Given an isoconjugation with pole W = (p : q : r) swapping M and M*, the locus of M such that the trilinear polars of M and M* are parallel is the pivotal isocubic whose pivot is the pole W. It always contains the centroid G and the tangents at A, B, C are the medians. W and W/G (cevian quotient) are two other points of the cubic. See Special Isocubics ยง1.4.2. The points at infinity of pK(W,W) are those of pK(G,W') where W' is the isotomic conjugate of W. The isotomic transform of pK(W,W) is the cubic pK(W',G). See CL048. The general equation of these cubics is :
 The following table gives a selection of pK(W,W).
 W cubic or X(i) on the cubic remark X(1) K101 X(2) union of the medians X(4) K181 X(6) K102 isogonal pK X(67) K103 the only circular pK(W,W) X(69) X(2), X(20), X(69) isotomic transform of K663 X(75) X(2), X(8), X(75), X(85) isotomic transform of K363 X(76) K184 isotomic transform of K002 X(11058) K104 the only equilateral pK(W,W) X(34) X(1), X(2), X(34), X(87), X(278), X(1722) X(56) X(1), X(2), X(56), X(57), X(87), X(509), X(978), X(1423) X(264) X(2), X(4), X(76), X(264), X(491), X(492) isotomic transform of K168 X(274) X(2), X(75), X(85), X(86), X(274), X(333), X(348) isotomic transform of K345 X(276) X(2), X(76), X(95), X(264), X(275), X(276) isotomic transform of K612 X(286) X(2), X(27), X(29), X(75), X(85), X(286) X(290) X(2), X(76), X(98), X(264), X(287), X(290), X(511) isotomic transform of K357, see CL032 X(292) K136 X(300) K342a isotomic transform of K341a X(301) K342b isotomic transform of K341b X(308) X(2), X(76), X(83), X(264), X(308), X(1799) isotomic transform of K836 X(313) X(2), X(10), X(76), X(264), X(306), X(313), X(1330) X(327) X(2), X(76), X(262), X(264), X(327), X(1352) X(331) X(2), X(75), X(85), X(92), X(273), X(331) X(334) K868 isotomic transform of K251 X(349) X(2), X(76), X(226), X(264), X(307), X(349) X(393) X(2), X(4), X(393), X(459), X(1123), X(1336) X(870) X(1), X(2), X(75), X(85), X(87), X(870) X(1220) X(1), X(2), X(10), X(87), X(894), X(1220) X(1221) X(2), X(37), X(75), X(85), X(1221), X(1909) X(1434) X(2), X(7), X(57), X(86), X(274), X(1434) X(1494) X(2), X(30), X(298), X(299), X(1494) isotomic transform of K472 X(1502) X(2), X(76), X(264), X(305), X(315), X(1502) isotomic transform of K177 X(1989) K278 X(2052) X(2), X(92), X(1585), X(1586), X(2052) isotomic transform of K857 X(2207) X(2), X(6), X(19), X(393), X(1611), X(2207) X(4590) X(2), X(99), X(4590), X(6189), X(6190) isotomic transform of K237 tX(44) X(2), X(75), X(85), X(320), X(903) isotomic transform of K453 tX(53) X(2), X(69), X(95), X(276), X(394) isotomic transform of K671 tX(187) X(2), X(76), X(264), X(316), X(671) isotomic transform of K043 tX(574) X(2), X(76), X(264), X(598) isotomic transform of K284 tX(1100) X(2), X(75), X(85), X(319), X(321), X(1268) isotomic transform of K637 tX(1691) X(2), X(76), X(141), X(264), X(325), X(334), X(1916) isotomic transform of K252 tX(2092) X(2), X(76), X(86), X(264), X(274), X(314), X(1240) isotomic transform of K253 tX(5019) X(2), X(76), X(264) isotomic transform of K321 tcX(11057) X(2) isotomic transform of K485
 tcX(11057) : SEARCH = 9.67361214847370. tX(i) is the isotomic conjugate of X(i). Remark : Any pK(W,W) which passes through a point P also contains Q, the P-isoconjugate of the anticomplement of the isotomic conjugate of P. For example, if P = X(1) then Q = X(87). Generally, if P = u : v : w , then Q = u / (uv + uw - vw) : : . Furthermore, pK(W,W) passes through P = u : v : w if and only if W lies on the circum-conic C(P) passing through P and its barycentric square P^2. This is the circum-conic with perspector S = u^2(v - w) : : , the intersection of the trilinear polar of P and the polar of P in the Steiner ellipse. C(P) is the isotomic transform of the line G-tP.