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Let M be a point and denote by K(M) the nodal cubic psK+(M, tcM, M).

With M = u : v : w, the equation of K(M) is :

∑ u (v + w)[(u + w) y – (u + v) z] y z = (u – v) (v – w) (w – u) x y z

When M = G, K(M) is the union of the medians of ABC. When M lies on a sideline or on a median of ABC, K(M) splits into this same line and a (possibly decomposed) conic. This is excluded in the sequel.

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Geometric properties

• The node of K(M) is M with nodal tangents parallel to the asymptotes of the circum-conic C(M) passing through M and cM. These nodal tangents are perpendicular if and only if M lies on the Lucas cubic K007.

• K(M) has three asymptotes concurring at the midpoint X of GM. These are parallel to those of pK(M x ccM, M), pK(ctM, cM), pK(cM x ccM, aM) and more generally to those of a pK with pole on psK(ctM x ccM, G, atM) and pivot on the complement of psK(ctM x ccM, G, M).

• K(M) is globally invariant under the product of M-isoconjugation and tcM-Ceva conjugation, in both orders.

• K(M) passes through A, B, C, M, cM, the vertices or the cevian triangle of tcM, the two points M1, M2 deduced from the transformations above .

M1 = M ÷ (tcM / cM) and M2 = tcM / (M ÷ cM) where ÷ and / denote M-isoconjugation and tcM-Ceva conjugation respectively. This can be repeated thus giving other points on K(M).

The tangents at A, B, C to K(M) concur at the pseudo-isopivot which is ctM = M ÷ tcM.

• The tangential T1 of cM lies on the line M1M2 and on the parallel at M to the trilinear polar L(M) of M.

It folllows that T2 = M ÷ (tcM / T1) and T3 = tcM / (M ÷ T1) are two other points on K(M). Note that cM, M1, T3 and cM, M2, T2 are two triads of collinear points on K(M).

• If P = p : q : r is any point different of M then Q = (v+w) (q u-r u-p v-r v+p w+q w) / (r v-q w) : : lies on K(M).

Q = cM x aZ ÷ Z where Z is the isotomic conjugate of the trilinear pole of the line MP.

Construction of K(M)

Let M be a point different of G.

Denote by MaMbMc the cevian triangle of tcM and by (L) the line passing through G and tcM.

The trilinear pole S of (L) lies on the Steiner ellipse hence the inscribed conic with perspector S is a parabola (P).

A variable tangent (T) to (P) meets the lines MMa, MMb, MMc at Qa, Qb, Qc respectively.

The triangles ABC and QaQbQc are perspective at Q which lies on K(M).

Note that the nodal tangents at M to K(M) are those drawn from M to (P).

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When M = X(1), K(M) is the cubic K974 = psK(X1, X86, X1) and (P) is the Kiepert parabola.

More generally, for any M on the Wallace hyperbola, (P) is the Kiepert parabola again and tcM lies on the line X(2)X(6).

Recall that the Wallace hyperbola is the diagonal rectangular hyperbola passing through the in/excenters of ABC, G and many other centers such as X(i) for i = 20, 63, 147, 194, 487, 488, 616, 617, 627, 628, 1670, 1671, 1764, 2128, 2582, 2583, 2896, etc. Another example is K(X20) = K041.

Related cubics

• Under isotomic conjugation, K(M) is transformed into a cubic of the same type namely K(tM) = psK+(tM, tctM, tM).

• Under M-isoconjugation and/or tcM-Ceva conjugation, K(M) is transformed into another nodal cubic K'(M) = psK(M, tcM, G).

• The barycentric quotient of K(M) by M is psK(tM, tcM x tM, G).

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Strong and weak cubics

When M is a strong (resp. weak) center, K(M) is a strong (resp. weak) cubic.

Under the symbolic substitution SS{a -> a^2}, a weak cubic K(M) is transformed into a strong cubic K(M). For example, the weak cubic psK+(X8, X75, X1) is associated with the strong cubic psK+(X69, X76, X6) = K257.

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A selection of cubics

The following table presents a selection of cubics K(M) according to the node M. A weak cubic (highlighted in green) is associated with a strong cubic (highlighted in pink). Two strong cubics (in yellow) can also be associated.

Remarks :

• two isotomic conjugates M and tM correspond to K(M) and K(tM) = t K(M). See K028 and K257 for example with M = X4 and X69.

• the "orange" points M lie on K007 hence the nodal tangents at M to K(M) are perpendicular. The pseudo-pivot tcM lies on K184.

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The diagram below shows several cubics K(M) and other related cubics obtained through usual transformations. A blue cubic is weak and the red nearby cubic is that obtained under SS{a -> a^2}. Beware not all the cubics are members of the class.

Two cubics of the same color in a diagonal of an octogon have the same pseudo-pole Ω and the same pseudo-pivot P. They are exchanged by Ω-isoconjugation and by P-Ceva conjugation. For example, K009 = psK(X184, X2, X3) and K260 = psK(X184, X2, X6).

As usual, g and t denote isogonal and isotomic conjugations. X1 denotes X1-isoconjugation i.e. isoconjugation with pole the incenter X1.

Remarks :

• two cubics of the same color can be related either by other isoconjugations or by some barycentric products. For example, K260 is the X3-isoconjugate of K257, the X32-isoconjugate of K429 and the barycentric products X(3) x K028, X(6) x K935.

• the cubics K620, K724, K967 = g K971, K1069 = t K220 can also be related similarly to one or several of the sixteen cubics in the diagram above.

• the following table gives the unlisted isogonal and isotomic transforms of the cubics of the diagram.