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Let P = p : q : r be a point and denote by K(P) the cubic psK(P^3, G, P) where G is the centroid of ABC.

The equation of K(P) is : (q - r)(r - p)(p - q) x y z + ∑ p^3 (y -z) y z = 0.

When P lies on a sideline or on a median of ABC, K(P) splits into this same line and a conic. In particular, K(G) is the union of the medians of ABC. These special cases are excluded in the sequel.

Properties of K(P)

• K(P) is a circum-cubic of ABC and the tangents at A, B, C concur at P^3.

• K(P) is a circum-cubic of the medial triangle A'B'C'.

• K(P) passes through P which is a point of inflexion. The inflexional tangent T(P) passes through G and the polar conic of P splits into T(P) and the harmonic polar H(P) of P. H(P) is the image of the trilinear polar L(P) of P under the homothety h(P, 2/3). T(P) and H(P) meet at X, the center of the polar conic of P, hence X (and P) must lie on the (orange) Hessian of K(P).

• K(P) is globally invariant under two transformations 𝛗₁ and 𝛗₂ which are inverse of one another : for any point M on K(P), M₁ = 𝛗₁ (M) and M₂ = 𝛗₂ (M) are also two points on K(P).

M₁ is the G-Ceva conjugate of the P^3 isoconjugate of M and M₂ is the P^3 isoconjugate of the G-Ceva conjugate of M. With M = x : y : z, this gives :

M₁ = p^3 y z (-p^3 y z + q^3 x z + r^3 x y) : : and M₂ = p^3 y z (x-y+z) (x+y-z) : : .

In particular, K(P) passes through P₁ = 𝛗₁ (P) = p^2 (-p^2+q^2+r^2) : : and P₂ = 𝛗₂ (P) = p^2 (p+q-r) (p-q+r) : : , on a line passing through P.

• P₃ = 𝛗₁ (P₁) and P₄ = 𝛗₂ (P₂) are the respective tangentials of P₂ and P₁, also on a line passing through P.

• K(P) meets the line at infinity at the same points as pK(G/P, aP₂) where G/P is the G-Ceva conjugate of P and aP₂ the anticomplement of P₂.

These two cubics meet again at six finite points of a same conic C(P) namely A, B, C, P₄ and two points on the line GP₂.

L(P) and T(P) meet at Y.

X is h(P, 2/3)(Y), on the (orange) Hessian of K(P). Recall that X is the center of the (decomposed) polar conic of P.

K(P) with P on the line at infinity

When P is an infinite point, K(P) becomes a trident and its pseudo-pole P^3 lies on K656, the image of K015 = cK(#X2, X2) under h(G, 3/2).

K(P) has the line GP as an asymptote but it has now a parabolic asymptote (P) meeting GP at the image Z of P^3 under h(G, 1/3). It follows that Z lies on the image of K015 under h(G, 1/2). See green curve below.

Apart the points A, B, C, A', B', C', P mentioned in the general case, K(P) now also passes through P^4 and obviously its successive images under 𝛗₁ and 𝛗₂.

When P traverses the line at infinity, (P) envelopes a cubic which is the image under h(G, 1/2) of the Tucker cubic obtained when 𝝺 = – 4/11. See light blue curve.

(P) is tritangent to this cubic and the centroid GP of the triangle of contacts lies on GP and on the image of K015 under h(G, – 1/3).

***

See K1152 = K(X523) where further properties are given.

A generalization

If M is an infinite point, the cubic psK(Ω, P, M) is a trident if and only if :

• the pseudo-pole Ω lies on pK(M^4, M) passing through M, M^2, M^3,

• the pseudo-pivot P lies on pK(G, tM) passing through G, M and the isotomic conjugate tM of M.

This gives the cubics psK(M, tM, M) of CL063 and psK(M^3, G, M) of CL070.

psK(M^2, M, M) = pK(M^2, M) is not a proper trident since it is the union of three parallels passing through A, B, C.

Some examples

Peter Moses has computed a selection of cubics K(P) passing through at least three ETC centers. Those in the first table are tridents.

With P on the line at infinity L∞, K(P) contains the points P^4, 𝛗₁ (P^4) and 𝛗₂ (P^4).

𝛗₁ (P^4) is the G-Ceva conjugate of the isotomic conjugate tP of P. It is the center of the circum-conic with perspector tP and vice-versa. See red points.

𝛗₂ (P^4) is the perspector of the bicevian conic C(G, tP). See green points.

The only listed P^4 is X(36435) = X(30)^2. See blue point.

See K1152 = K(X523.

If Q is a finite point different from G, the locus of P such that K(P) passes through Q is a central cubic C(Q) with center Q where the inflexional tangent passes through G. With Q = p : q :r , the equation of C(Q) is :

p q r (y - z) (z - x) (x - y) + ∑q r (q - r) x^3 = 0.

For instance, C(X3) is K100 passing through X(i) for i in {1, 3, 40, 1670, 1671, 17749}.

C(Q) meets the line at infinity at the same points as pK(G/Q, Q) where G/Q is he G-Ceva conjugate of Q. The six other (finite) common points are Q (counted twice) and the four (real or not) square roots Ro, Ra, Rb, Rc of G/Q.

These latter points lie on the polar conic (D) of Q in pK(G/Q, Q)which is a diagonal conic in ABC. (D) also contains G/Q, the vertices of the anticevian triangles of Q and G/Q, the (real or not) square roots of every point on the line passing through Q and G/Q. The equation of (D) is :

∑q r (q - r) x^2 = 0.

C(Q) obviously passes through the reflections of Ro, Ra, Rb, Rc in Q.

C(Q) also passes through the six midpoints of every pair of points in {Ro, Ra, Rb, Rc}, the midpoint of RxRy being denoted by Rxy. These midpoints are two by two symmetric about Q and lie on the circum-conic of ABC with center Q and perspector G/Q.

***

The figure opposite is made with Q = X(1) hence G/Q = X(9) and pK(G/Q, Q) is K157.

Ro = X(188) and (D) is the diagonal rectangular hyperbola passing through X(i) for i = 1, 9, 40, 188, 191, 366, 1045, 1050, 1490, 2136, 2949, 2950, 2951, 3174, etc.

More generally, (D) is a diagonal rectangular hyperbola for every Q on the Thomson cubic K002 and then, the line Q, G/Q passes through the Lemoine point X(6), the isopivot of K002.

In this case, the circum-conic passing through Q and G/Q is also a rectangular hyperbola and the center of (D) lies on the circumcircle of ABC.

Note that the center of (D) lie on the circum-conic of ABC with center S (and perspector G/S) if and only if Q lies on pK(G/S, G). With S = X(3) hence G/S = X(6), we find the Thomson cubic as above.