points on an isogonal pK with pivot on the line GK.

Those in the green cells are focal cubics with singular focus F on the Fermat axis and same orthic line, namely the Euler line. These cubics are in a same pencil of circular cubics passing through X(30), X(31862), X(31863). The latter two points lie on the Fermat axis.

The red cubics are also in a same pencil of focal cubics with singular focus F on the line {3, 74, 110, 156, 246, 399, 1511, 1614, 2972, 3470, 5191, 5609, etc} which contains the circum-cubic K187 and the decomposed cubic, union of the line at infinity and the rectangular hyperbola with center G, passing through {3, 6, 381, 599, 2574, 2575} and the four foci of inSteiner. This hyperbola is the polar conic of X(30) in K003, of X(3) in K005 , of X(6) in K018. Every cubic of the pencil passes through X(30) and O, with tangent the line {3, 74, etc}, and orthic line the Euler line, except K038 which has a node at O.

The blue cubics are in a same pencil of stelloids with radial center X on the Euler line, which also contains the decomposed cubic above and the circum-cubic K358. Every cubic contains O, X(381) and the infinite points of the McCay cubic K003.