Any isotomic cubic is invariant under isotomic conjugation. If K is a non-isotomic circum-cubic, its isotomic transform is another circum-cubic K'. When K = pK(X6,P), we have K' = pK(X76,Q) where Q is the isotomic conjugate of the isogonal conjugate of P.
 P K Q K' passes through X(i) for i = X(30) K001 Neuberg X(3260) K276 X(2) K002 Thomson X(76) K184 X(3) K003 McCay X(69) 69, 75, 264 X(20) K004 Darboux X(14615) K183 X(5) K005 Napoleon X(311) 69, 75, 95, 264, 302, 303, 311 X(4) K006 Orthocubic X(264) 69, 75, 264, 491, 492 X(384) K020 Brocard 4th cubic ? K743 X(512) K021 Brocard 5th cubic X(523) 75, 523, 670 X(99) K035 Steiner cubic X(670) 75, 141, 308, 523, 670 X(6) K102 Grebe cubic X(2) 2, 75, 76 X(69) K169 X(305) 2, 4, 75, 76, 253, 305, 341, 1088, 1370 ? K173 OXI cubic ? 4, 5, 305 X(376) K243 ? 69, 75, 264 X(515) K269 ? 75, 309, 320, 322 X(1503) K270 ? 75, 253
 Other cubics
 K K' K024 Kjp K196 nK0+(X76, X2) K028 Musselman (third) cubic K257 pK(X69, X2) K181 pK(X4, X4) K168 = pK(X3, X2) pK(X264, X264) K199 Soddy-Nagel cubic pK(X85, X75) K308 pK(X1, X8) pK(X75, X312) K323 pK(X1, X239) K766 pK(X75, X350)