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Let P = u:v:w be a point and k a real number.

The generalized Tucker cubic T(P, k) is the cubic with equation :


T(P, k) is a non-pivotal cubic nK(#P, P) with pole the barycentric square Ω of P and root P.

Its equation may be written under the following equivalent forms :

(1) : ∑ v w x^2 (w y + v z) + (3 - k) u v w x y z= 0 or u^2 y z (w y + v z) + (3 - k) u v w x y z = 0,

(2) : ∑ u x (w y - v z)^2 + (9 - k) u v w x y z = 0.

(3) : ∑ u x (w y + v z)^2 - (k + 3) u v w x y z = 0,


Singular cases and consequences

T(P, 0) decomposes into the trilinear polar L(P) of P and the circum-conic C(P) with perspector P.

T(P, 1) decomposes into the sidelines of the anticevian triangle of P.

It follows that any cubic T(P, k) is a member of the pencil of circum-cubics generated by these two decomposed cubics. For a given P, all these cubics pass through the traces U, V, W of L(P) on the sidelines of ABC and the tangents at A, B, C to T(P, k) also contain U, V, W.

If Q is another point, the isoconjugation that swaps P and Q maps T(P, k) onto T(Q, k). The pencil of cubics generated by T(P, k) and T(Q, k) contains the pivotal cubic with pole Ω and pivot the intersection of L(P) and L(Q).


Special cases

• T(P, k) contains P if and only if k = 9 and then, T(P, 9) is cK0(#P, P), a nodal cubic with node P. This is obvious from equation (2).

For instance, K015 = T(X2, 9), K228 = T(X1, 9), K229 = T(X6, 9).

T(P, 3) is a nK0, see equation (1).

For instance, K016 = T(X2, 3).

T(P, 2) is also remarkable.

For instance, K327 = T(X2, 2).


Circular cubics

T(P, k) can be a circular cubic (for some k) if and only if P lies on the Grebe cubic K102 = pK(X6, X6).

Two isogonal conjugates on K102 correspond to two cubics, each being the isogonal transform of the other.

In particular, when P = X(1), both cubics coincide giving the isogonal focal cubic K721.


Equilateral cubics

T(P, k) can be an equilateral cubic (for some k) if and only if P lies on K278 = pK(X1989, X1989).

The corresponding cubics are very complicated.


Other remarkable cubics

See the related cubics K214 and K721.