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, X3563 X232 X115 X512 X4, X112, X115, X512 X2489 X114 X513 X4, X11, X108, X513 X6591 X119 X514 X4, X116, X514, X26705 X7649 X118 X515 X4, X117, X515, X32706 X8755 X124 X516 X4, X118, X516, X917 X1886 X116 X517 K334 X1, X4, X46, X80, X119, X517, X915, X1785, X1845 X14571 X11 X518 X4, X120, X518, X15344 X5089 X5511 X519 X4, X121, X519, X40101 X8756 X5510 X520 X4, X122, X520, X1301 X647 X133 X521 X4, X123, X521, X40097 X650 X25640 X522 X4, X124, X522, X26704 X3064 X117 X523 X4, X107, X125, X523 X2501 X113 X524 K209 X2, X4, X126, X193, X468, X524, X671, X2374 X468 X5512 X525 X4, X127, X525, X1289 X523 X132 X526 X4, X526, X1304, X3258 X47230 X25641 X690 X4, X690, X935, X5099 X14273 X16188 X698 X4, X76, X698, X1916 ? ? X740 X4, X10, X242, X740 ? X44950 X900 X4, X900, X1309, X3259 ? X31841 X924 X4, X110, X136, X924 X6753 X131 X926 X4, X926, X1566, X40116 ? X33331 X1154 K050 X4, X5, X15, X16, X52, X128, X186, X1154, X1263, X2383, X2902, X2903, X2914, the Neuberg cubic of the orthic triangle X11062 X137 X1503 X4, X98, X132, X1503 X16318 X127 X1510 X4, X137, X933, X1510 ? X128 X2390 X4, X106, X2390, X20619 ? ? X2393 K475 X4, X6, X25, X67, X111, X858, X1560, X2393 X14580 X14672 X13754 K339 X3, X4, X131, X155, X265, X403, X1299, X1986, X13754 X3003 X136
 Remark : the centers Oa, Ob, Oc of the osculating circles at A, B, C to any pK(W, H) above lie on three perpendiculars to the real asymptote. In other words, the triangles ABC and OaObOc are perspective with perspector at infinity. OaObOc is also perspective to HaHbHc with perspector on a complicated sextic.
 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. The isogonal transform of pK(W, H) is a pK with isopivot O, a member of CL074. • K is a pK+ i.e. has concurring asymptotes if and only if W lies on K208. In this case, H* (blue cells in the table) lies on K071 and the asymptotes concur on K412. • When the pole W (green cells 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). This Brocard axis passes through X(i) for i = 5, 53, 216, 233, 3199, 6116, 6117, 6750, 8887, 10003, 10600, 11062, 14576, 14577, 14640, 27358, 27359, 27371, 32428, 35719, 36412, 39530, 39569. • 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 cells 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 X37669 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 X14576 K415 the Orthocubic of the orthic triangle X143 X14577 K416 the Napoleon cubic of the orthic triangle X25 X2207 K445 X1495 X14581 K496 X141 X427 K517 X1 X19 K691 X40 X1249 X6525 K709 X325 X297 K718 X40 X2331 K807 X1 X343 X5 K1087 X27374 ? K1088 X1511 X39176 K1157 X6467 X1196 K1165 X942 X1841 K1184 X1936 X2202 K1185 X55 X607 K1186 X3690 X1500 K1187 X1147 X571 K1318 X4 X393 union of the altitudes X20 a pK+ X76 X264 X(4), X(76) a pK+ ? X233 X(4), X(5), X(52), X(140) X1568 X52945 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 X45141 X(4), X(371), X(372), X(1350) X6 X376 X40138 X(4), X(376) X2 X550 ? X(4), X(550) X5 X22 X8743 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. *** Additional cubics by Peter Moses
 Ω X(i) on pK(Ω, H) for i = 3 2, 3, 4, 155, 193, 394, 5374, 6391, 6504 39 3, 4, 69, 155, 427, 1843, 3917, 19583 235 2, 4, 185, 193, 235, 2052, 9307, 13567 325 4, 76, 114, 487, 488, 511, 6393, 8781 343 4, 264, 487, 488, 5562, 8905, 11090, 11091 430 2, 4, 10, 193, 430, 1213, 1839, 2901 1184 4, 6, 25, 487, 488, 5286, 7386, 19459, 34817 1495 4, 371, 372, 3284, 6110, 6111, 36296, 36297 1824 4, 10, 19, 33, 37, 209, 226, 2901 3172 4, 6, 20, 25, 154, 1249, 3532, 32605 4185 1, 2, 4, 46, 193, 940, 4185, 5307 6620 2, 4, 193, 263, 393, 3424, 6525, 6620, 6776, 7735, 9292, 9752 7473 2, 4, 193, 542, 648, 7473, 14999, 16188 8735 4, 11, 513, 2588, 2589, 3307, 3308, 34434 8739 4, 13, 15, 18, 371, 372, 2902, 10633, 15441, 15445 8740 4, 14, 16, 17, 371, 372, 2903, 10632, 15442, 15444 8754 4, 115, 512, 2039, 2040, 3413, 3414, 14618, 27375 14165 4, 340, 470, 471, 2992, 2993, 38427, 38428 18487 4, 30, 113, 265, 381, 1531, 4846, 40909 34854 4, 6, 25, 232, 297, 1987, 6530, 16081, 17980, 39931 36417 4, 6, 25, 1974, 2207, 13854, 17409, 34207 39008 4, 30, 113, 122, 520, 1650, 9033, 16177, 38956 40938 4, 22, 315, 371, 372, 427, 1843, 3313
 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. The isopivot is the H-Ceva conjugate of H*. It is the tangential of H* in pK(W, H). 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.
 Points of pK(W, H) on (L∞) and (O) Every cubic pK(W, H) meets the line at infinity (L∞) and the circumcircle (O) at the same points as two isogonal pKs with respective pivots P1, P2. Obviously, when W = X(6), P1 = P2 = H corresponding to the Orthocubic K006. When W ≠ X(6), P1 = a(Ω ÷ H) and P2 = at(cevapoint(Ω ÷ X6, H)). See CL073 for a generalization. pK(W, H) and pK(X6, P1) meet at three points on (L∞) and six points on a circum-conic passing through X(2). pK(W, H) and pK(X6, P2) meet at six points on (O) and three points on a line passing through X(6), the isogonal transform of the circum-conic above. Let Q be the midpoint of P1, P2. Q is the X(25)-isoconjugate of the Ceva-point of X(6) and W. The cubic spK(P1, Q) meets (L∞) and (O) at the same points as pK(X6, P1) and pK(X6, P2) respectively, hence pK(W, H) and spK(P1, Q) share the same points on (L∞) and (O). spK(P1, Q) is a pK if and only if W lies on the bicevian conic C(X4, X112), then P1 lies on the Jerabek hyperbola, P2 lies on the bicevian conic C(X4, X648), and X(6), P1, P2 are collinear. In such case, this pK is an isogonal cubic with respect to the orthic triangle and its pivot is H*. Example : K1184 = pK(X1841, X4) = spK(X72, X44547) and P2 = X(14054), H* = X(942). With W = X6, X25, X53, X427, X571, we obtain K006, K233, K049, K517, K1318 respectively.