Consider a circum-conic (C) with center T and perspector S = G/T (Ceva conjugate). A variable line (L) passing through T meets (C) at two points M, N. Let C(M, N) be the bicevian conic passing through the vertices of the cevian triangles of M and N. (C) and (L) meet at two points P, Q and, when (L) rotates around T, the locus of P, Q is a cubic K(T). K(T) is a pivotal cubic with pivot T and pole the barycentric square Ω of G/T. Hence, K(T) = K(G/S) = pK(S^2, G/S) = pK((G/T)^2, T). The isoconjugation with pole S transforms K(T) into the isotomic pivotal cubic pK(G, aS) with pivot aS, the anticomplement of S. The most remarkable example is K172 = pK(X32, X3) since (C) is the circumcircle of ABC. The corresponding isotomic cubic is the Lucas cubic K007. Here is a selection of some cubics K(T) and K'(T) = pK(G, aS) with a list of centers X(i) when the cubic is not listed in CTC (updated by Peter Moses, 2021-12-21). Remark : when T = G, the cubics K(T) and K'(T) are the union of the medians of ABC.
 T K(T) = pK(Ω, T) K'(T) = pK(G, aS) 1 1, 9, 200, 3158, 3680 K1078 3 K172 K007 5 5, 51, 216, 418, 5562 K045 6 K1047 K170 9 K351 K200 10 10, 37, 42, 65, 71, 210, 227, 1826 K034 37 10, 37, 321, 3971, 22028, 42027 1, 2, 75, 192, 330, 3223, 17149 39 39, 141, 8024, 42551 K659 113 113, 3003, 13754, 44084 K279 115 2, 115, 523, 9293, 10278, 14086 2, 99, 148, 523, 35511 141 39, 141, 1843, 3051, 3917, 23208 K141 206 32, 184, 206, 1974, 22262, 40146 2, 4, 66, 69, 315, 1370, 5596, 13575 214 1, 44, 214, 678, 1319, 3689, 40172 K311 216 5, 216, 324, 42453 K146 523 115, 523, 8029, 13636, 13722 K242 618 396, 618, 8014, 15768, 34296 K264a 619 395, 619, 8015, 15769, 34295 K264b 690 690, 1648, 1649, 14443, 14444, 23992, 33915, 33919, 46049 K240 891 891, 1646, 14434, 14441, 33917, 39011 2, 513, 536, 668, 889, 891, 3227, 41314, 43928 900 900, 1647, 6544, 6550, 14442, 33922, 35092, 46050 2, 190, 514, 519, 900, 903, 4555, 6548, 17780 942 942, 2294, 18591, 23207, 40952, 41393 K610 946 946, 2262, 40943 K133 960 960, 2092, 2292, 2300, 3725, 21810, 22076, 42550, 44092 K254 1015 2, 513, 1015, 9267, 14079, 38238 2, 513, 668, 9263, 9295 1086 2, 514, 1086, 14078, 21204, 42555 2, 190, 514, 4440, 6630 1511 3, 1495, 1511, 3284 K611 2883 185, 800, 2883, 44079 K235 3163 2, 30, 3163, 34582 K860 3647 1100, 1962, 3647, 3683, 32636 K455 6184 2, 241, 518, 3693, 6184, 8299, 22116 2, 7, 8, 239, 335, 518, 2481, 10025, 17794, 39350 6260 1108, 1864, 6260, 37566 K154 6593 6, 187, 3292, 6593, 10417, 39689, 44102 K008 11672 2, 232, 511, 11672, 36212, 36213, 40804, 40810 K355