Let W = (p : q : r) be a point. Kp(W) is the locus of pivots of all pK+ invariant under the isoconjugation with pole W. These cubics form the class CL014 of cubics. The locus of the common point Q of the three asymptotes is Kc(W), member of the class CL015.
Now, instead of fixing the pole W, we fix the pivot P = (u : v : w). The locus of poles of all pK+ with pivot P is Kw(P) and the locus of the common point Q of the three asymptotes is Kc'(P).
The cubics Kw(P) and Kc'(P) form the classes CL017 and CL018 respectively.
Kc'(P) is always a K+ i.e. its asymptotes are always concurrent at the point with coordinates :
(v + w)(v + w + 2u)[5u (u + v + w) + 3vw] : : .
The general equation of Kc'(P) is :
Kc'(P) contains the points Qi intersections of the asymptotes of the cubic Kw(Wi) of the class CL017.
Kc'(P) contains :
• A, B, C.
• Qa = 0 : 2v + w : v + 2w, homothetic of the cevian Pa of P under h(Ma,1/3) where Ma is the midpoint of BC. Qb, Qc similarly.
• G (centroid of ABC) = Q0.
• Q1 = v + w + 2u : : , homothetic of P under h(G,1/4) and complement of complement of P.
• Q3 = P (this is the case of the degenerate cubic of CL017).
• Q4 = u(v + w)(v + w + 2u) : : , the centroid of the cevian triangle of P.
• Q5 = (v^2 + w^2 - u^2 + vw) / (v^2 + w^2 - u^2 + vw + uv + uw) : : .
• Q6 = (v + w + 2u)(v^2 + w^2 - u^2 + vw) / (v^2 + w^2 - u^2) : : .
• Q7 = u / (u^2 + vw + 2uv + 2uw) : : .
• Q8 =
• Q9 = (v + w + 2u) / [u(v + w - u) + 2vw] : : .
• Q10 = u(v + w - u) / (v + w) : : .
• Q11 = u(v + w + 2u)(v^2 + w^2 + vw -uv - uw) / (v + w - u) : : .
1. This class contains only one equilateral cubic : Kc'(X4) = K412.
2. Kc'(G) decomposes into the medians of ABC.
3. Kc'(P) is a nK if and only if either :
– P lies at infinity but the cubic degenerates.
– P lies on the Steiner ellipse. The pole of the nK lies on the homothetic of the Steiner ellipse under h(G, 1/4) and on the cubic. The root is the infinite point of the conjugated diameter of the line tP-G in the Steiner ellipse.
For axample, Kc'(X99) = nK(X620, X524, X2).
4. For any point P on K080 = KO++, there is always an isogonal pK with the same asymptotic directions as Kc'(P).