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X(2), X(4240), X(5466), X(5468), X(6548), X(17780), X(30508), X(30509), X(34760), X(34761), X(34762), X(34763), X(34764), X(34765), X(34766), X(34767), X(41314), X(43928), X(47070), X(47071)

points at infinity of ABC sidelines (inflexion points)

See "Tucker cubics" in the Downloads page and Special Isocubics §8.3.

The Tucker nodal cubic T(G) = cK(#X2, X2) is the only unicursal Tucker cubic. It is an isotomic conico-pivotal cubic with pivotal conic the Steiner ellipse. See also isotomic nK0 cubics. Compare K015 with K228, isogonal conico-pivotal cubic.

The complement of K015 is K219 = A1(G), an Allardice cubic (see CL010). The isogonal transform of K015 is K229.

K015 and K229 are two members of CL064.

The Hessian of K015 is its homothetic under h(G, -1/3).

More generally, the n^th Hessian of K015 is the homothetic of K015 under the homothety with center G, ratio (-1/3)ⁿ.

Locus properties :

1. For any point P, the parallel at G to BC meets AP at Pa. Define Pb, Pc similarly. The locus of P such that Pa, Pb, Pc are collinear is T(G).
2. Let P be the perspector of an inscribed conic (C) with center Q. The circum-conic (C') passing through P and Q is a parabola if and only if P lies on T(G).
3. Let M be the Miquel point of the quadrilateral formed by the sidelines of ABC and the trilinear polar L of P. Let M' be the second intersection of the line KM with the circumcircle. The Simson line of M' is perpendicular to L if and only if P lies on K015 (after Philippe Deléham). These lines are parallel if and only if P lies on K295.
4. Two parabolas, one inscribed and one circumscribed, with the same infinite point meet at two other isotomic conjugate points M1, M2 lying on K015. See figure 1 below. See also the related property 22 in K002 and Q154.
5. The arithmetic and harmonic means of the barycentric coordinates of a point are equal if and only if this point lies on K015. When "harmonic" is replaced by "geometric", we find the cubic K656.
6. See also the generalizations in the pages K052 and K406.
7. Locus of P whose tripolar centroid lies on the Steiner inscribed ellipse. See CL045.
8. Locus of P whose tripolar centroidal conjugate is P. See CL045.

Let M = u : v : w be a point and M' its isotomic conjugate. The polar lines of M and M' in the Steiner circum-ellipse intersect at f(M) = f(M') which is not defined when M is the centroid G or one of the vertices of the reference and antimedial triangles.

This transformation is given by : f(M) = (v - w) / (v + w) : (w - u) / (w + u) : (u - v) / (u + v).

f transforms the line at infinity and the Steiner circum-ellipse into K015 which gives a lot of points on K015 with rather simple coordinates. Very few of these points are listed in ETC.

For example : f(X99) = f(X523) = X5468, f(X648) = f(X525) = X4240, f(X671) = f(X524) = X5466, f(X903) = f(X519) = X6548.

Figure 1

Pc, Pi are the circumscribed and inscribed parabolas,

Fc, Fi are their foci with Fi on the circumcircle (O),

Dc, Di are their directrices with Di passing through H.

The line M1M2 is tangent to the Steiner ellipse at M, the isotomic conjugate of the common infinite point of Pc, Pi.

Recall that the parameter of Pi is 4 times that of Pc.

Q077 is the locus of Fc, Q080 is the envelope of Dc,

Q078 and Q079 are the loci of the vertices of Pi, Pc.

Pc and Pi are obviously homothetic at a point Q which lies on the Bataille acnodal cubic K656.

Remarks :

• The tangents at M1, M2 to Pi concur at G and the tangents to Pc concur at a point lying on the homothetic of K015 under h(G, 3).
• The midpoint of M1M2 lies on the reflection of K015 about G.