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Let us consider a pivotal cubic pK(Ω,P) with pole Ω = (p:q:r) and pivot P = (u:v:w).

pK is a circular cubic if and only if

– either its pivot P is the transform of its pole Ω under the mapping ΩPc : Ω -> P, where P is the reflection of the trilinear pole S of the line ΩK with respect to the trilinear polar of tgΩ, the isotomic conjugate of the isogonal conjugate of Ω,

– or its pole Ω is the transform of its pivot P under the mapping PΩc : P -> Ω, where Ω is the barycentric product of P and igP, the inverse in the circumcircle of the isogonal conjugate of P.

These two mappings have the same fixed points namely A, B, C and X(67).

Each mapping has three singular points

– for PΩc : H (orthocenter) and J1, J2 (circular points at infinity),

– for ΩPc : K (Lemoine point) and O1, O2 which are two imaginary points on the orthic axis and on the circum-conic with perspector X(184) passing through X(112), X(248), X(906), X(1415), X(1576), X(2966).

These two pairs of points are very much alike :

– the barycentric product J1 x J2 is K and O1 x O2 is X(32), thus O1, O2 lie on any pivotal cubic with pole X(32) and pivot on the orthic axis, see K478 for instance,

– with suitably chosen coordinates, we have J1, J2 = X(30) ± i ∆ X(523) and O1, O2 = X(230) ± i ∆ X(523), where ∆ is the area of ABC.

Furthermore, O1 and O2 are the barycentric squares of J2 and J1 respectively hence O1 and O2 lie on the Steiner inellipse.

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ΩPc maps any point on the orthic axis onto H and PΩc maps any point on the line at infinity onto K. In other words, there are infinitely many circular pKs with pivot H and pole on the orthic axis and infinitely many circular isogonal pKs with pivot on the line at infinity. See CL035 and Table 12.

From this, we see that ΩPc transforms a line (L) into a circle (C) passing through H and PΩc transforms (L) into a conic (C') passing through K, O1, O2.

These two correspondences lead to two families of interesting cubics when we look at the intersections of a line passing through a fixed point Q and (C) or (C'). This has to be compared with the definition of pivotal isocubics although the two mappings above are not involutions but simply reciprocal transformations.

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A family of circular cubics

These are the cubics K(Q) obtained when we intersect the line (L) and the circle (C). For a given point Q, the cubic K(Q) is a circular circum-cubic that also contains H, X(67), Q, the infinite point of the line KQ (when Q≠K) and ΩPc(Q). K(Q) meets the sidelines of ABC at U, V, W.

One very remarkable fact to observe is that the orthic line of any K(Q) with Q≠K always contains the nine point center X(5) and obviously the infinite point of the line KQ. Recall that the polar conic of any point on this line is a rectangular hyperbola.

With Q = (p:q:r), the equation of K(Q) is:∑[a^4(x+y)(x+z) + b^2c^2 x(y+z)](ry - qz) = 0.

Hence, these cubics form a net that contains several remarkable examples.

– cubics K0 without term in xyz when Q lies on the Euler line. In this case, the cubic contains the complement X(858) of X(23) also on the Euler line.The tangent at H is the same for all these cubics and is parallel to the line X(6)-X(25).

The singular focus lies on the circle passing through

• X(114) the focus of K474 = K(O),

• X(1352) the focus of all K(Q) with Q at infinity (see below),

• the Droussent focus (that of the Droussent cubic K008), intersection of the lines X(3)X(126), X(5)X(111), X(30)X(1296).

– cubics such that the triangles ABC and UVW are perspective when Q lies on a cubic through X(2), X(25), X(251), X(427). It follows that there are exactly three pKs in the pencil : K(X2) = K008 (Droussent cubic), K(X25) = K475 and K(X427).

– cubics such that the points U, V, W are collinear when Q lies on a cubic passing through X(6), X(32), X(112), X(523) giving the two nKs K(X32) = K433 with focus X(98) and K(X523) = K476 with focus X(1352). K(X6) is the union of the line at infinity and the Jerabek hyperbola, K(X112) is the union of the circumcircle of ABC and the line X(4)-X(67).

– cubics with concurring asymptotes when Q lies on a non-circum-cubic through X(6), X(2076), X(2079). The focus of K480 = K(X2079) is X(5), the nine point center. More generally, any K(Q) with Q on the circle through X(6), X(111), X(112), X(115), X(187), X(1560), X(2079) has its focus on the Euler line.

– focal cubics when Q lies on a focal non-circum-cubic through X(6), X(32) already mentioned, X(115) giving the cubic K(X115) = K301 with focus X(4), X(187) giving the cubic K(X187) = K473 with focus X(2). The locus of the foci of all these focal K(Q) is K477, a focal circum-cubic with focus X(67) and a member of this same net.

– nodal cubics among them K(A), K(B), K(C) – these are the isogonal transforms of circles passing through X(3), X(23) and the corresponding vertex of ABC – that can be used to generate the net of cubics K(Q), K(X4) = K288, K(X67).

– cubics with singular focus X(1352), the reflection of X(6) in X(5), when Q lies on the line at infinity. See K(X523) = K476 and K(X30) = K479 for instance.

The following table sums up these cubics with additional less interesting members of the net.

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A family of pseudo-circular cubics

These are the cubics K'(Q) obtained when we intersect the line (L) and the conic (C'). For a given point Q, the cubic K'(Q) is called a pseudo-circular circum-cubic passing through K, X(67), Q, PΩc(Q), these pseudo-circular points O1, O2 and meeting the sidelines of ABC at U, V, W.

These cubics also form a net of cubics and the general equation of K'(Q) is :

∑[a^4(x+y)(x+z) + b^2c^2 x(y+z)](ry - qz) + ∑(-a^4+b^4+c^4-b^2c^2)x^2(ry - qz)= 0,

showing that K'(Q) belongs to the pencil of cubics generated by K(Q) and pK(Q x X67, X67). With Q = G, we obtain the Droussent pencil. See Table 36.

Although less interesting than the previous net of circular cubics, this net contains :

– cubics K0 (without term in xyz) when Q lies on the Euler line again. In such case, K'(Q) contains X(3) and X(468) both on the Euler line. These form a pencil containing the Droussent medial cubic K043 and the cubic K'(X4) that decomposes into the orthic axis and the Jerabek hyperbola.

– cubics such that the triangles ABC and UVW are perspective when Q lies on pK(X25, X427) passing through X(2), X(25), X(251), X(427). It follows that the net contains exactly three pKs namely K'(X2) = K043, K'(X25) = pK(X32, X468) and K'(X427) = pK(X3455, X3455 ÷ X3). X(3455) is the isogonal conjugate of the Droussent pivot X(316) and the inverse in the circumcircle of X(67).

– cubics such that the points U, V, W are collinear when Q lies on the cubic nK(X25, X468, X4) passing through X(4), X(6), X(112), X(523). Hence the net contains exactly three nK0s namely K'(X4), decomposed as seen above, and two other cubics.

Note that K'(X112) is also a decomposed cubic, the union of the line X(6)-X(67) and the circum-conic through the points O1, O2 mentioned at the top of this page.

K'(X6) is K381 and K'(X523) is nK(X3455, X67, X6).

– one circular cubic – K043 already mentioned – and one equilateral cubic.

Here is a selection of these cubics K'(Q).