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Circular pK(W, H) Let P be the pivot of a pK. When P is different of H, there is one and only one circular such pK but when P = H, there are infinitely many such cubics. These cubics have their singular focus F on the nine point circle and their pole W on the orthic axis. The isoconjugate H* of H is the point at infinity of the cubic. pK(W, H) is also the locus of M whose orthotransversal has a fixed direction, that of any perpendicular to the direction given by H*. The intersection with the real asymptote is X, the antipode of F on the nine point circle and this asymptote envelopes the Steiner deltoid H3. The "last" point on the circumcircle is E orthoassociate of X. These cubics are isogonal pivotal circular cubics with respect to the orthic triangle. They are also invariant under orthoassociation i.e. inversion in the polar circle and under three other inversions with poles A, B, C swapping H and the feet Ha, Hb, Hc of the relative altitude. They are also antigonal cubics i.e. cubics invariant under antigonal conjugation. They are the isogonal transforms of the inversible cubics. All circular pK(W, H) form a pencil of circular cubics which contains three focal cubics : the pole must be a common point Wa, Wb, Wc of the orthic axis and a symmedian of ABC. For example pK(Wa, H) has singular focus Ha, meets its real asymptote at Xa antipode of Ha on the nine point circle. This asymptote is parallel to OA. See also the related quartics at Q107. The following table gives a selection of circular pK(W, H) with corresponding X and E. 



E659 is the intersection of the lines X3X49, X4X52, X5X389, X51X381, X55X500, X56X1069, X74X323, etc. SEARCH = 1.02294526418116 



Other remarkable pK(W, H) Let K = pK(W, H) be a pivotal cubic with isopivot H* not lying on the line at infinity. Recall that K is an isogonal pivotal cubic with pivot H* with respect to the orthic triangle. • K is a pK+ i.e. has concurring asymptotes if and only if W lies on K208. In this case, H* (yellow point in the table) lies on K071 and the asymptotes concur on K412. • When the pole W (green point in the table) lies on the Brocard axis of the orthic triangle T, H* lies on the Euler line of T and the cubic is a member of the Euler pencil of T. It contains X(5) and X(52). • pK(W, H) contains its pole W if and only if W lies on the Euler line and then H* lies on the line GK (pink points in the table). These cubics belong to a same pencil and always contain G and X(193). 



MOAP points These points are studied in Table 53. We simply recall that any pK with pivot H, isopivot H* (not lying on the line at infinity) contains the four MOAP points where M is the reflection of H* about the circumcenter O. 



Biisogonal pK(W, H) Recall that K= pK(W, H) is an isogonal pivotal cubic with pivot H* = W÷ H with respect to the orthic triangle. When W lies on K627 = pK(X3199, X393), this cubic is also an isogonal pK with respect to another triangle T and then the cubic is said to be a biisogonal pK with a second pivot Q. Q is the tangential of H in pK(W, H). 
