Any isogonal cubic is invariant under isogonal conjugation. If K is a non-isogonal circum-cubic, its isogonal transform is another circum-cubic K'. When K = pK(X2,P), we have K' = pK(X32,Q) where Q is the isogonal conjugate of the isotomic conjugate of P.
 P K Q K' X(69) K007 Lucas cubic X(3) K172 X(316) K008 Droussent cubic, circular cubic X(23) K108 X(75) K034 Spieker perspector cubic X(1) K175 X(264) K045 Euler perspector cubic X(4) K176 X(11057) K092, a pK60 X(7712) K922 X(309) K133 X(84) K180 X(76) K141 X(2) K177 X(3) K146 X(184) X(322) K154 X(40) K179 X(4) K170 X(25) K171 X(8) K200 X(55) X(14615) K235 X(20) K236 X(892) K240 X(691) X(99) K242 X(110) X(314) K254 X(21) K430 X(298) K264a X(15) X(299) K264b X(16) X(3260) K279 X(30) K495 X(320) K311 X(36) K312 X(22) X(206) K160 X(315) X(22) K174 X(305) X(69) K178
 Other cubics
 K K' K009 Lemoine cubic K028 Musselman (third) cubic K010 Simson cubic K162 cK(#X6, X3) K015 Tucker nodal cubic K229 nK(X32, X6, X6) K039 Jerabek strophoid K025 Ehrmann strophoid K043 Droussent medial cubic K273 pK(X111,X671) K060 Kn = O(X5) orthopivotal K073 Ki K135 pK(X1911, X291) K251 pK(X238, X2) K155 pK(X31, X238) K323 pK(X1, X239) K167 pK(X184, X6) K181 pK(X4, X4) K184 pK(X76, X76) K346 pK(X1501, X6) K185 nK0(X2, X523) K222 nK0(X32, X512) K199 Soddy-Nagel cubic K632 pK(X604, X1) K229 nK(X32, X6, X6) K015 nK(X2, X2, X2) K233 pK(X25, X4) K168 pK(X3, X2) K261a O(X62) orthopivotal K261b O(X61) orthopivotal K262a O(X15) orthopivotal K262b O(X16) orthopivotal K263 O(X511) orthopivotal K292 O(X182) orthopivotal K278 pK(X1989, X1989) pK(X50, X6) K290 O(X39) orthopivotal K291 O(X32) orthopivotal K308 pK(X1, X8) pK(X31, X9) K317 pK(X81, X86) K362 pK(X213, X1) K319 pK(X1333, X81) K345 pK(X37, X2)