These two triangles are very frequently mentioned specially in pages devoted to equilateral cubics. Their vertices lie on many interesting curves as seen below. First we recall some of their properties as in TCCT p.166.
 CircumTangential Triangle This triangle T1T2T3 is formed by the points on the circumcircle such that the line passing through one point and its isogonal conjugate is tangent to the circumcircle. It is equilateral and homothetic to the Morley triangle. The homothety h(X20, 3/2) maps its vertices to the cusps of the Steiner deltoid. T1, T2, T3 lie on the Kjp cubic and other curves as seen below.
 CircumNormal Triangle This triangle N1N2N3 is formed by the points on the circumcircle such that the line passing through one point and its isogonal conjugate is normal to the circumcircle i.e. passes through O. It is equilateral and homothetic to the Morley triangle. The homothety h(X631, -3/2) maps its vertices to the cusps of the Steiner deltoid. N1, N2, N3 lie on the McCay cubic and other curves as seen below. *** These CircumTangential and CircumNormal triangles are symmetric in O. See also the related FG paper "A Morley configuration".
 Related Conics Any conic passing through O and the vertices of one of these two triangles is obviously a rectangular hyperbola. This gives two pencils of rectangular hyperbolas we call CircumTangential and CircumNormal rectangular hyperbolas. Each conic of each pencil is entirely characterized by a fifth point P = u:v:w on the curve. Their equations are rather complicated and can be downloaded here :
 The following tables give a selection (some of them highlighted) of such hyperbolas with given P on the circumcircle.
 CircumTangential rectangular hyperbolas CircumNormal rectangular hyperbolas
 P centers on the curve X(74) X(3), X(74), X(2574), X(2575) X(98) X(2), X(3), X(98), X(3413), X(3414) X(102) X(3), X(40), X(102) X(103) X(3), X(103), X(165) X(104) X(3), X(21), X(104), X(3307), X(3308) X(105) X(3), X(55), X(105) X(106) X(1), X(3), X(106) X(111) X(3), X(111), X(574), X(1995) X(477) X(3), X(30), X(477), X(523) X(699) X(3), X(32), X(699) X(713) X(3), X(713), X(1333) X(727) X(3), X(58), X(727) X(729) X(3), X(6), X(729) X(733) X(3), X(39), X(733) X(741) X(3), X(171), X(741) X(759) X(3), X(35), X(759) X(840) X(3), X(840), X(1155) X(842) X(3), X(23), X(842) X(915) X(3), X(28), X(915) X(917) X(3), X(27), X(917) X(933) X(3), X(933), X(1624) X(934) X(3), X(934), X(2283) X(953) X(3), X(513), X(517), X(859), X(953) X(972) X(3), X(972), X(1817) X(1141) X(3), X(5), X(1141) X(1289) X(3), X(1289), X(2409) X(1294) X(3), X(20), X(1294) X(1297) X(3), X(22), X(1297) X(1298) X(3), X(418), X(1298), X(2979) X(1299) X(3), X(24), X(1299) X(1300) X(3), X(4), X(1300) X(1477) X(3), X(57), X(1477) X(2371) X(3), X(9), X(2371) X(2687) X(3), X(1325), X(2687) X(2693) X(3), X(2071), X(2693) X(2697) X(3), X(858), X(2697) X(2698) X(3), X(237), X(511), X(512), X(2698) X(2716) X(3), X(2077), X(2716) X(2718) X(3), X(36), X(2718) X(2724) X(3), X(514), X(516), X(2724) X(2734) X(3), X(515), X(522), X(2734) X(2757) X(3), X(8), X(2757)
 P centers on the curve X(99) X(3), X(99), X(376), X(3413), X(3414) X(100) X(3), X(100), X(3307), X(3308) X(107) X(3), X(4), X(107), X(1075) X(108) X(3), X(56), X(108) X(109) X(1), X(3), X(109), X(1745) X(110) X(3), X(54), X(110), X(182), X(1147), X(1385), X(2574), X(2575), X(6759), X(8717), X(8718), X(8723), X(8907), X(9932), X(11935), X(12584), X(12893) X(112) X(3), X(32), X(112), X(378) X(476) X(3), X(30), X(476), X(523) X(691) X(3), X(691), X(2080) X(805) X(3), X(511), X(512), X(805) X(901) X(3), X(513), X(517), X(901) X(925) X(3), X(20), X(925) X(927) X(3), X(514), X(516), X(927) X(930) X(3), X(550), X(930) X(934) X(3), X(934), X(999) X(1291) X(3), X(1157), X(1291) X(1293) X(3), X(40), X(1293) X(1301) X(3), X(24), X(1301) X(1302) X(2), X(3), X(1302) X(1304) X(3), X(186), X(1304) X(1309) X(3), X(515), X(522), X(1309) X(2222) X(3), X(36), X(2222) X(2715) X(3), X(1691), X(2715) X(2720) X(3), X(1319), X(2720) X(2731) X(3), X(944), X(2731) X(2743) X(3), X(2077), X(2743) X(2867) X(3), X(525), X(1503), X(2867)

Remark 1 : the most interesting hyperbola is probably the CircumNormal rectangular hyperbola passing through X(110) since it contains seventeen ETC centers. Its center is X(1511) and its asymptotes are parallel to those of the Jerabek hyperbola.

We meet this hyperbola in table 16 : it is the locus of points whose polar conic in the Neuberg-Lemoine pencil is a circle. It is also the polar conic of X(3) in K361.

Its inverse in the circumcircle is the strophoid K725.

***

Remark 2 : the yellow, light blue, orange cells correspond to hyperbolas homothetic to the Jerabek, Kiepert, Feuerbach hyperbolas respectively.

 Note that the inversive image in the circumcircle of any of the rectangular hyperbolas gives a strophoid passing through the vertices of one of the two triangles. See K725 for example in remark 1 above. The CircumTangential rectangular hyperbolas are the images of lines passing through G under the involution f defined in the page K024.
 Cubics The following tables give a selection of remarkable cubics through the vertices of these triangles.
 CircumTangential cubics CircumNormal cubics
 cubic remark / other centers on the curve K024 none, a nK0 K078 X(1), X(2), X(3), X(165), X(5373) K085 X(1), a cK K098 none K105 X(3), X(4), a nK60 K403 none, a psK K409 none, a nK K686 X(2), X(6), a nK60 K723 X(3) K726 X(6) K727 X(2), X(3) K728 X(2), X(3), X(23), X(111), X(187), X(2930), X(3098) K734 X(2), X(3), X(4230), a nK K735 X(3), X(182), X(3098), X(8666), X(8715) K896 X(6), X(111), X(368), X(511), X(3098), X(5640), X(6194)
 cubic remark / other centers on the curve K003 McCay cubic, a pK K009 Lemoine cubic, a psK K227 only one vertex on each curve K361 X(3), X(4), X(54), X(1342), X(1343), a psK K373 X(3), X(54), X(96), X(1147), a pK K404 nK0(X924, X323) K405 McCay-Lemoine cubic, a spK. D(1/2), see Table 33 K519 an Orion cubic. See Table 11 K664 isogonal transform of K665, a spK K725 Neuberg strophoid K735 X(3), X(182), X(3098), X(8666), X(8715) K736 X(3), X(40), X(376), X(7709) K931 X(3), X(6), X(30), X(67), X(182), X(186)