Let (L) be a line and let A', B', C' be the reflections of A, B, C about (L). There is a unique circumconic (C) passing through A', B', C'.
Let T be the trilinear pole of (L) and let Z be the perspector of (C). The isoconjugation f that swaps T and Z transforms (L) into (C) and vice versa. Let Ω be the pole of f i.e. the barycentric product of T and Z. In the sequel, we shall write M* = f(M).
Let X be the infinite point of (L) and Y that of any perpendicular to (L). The reflection of Y* about (L) is P. Note that P is also the reflection of T about Y*.
Theorem : pK(Ω, P) = pK(L) is an axial pivotal cubic invariant in the symmetry with axis (L).
Note that pK(L) meets the line (L) at three real points A", B", C" which are the isoconjugates of A', B', C'.
pK(L) has always one real asymptote, the perpendicular to (L) at Z. The two other asymptotes are not necessarily real.
When (L) passes through O, the conic (C) is the circumcircle. In this case,
• Ω lies on the circumconic with perspector X(184) passing through X(112), X(248), X(906), X(1415), X(1576), X(2966),
• the real asymptote passes through K,
• the two other asymptotes are real if and only if ABC is obtusangle,
• the orthic line passes through X(184).
Two other axial pK(L) are presented in CL056 when (L) is the orthic axis and the antiorthic axis.