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Let Q be a point and K(Q) = spK(cQ, Q), where cQ is the complement of Q. See CL055 for general properties of spK cubics.

When Q is the centroid G of ABC, K(Q) = spK(G, G) = pK(X6, G) = K002. This case is excluded in the sequel.

When Q lies on the line at infinity, here again cQ = Q and K(Q) is a circular cubic. See below.

For any point Q, K(Q) is a circum-cubic that must pass through :

• four fixed points R0, R1, R2, R3 (independent of Q) described in the page K755. These are the common points of all the polar conics of the circumcenter O of ABC in the isogonal circular pKs. See CL035. Since these polar conics are rectangular hyperbolas, R0, R1, R2, R3 are the vertices of an orthocentric quadrilateral with diagonal triangle T whose circumcircle is C(X4, 2R) and whose in/excenters are these four points. Further details are given at K755.

• the isogonal conjugate gcQ of cQ.

• the reflection scQ of cQ in Q.

• the infinite points of pK(X6, cQ).

• the points on (O) of pK(X6, scQ).

• the four foci of the inconic I(Q) with center Q (when Q is not an in/excenter nor a point at infinity). These foci lie on pK(X6, Q). The real foci are denoted F1, F2.

• the two remaining common points S1, S2 of the axes of I(Q). These lie on a line passing through gcQ.

• the traces on the sidelines of ABC of the reflections (Da), (Db), (Dc) in Q of the corresponding cevian lines of cQ. This gives a form of the equation of K(Q) which is ∑ a^2 y z (Da).

• the common points Q1, Q2 of the line GQ and the conic which is its isogonal transform. Obviously, Q1, Q2 lie on K002.

Special cubics spK(cQ, Q)

• Circular cubics : these are the cubics obtained when Q lies on the line at infinity. K(Q) passes through Q, gQ. The singular focus is the homothetic of gQ under h(O, 5) hence it lies on the circle C(O, 5R).

• Equilateral cubics : there is only one such cubic since cQ must be O hence Q = H. The cubic is the stelloid K525.

• K0s : the term in xyz vanishes if and ony if Q lies on the line GK in which case K(Q) passes through G and K. See yellow cells in the table below and specially K755.

• psKs : Q must lie on one median of ABC. The only pK is K(G) = K002.

• nKs : cQ must lie on a nK(X2, X2, ?) passing through no listed ETC centers.

• nodal cubics : Q must lie on a very complicated 12th degree curve passing through the in/excenters.

• K+ : K(Q) has concurring asymptotes if and only if Q lies on K917, a circum-cubic with node H. K(H) is K525 as already mentioned. K(A), K(B), K(C) are central psK cubics with centers A, B, C respectively.

• K(Q) passes through cQ when Q lies on K007.

• When cQ lies on K007, K(Q) meets the sidelines of ABC again at the vertices of a pedal triangle. Some examples are given in the table below.

The following table gives a selection of cubics K(Q), with contributions by Peter Moses.

Remark : the cubics K(Q) are the members of a same net which contains several remarkable pencils corresponding to Q lying on a same line passing through G. Each pencil contains K002 and a circular pK. They are denoted by cells of the same color in the table.