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.
 H* cubic centers on the curve W F X30 K059 X4, X13, X14, X30, X113, X1300 X1990 X125 X511 K337 X4, X114, X371, X372, X511, X2009, X2010, E487 X232 X115 X512 X4, X112, X115, X512 X2489 X114 X513 X4, X11, X108, X513 X119 X514 X4, X116, X514 X118 X515 X4, X117, X515 X124 X516 X4, X118, X516, X917 X1886 X116 X517 K334 X1, X4, X46, X80, X119, X517, X915, X1785, X1845 X11 X518 X4, X120, X518 X5089 X519 X4, X121, X519 X520 X4, X122, X520, X1301 X647 X133 X521 X4, X123, X521 X650 X522 X4, X124, X522 X3064 X117 X523 X4, X107, X125, X523 X2501 X113 X524 K209 X2, X4, X126, X193, X468, X524, X671, X2374 X468 X525 X4, X127, X525, X1289 X523 X132 X526 X4, X526, X1304 X690 X4, X690, X935 X698 X4, X76, X698, X1916 X740 X4, X10, X242, X740 X900 X4, X900, X1309 X924 X4, X110, X136, X924 X131 X926 X4, X926, X1566 X1154 K050 X4, X5, X15, X16, X52, X128, X186, X1154, X1263, X2383, X2902, X2903, X2914 X11062 X137 X1503 X4, X98, X132, X1503 X127 X1510 X4, X137, X933, X1510 X128 X2390 X4, X106, X2390 X2393 K475 X4, X6, X25, X67, X111, X858, X1560, X2393 E659 K339 X3, X4, X131, X155, X265, X403, X1299, X1986, E659 X3003 X136
 E659 is the intersection of the lines X3-X49, X4-X52, X5-X389, X51-X381, X55-X500, X56-X1069, X74-X323, 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).
 H* W Cubic or X(i) on the cubic M for OAPs Remarks X3 X6 K006 Orthocubic X3 isogonal pK X5562 X216 K044 Euler central cubic X185 the only central cubic, a pK+, the Darboux cubic of the orthic triangle X5 X53 K049 McCay orthic cubic X550 the only equilateral cubic, a pK+, the McCay cubic of the orthic triangle X69 X2 K170 X6776 isotomic pK X184 X32 K176 X2 X4 K181 X376 ? X20 K182 X6 X25 K233 X1350 X125 X115 K238 X20 X1249 K329 X4 a pK+ X51 X3199 K350 the Thomson cubic of the orthic triangle X52 X52 x X4 K415 the Orthocubic of the orthic triangle X143 X143 x X4 K416 the Napoleon cubic of the orthic triangle X25 X2207 K445 X1495 ? K496 X141 X427 K517 X1 X19 K691 X40 X1249 X6525 K709 X325 X297 K718 X40 X2331 K807 X1 X4 X393 union of the altitudes X20 a pK+ X76 X264 X(4), X(76) a pK+ X343 X5 X(2), X(4), X(5), X(52), X(193), X(343) ? X233 X(4), X(5), X(52), X(140) X1568 ? X(4), X(5), X(30), X(52), X(113), X(1568) X1213 X430 X(2), X(4), X(10), X(193), X(430), X(1213), X(1839), X(2901) X1350 ? X(4), X(371), X(372), X(1350) X6 X376 ? X(4), X(376) X2 X550 ? X(4), X(550) X5 X22 ? X(4), X(22) X378 X378 ? X(4), X(378) X22
 M-OAP 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 M-OAP points where M is the reflection of H* about the circumcenter O.
 Bi-isogonal 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 bi-isogonal pK with a second pivot Q. Q is the tangential of H in pK(W, H).