The vertex conjugation is defined in ETC just below X(3414) as follows. Let U and X be two points with circumcevian triangles AuBuCu and AxBxCx respectively. The lines AuAx, BuBx, CuCx bound a triangle which is in perspective with ABC. The perspector is called the vertex conjugate of U and X which is here denoted by U&X. Obviously U&X = X&U.
These two points U and X are not necessarily distinct. It is easy to see that U&U = gagU, the isogonal of the anticomplement of the isogonal of U.
When X lies on the circumcircle (O) of ABC, U&X = X.
Now, if U is a given point not lying on (O), we consider the mapping f : X -> U&X. f is a fourth degree transformation with respect to the coordinates of X.
It has six singular points namely A, B, C and three other points Qa, Qb, Qc which are the common points of the trilinear polar of the cevapoint of U and the Lemoine point K with the cevian lines of F = gcgU, the isogonal of the complement of the isogonal of U.
F is the only fixed point of this transformation and is called the first Saragossa point of U in ETC.
U&X coincide with the isogonal conjugate of X if and only if X is a focus of the inscribed conic with center cgU, the complement of the isogonal of U i.e. the isogonal conjugate of the fixed point F.
This shows that, in general, f transforms a circumcubic (K) into a sextic unless (K) contains Qa, Qb, Qc. In this latter case, f transforms (K) into another circumcubic through Qa, Qb, Qc. These two cubics coincide if and only if (K) contains F and, in this case, one can verify that (K) is the isogonal transform of a central cubic (K') with center N = cgU, the isogonal conjugate gF of F. This gives the following
Theorem : A circumcubic (K) is invariant under f if and only if its isogonal transform (K') is a central cubic with center N = gF = cgU passing through the isogonal conjugates of Qa, Qb, Qc which are the reflections of A, B, C in N.
For a given U, all these central cubics (K') contain seven fixed points and therefore form a net of cubics. Consequently, the cubics (K) also form a net of cubics which is generated by the three independent cubics (Ka), (Kb), (Kc) defined as follows. (Ka) is the union of the line AF and the circumconic passing through Qb and Qc, (Kb) and (Kc) in the same way.
With U = u:v:w, the barycentric equation of AF is : c^2 v (c^2 u + a^2 w) y – b^2 (b^2 u + a^2 v) w z = 0 and that of the conic is : u (c^2 v + b^2 w) x (c^2 y + b^2 z) – a^4 v w y z = 0.
Note that the net of cubics (K') is generated by the isogonal transforms of (Ka), (Kb), (Kc) which are each the union of three very simple lines. The isogonal transform (Ka') of (Ka) is the union of AN, BC and the reflection of BC in N. With N = p:q:r, this gives the equation : x (p x - q x - r x + 2 p y + 2 p z) (r y - q z) = 0 for (Ka').
In his ETC, Clark Kimberling observes that the Darboux cubic K004 is invariant under O – vertex conjugation. Here we have N = O hence U = O as well which gives a very special example of such cubic since (K) and (K') coincide. Recall that K004 is the only central isogonal pK.
More generally, for a given pole W which is not G, there is one and only one central pK with center N = G/W (the center of the circumconic with perspector W), with pivot P = aaN (the anticomplement of the anticomplement of N). See central cubics. The isogonal transform of this pK is another pK invariant under U – vertex conjugation with U = gtaW. Its pole W' is the X(32)–isoconjugate of W hence K004 is the only pK such that (K) and (K') coincide. Its pivot Q is the barycentric product of P and the isogonal conjugate of W.
In other words, for any point U, there is one and only one pK invariant under U – vertex conjugation. In particular, when U lies on the Euler line, W lies on the Brocard axis.
If we consider two isoconjugate points M1, M2 on the central pK (hence collinear with P) and their isogonal conjugates N1, N2 on (K'), we have the following properties :
We now consider the isogonal central nKs. The center N lies on (O). The root R is the complement of the isotomic conjugate of the trilinear pole of NN*. All these cubics form a family of cubics invariant under gaN – vertex conjugation. Here gaN denotes the isogonal of the anticomplement of N.
For example, K084 has center X(99) and is invariant under U – vertex conjugation with U isogonal conjugate of X(148).
The Darboux cubic K004 is not the only isocubic invariant under O – vertex conjugation but it is the only pK. There is a family of nKs with this same property. These are the isogonal transforms of nK(Ω, R, H) with pole Ω on the circum-conic with perspector X(25) and with root R on the Simson cubic K010, for example, nK(X112, X2407, H), nK(X648, X2396, H), nK(X1783, X2397, H).