Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

too complicated to be written here. Click on the link to download a text file.

X(2), X(4), X(3434), X(4846), X(11185), X(14593), X(30513), X(34029), X(41370)

X(52448) → X(52456)

X(52483) → X(52489)

A' on BC and on the line through G and the reflection of H in BC, B' and C' likewise.

infinite points of K243 = pK(X6, X376)

points on (O) and pK(X6, X3060)

Geometric properties :

K1301 is a member of the pencil of cubics generated by K025 and K028. See Table 43.

It is the locus of the intersection of the Steiner line of M and the line GM when M is a variable point on the circumcircle.

K1301 is a nodal cubic with node H and nodal tangents parallel to the asymptotes of the Jerabek hyperbola.

***

Let HaHbHc be the orthic triangle and PaPbPc the cevian triangle of P. The triangle formed by the midpoints A', B', C' of HaPa, HbPb, HcPc is perspective to ABC if and only if P lies on K616 and then, the perspector lies on K1301 (based on notes by César Lozada).

Every cubic of the pencil above can be characterized analogously. Let A' be the homothetic of Pa under h(Ha, t) and define B', C' cyclically. The triangles ABC and A'B'C' are perspective if and only if P lies on a cubic K(t) and then, the perspector lies on a similar cubic K(1/t). Both cubics are similar to K616 (t = 1/2) and K1301 (t = 2).

K617 and K028 are limit cases K(t → 0) and K(t → 1) respectively. K(t → ∞) is the union of the altitudes of ABC.

K(-1) = K028, K(1/3) = K618, K(2/3) = K917. The cubic K1302 is obtained with t = 1 - (a^2b^2c^2) / (8 SA SB SC). It is the only proper cK of the pencil.

If E(t) is the homothetic of O under h(H, 2 / (1+t)), K(t) is the locus of the intersection of the Steiner line of M and the line E(t)M when M is a variable point on the circumcircle.

***

Generalization

Let SaSbSc be the cevian triangle of a fixed point S and PaPbPc the cevian triangle of a variable point P. Let A' be the homothetic of Pa under h(Sa, t) and define B', C' cyclically. The triangles ABC and A'B'C' are perspective if and only if P lies on a cubic K(S, t) and then, the perspector lies on a similar cubic K(S, 1/t).

K(G, t) is the union of the medians of ABC for every t. This is excluded in the sequel.

The isotomic transform of K(S, t) is K(S', t) where S' is the isotomic conjugate of S.

For a given point S, these cubics K(S, t) form a pencil F(S) of nodal circum-cubics with node S, all having the same nodal tangents (T1), (T2). These are the parallels at S to the asymptotes of the circum-conic C(S) that passes through S and S'.

C(S) is the isotomic transform of the line L(S) that passes through S and S'. Hence, the nodal tangents are perpendicular if and only if C(S) contains H and then L(S) contains H' = X(69). It follows that S must be a point on the Lucas cubic K007 and then, C(S) is the Jerabek hyperbola.

Each cubic K(S, t) is then characterized by its "last" point T on the line GS. T is the homothetic of G under h(S, 3 / (1+t)).

Limit cases :

• K(S, t → ∞) is the union of the cevian lines of S.

• K(S, t → 1) is a psK+ with asymptotes concurring at the midpoint of GS. It is psK(S, tcS, S) passing through cS, the complement of S. These are the cubics of CL067. [K028]

• K(S, t → 0) passes through aS (anticomplement of S), the vertices of the antimedial triangle. [K617] . It is the anticomplement of psK(cS x ctS, G, S). [K009]

Other cases :

• K(S, t = -1) passes through the infinite points of the line GS and the circum-conic with center cS. [K025]

• K(S, t = 2) passes through G. [K1301]

• K(S, t = 1/2) passes through the reflection of S in G. [K616]

Special cubics :

The pencil F(S) contains :

• a second psK obtained when t = 1 - (v+w)(w+u)(u+v) / (2uvw). It is psK(S^3, S x tcS, S), [K620] , the barycentric product S x psK(S, tcS, G). [K555]

• a proper cK obtained when t = 1 - (v+w)(w+u)(u+v) / (4uvw). It is cK(#S, R) where R is the barycentric product of S and the intersection of the polar lines of S and S' in the Steiner ellipse. R = u (v-w)) / (v+w) : : . [K1302]

• two decomposed cKs. Each is the union of a nodal tangent and its S^2-isoconjugate, the circum-conic passing through S and tangent at S to the nodal tangent.

The pencil F(S) contains one circular cubic and, at the same time one equilateral cubic, if and only if S lies on Q050. In this case, the line GS passes through igS, the inverse in (O) of the isogonal conjugate of S. [K025] and [K028]. Q050 passes through X(i) for i = 2, 3, 4, 13, 14, 67, 1113, 1114, 11058.

The cubics [Knnn] are those obtained when S = H as above.