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Let ABC be the reference triangle and XYZ another triangle. For any point M, denote by Ma, Mb, Mc the intersections of the lines MX, MY, MZ with the sidelines BC, CA, AB respectively. These three points Ma, Mb, Mc are collinear if and only if M lies on a cubic passing through A, B, C, X, Y, Z and many (easy to draw) other points such as :

U = BC /\ YZ, V = CA /\ ZX, W = AB /\ XY,

U' = BZ /\ CY, V' = CX /\ AZ, W' = AY /\ BX,

U1 = VW /\ V'W', V1 = WU /\ W'U', W1 = UV /\ U'V', etc.

This type of cubic is called a (general) Grassmann cubic (see for example : Brocard H. and Lemoyne T. : Courbes Géométriques Remarquables. Librairie Albert Blanchard, Paris, third edition, 1967, tome 3, p.69, §52).

Now, the locus of M such that the triangles ABC and MaMbMc are perpective is another cubic also passing through A, B, C, X, Y, Z.

***

From now on, we will consider a more particular configuration which gives interesting nK0 and pK and generalizes the two Brocard cubics K017, K020 and the related cubic K322.

Let Q = p : q : r be a triangle center and tQ its isotomic conjugate (Q is not the centroid G or one of its harmonic associates). Consider the related points X = p : r : q, Y = r : q : p, Z = q : p : r. X is the intersection of the parallel at Q to BC with the line tQA, Y and Z similarly (see "Tucker Cubics" in the downloads page). The triangles ABC and XYZ are triply perspective.

Let X' = BZ /\ CY = qr : q^2 : r^2, Y' and Z' similarly. The perspector of the triangles ABC and X'Y'Z' is Q^2, the barycentric square of Q.

We denote by :

• GnK(Q) the locus of M such that Ma, Mb, Mc are collinear,
• GpK(Q) the locus of M such that ABC and MaMbMc are perspective,
• GpK'(Q) the locus of the perspector.

GnK(Q) = nK0(Q, R) where the root R = p^2 - qr : q^2 - rp : r^2 - pq. R is the Hirst transform of Q with respect to the pole G and the Steiner ellipse i.e. the intersection of the line GQ with the polar line of Q in this ellipse.

GpK(Q) = pK(Q, P) where the pivot P = p^2 + qr : q^2 + rp : r^2 + pq. P is the perspector of the triangles XYZ and X'Y'Z'. It is the harmonic conjugate of R with respect to Q^2 and tQ and also the intersection of the lines G-G/Q, Q-Q/G (cevian quotients).

GpK'(Q) = pK(tR, R*) where tR is the isotomic conjugate of R and R* is the Q-isoconjugate of R.

Their equations are : ∑ (p^2 - q r) x (r y^2 + q z^2) = 0; ∑ (p^2 + q r) x (r y^2 - q z^2) = 0; ∑ (p^2 - q r) x^2 (q y - r z) = 0 respectively.

Points on these cubics

GnK(Q) contains the following points :

• A, B, C, X, Y, Z, X', Y', Z', G, Q,
• U, V, W feet of the trilinear polar of R,
• the infinite point of the trilinear polar of Q and the infinite points of the Steiner ellipse (hence, GnK(Q) has always only one real asymptote),
• S the isotomic conjugate of the infinite point of the trilinear polar of tQ (this is the sixth intersection of the cubic with the Steiner ellipse),
• Ga = QU /\ GA, Gb = QV /\ GB, Gc = QW /\ GC (third points on the medians),
• G'a = QA /\ GU, G'b = QB /\ GV, G'c = QC /\ GW (third points on the cevians of Q),
• QX /\ GX', GX /\ QX', etc.

GpK(Q) contains the following points :

• A, B, C, X, Y, Z, X', Y', Z', P, P* (Q-isoconjugate of P),
• Pa, Pb, Pc vertices of the cevian triangle of P,
• tQ, Q^2 (tQ, Q^2 and P are obviously collinear on the cubic).

GpK'(Q) contains the following points :

• A, B, C, G, P, R, R*, tR, tQ, cQ (complement of Q), (tR)^2
• the infinite point of the line tQ-cQ i.e. q^2(r-p)-r^2(p-q): ... : ...,
• the infinite points of the Steiner ellipse (hence, GpK'(Q) has always only one real asymptote),
• S' the isotomic conjugate of (q+r)(p^2-qr) : ... : ... or tR-isoconjugate of cQ. S is the sixth intersection of the cubic with the Steiner ellipse,
• T the isotomic conjugate of the anticomplement of the isotomic conjugate of Q,
• the barycentric product of tP and tR.

Remarks :

1. GnK(Q) degenerates when Q lies at infinity or on the Steiner ellipse.
2. When Q lies at infinity, GpK(Q) is a central pK++ with center Q^2 ( a point on the Steiner in-ellipse).
3. When Q lies on the Steiner ellipse, GpK(Q) is a pK+ with asymptotes concurring at G. Wilson Stothers observes that, in this case, tQ is a flex (at infinity) on the cubic which has an oblique symmetry axis T(Q) (trilinear polar of Q), direction tQ. The symmetry becomes an orthogonal symmetry if and only if Q is one of the two intersections Q1, Q2 of the Steiner ellipse with the line GK. The two corresponding cubics GpK(Q1), GpK(Q2) are such that the real asymptote of one of them is the axis of symmetry of the other. These two lines are the axes of the Steiner ellipse. The corresponding pivots P1, P2 lie on the parallel to GK at X(2076), the inversive image of the Brocard midpoint X(39) in the circumcircle. These two cubics generate a pencil containing another pK which is K1 = pK(X99<->X148, X99) and K2, a nK0 whose pole and root are both the barycentric square of X(99).

Additional properties of the cubic GnK(Q)

GnK(Q) has always three real prehessians P1, P2, P3. The polar conics of the infinite point of the trilinear polar of Q with respect to these prehessians are decomposed into three pairs of secant lines intersecting at Q, S, G respectively.

These conics meet at four points lying on GnK(Q) and with tangents parallel to the real asymptote. QSG is the diagonal triangle of the quadrilateral formed with these four points.

The polar conic of this same infinite point with respect to GnK(Q) is the hyperbola (C) passing through this infinite point, Q, S, G and E which is the intersection of GnK(Q) with its real asymptote. E is the common tangential of the former four points.

It follows that GnK(Q) is a pivotal cubic with respect to QSG with pivot this infinite point and isopivot E.

Selected examples of cubics GnK(Q), GpK(Q), GpK'(Q)

The first table gives a selection of the three cubics for a given Q and the second table gives a selection of cubics GpK'(Q) according to their pivot.

Remarks :

1. the G-Hirst transform of pK(X, Y) is pK(Y, X) where X and Y are G-Hirst conjugate points.

2. in particular, when the (green) pivot lies on the Steiner ellipse, the cubic is invariant by G-Hirst conjugation. The cubic is a pK(X, X), a member of CL007.

3. when the pivot lies on the line at infinity, the pole is G. See CL048.

4. in general, there are six related cubics since by applying alternatively isotomic conjugation (denoted t) and G-Hirst conjugation (denoted h) we obtain a "ring" of six related centers according to the following hexagram. The six related cubics are those represented in the lines with same color in the table. When the pivot lies on the Steiner ellipse, there are only three such cubics.

Note that these six centers lie on a same isotomic non-pivotal cubic nK(X2, S, P) with root S on the Steiner ellipse. More details below.

Recall that these six cubics already pass through six known points namely A, B, C, G and the (imaginary) infinite points S1, S2 of the Steiner ellipses.

Three cubics with pivots the three vertices of one of the two dashed triangles belong to the same pencil of cubics passing through these same vertices.

Two cubics with pivots on a t-side contain the infinite point of this side and the two consecutive vertices of the hexagram.

Two cubics with pivots on a h-side contain the extremities of this side and the isotomic conjugate of the infinite point of the opposite t-side. Each cubic is the G-Hirst inverse of the other.

Two cubics with pivots two opposite vertices of the hexagram contain the two vertices of the parallel t-side and have the same tangent at G. Each cubic is the isotomic transform of the other. These twoopposite vertices are said to be antitomic conjugates. See ETC, preamble just before X(14941).

The corresponding equations of these six cubics are very similar. They are given in the following table. Each cubic passes through A, B, C, G, S1, S2 and four other points as in the last column. These six cubics are anharmonically equivalent.

Remark 1 : the cubics above with pivots P' and hP are called cevian/anticevian-associated cubics in ETC. See preambles just above X(40842) and X(40853) where some other examples are given.

Remark 2 : there are two other related pivotal cubics which are also equivalent to the previous six cubics. The pivots are the points rP = p^2 + qr : : and sP = qr(p^2 + qr) : : . Each cubic is the isotomic transform of the other.

With P = X(6), these cubics are K020 and its isotomic transform K743 whose isogonal transform is K1008 = pK(X1501, X1915) passing through X(2), X(25), X(31), X(184), X(251), X(1501), X(1613), X(1915), X(3051), etc.

With P = X(1), they are K132 and K744 whose isogonal transform is K1006 = pK(X560, X172) passing through X(1), X(32), X(41), X(56), X(58), X(172), X(213), X(904), X(2176), etc.

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The isogonal transforms of the six former cubics (but also the two latter) are also equivalent to the previous ones. This gives a batch of 12 cubics all related between themselves through isogonal, isotomic, G-Hirst conjugations or a product of these such as e = gtg which is X(32)-isoconjugation..

With P = X(1), this is the (light yellow) batch of weak cubics detailled in the page K323.

With P = X(4), this is the (pink) batch of strong cubics detailled in the page K718.

The diagram below shows the 11 (since K128 is self-isogonal) cubics obtained with P = K = X(6), the Lemoine point. These are the cubics associated with red pivots in the table above and an (orange) batch of strong cubics again.

K787

K739

K128

K788

K322

K738

K789

K354

K356

K252

K532

Cubics GpK'(Q) passing through a given point P

In the second table above, we notice the frequent appearance of some usual centers (apart G) on the cubics GpK'(Q).

For a given P ≠ G with isotomic conjugate tP and G-Hirst inverse hP, the cubic GpK'(Q) contains P if and only if Q lies on the cubic KQ(P) = pK(htP, tP) = GpK'(thP).

In this case :

• the pole of GpK'(Q) lies on Kpole(P) = pK(P, hP) = GpK'(htP),

• the pivot of GpK'(Q) lies on Kpivot(P) = pK(hP, P) = GpK'(hthP)

• the isopivot of GpK'(Q) lies on Kisopivot(P) = pK(thtP, thP) = GpK'(tP).

Note that Kisopivot(P) is the isotomic transform of KQ(P) and that Kpivot(P) is the G-Hirst inverse of Kpole(P).

We recognize the six related points we have met in the hexagonal diagram above namely P, tP, hP, htP, thP and thtP = hthP.

For instance, GpK'(Q) contains the incenter X(1) if and only if Q lies on pK(X350, X75) passing through X(2), X(8), X(75), X(239), X(256), X(274), X(291), X(350), X(740), X(1281), X(2481). In this case,

• its pole is on pK(X1, X239) = K323 passing through X(1), X(2), X(6), X(75), X(239), X(291), X(366), X(518), X(673), X(1575), X(2319), X(2669), X(3212), X(3226),

• its pivot is on pK(X239, X1) passing through X(1), X(2), X(86), X(192), X(239), X(257), X(335), X(350), X(385), X(740), X(3226),

• its isopivot is on pK(X291, X335) passing through X(1), X(2), X(7), X(37), X(291), X(335), X(350), X(518), X(694), X(1909).

Another remarkable property of GpK'(Q)

In a previous section, we have seen that GpK'(Q) = pK(tR, R*) contains G, R, tR, (tR)^2 and also √(tR), in other words five powers of its pole.

More generally, the pivotal cubic pK(Ω, P) with pole Ω ≠ G, pivot P,

– contains its pole if and only if Ω, G, P are collinear and, in this case, it must contain G.

– contains the barycentric square Ω^2 of its pole if and only if Ω^2, tΩ = Ω^(-1), P are collinear and, in this case, it must contain tΩ.

The two lines Ω, G and Ω^2, tΩ intersect at the G-Hirst inverse of Ω which turns out to be the pivot of the pivotal cubic which contains G, Ω, Ω^2, Ω^(-1) and Ω^(1/2).

Hence, any pK(Ω, G-Hirst Ω) contains the five mentioned points and it is the only pK with this particularity. Furthermore, when Ω lies on K327, the points Ω^(-3) and Ω^4 also lie on the cubic. Unfortunately, K327 does not contain any known center.

This is easily adapted for any two distinct numbers m, n such that m + n ≠ 1 : any pK(Ω, P) with P = Ω^nΩ^(1-n) /\ Ω^mΩ^(1-m) contains Ω^n, Ω^(1-n), Ω^m, Ω^(1-m) and Ω^(1/2).

Note that GpK'(Q) is obtained with m = 2 and n = 1. See K623 where Ω = X(7).

The cubic nK(X2, S, P)

Recall that this cubic contains the pivots of six related pKs namely the points P, tP, hP, htP, thP and thtP = hthP mentioned above.

Now, let us denote by fP or f(P) the intersection of the lines P-hP and tP-thP, a point with first coordinate : q^2 + r^2 + q r. Clearly, f(P) = f(hP).

This transformation gives six other points on nK(X2, S, P) namely fP, ftP, fthP and their isotomic conjugates tfP, tftP, tfthP. Each of these new points is the intersection of two lines passing through two pairs of points of the first set of points.

Note that the third points of the cubic on the lines P-tP, hP-thP, htP-thtP (with rather complicated coordinates) are not included in the list of twelve points above.

***

The root S of the cubic nK(X2, S, P) lies on the Steiner ellipse. Its isotomic conjugate (on the line at infinity) has first coordinate : p(q - r)(p^2 - qr)(q^2 + r^2 + qr).

S is the trilinear pole of the line passing through X(2) and the barycentric product P x thtP, this latter point being also the isoconjugate of hP in the isoconjugation with fixed point P.

Example 1 : with P = X(1), the cubic passes through X(1), X(75), X(239), X(291), X(335), X(350), X(2276), X(3661), tX(2276), tX(3661). Its root S is the trilinear pole of the line X(2)X(292), SEARCH = 0.588293281202481.

Example 2 : with P = X(6), the cubic passes through X(6), X(76), X(385), X(694), X(1916), X(3117), X(3314), X(3407), tX(3407). Its root S is the trilinear pole of the line X(2)X(3114), SEARCH = 0.753832565170757.

Other examples computed by Peter Moses are given in the table below.