   Reference : Stuyvaert, Point remarquable dans le plan d'une cubique, Nouvelles Annales de Mathématiques, Série. 3, 18 (1899), pp. 275-285. In general, there is one and only one point S whose polar conic in a cubic is a (possibly degenerate) circle. In such case, we call S the Stuyvaert point of the cubic. When the cubic is circular, it is the singular focus. When the cubic is a K+, it is the common point of the asymptotes and the circle splits into the line at infinity and another line. On the other hand, a Stuyvaert cubic SK is a cubic having a pencil of circular polar conics. In other words, there is a line L (we call the circular line of the cubic) such that the polar conic of each of its points is a circle. These circles are obviously in a same pencil. SK must be a K+ with asymptotes (one only is real) concurring at X and then the polar conic of X must split into the line at infinity and the radical axis of the circles. The poloconic of the line at infinity splits into two perpendicular lines secant at X which are the bisectors of the circular line L and the orthic line. Recall that this orthic line is the locus of points whose polar conic is a rectangular hyperbola (it would be undefined if the cubic be a K60+ which is not the case here). The following table gathers together several examples of these cubics.  cubic type centers on the cubic X L remarks K392 pK X(99), X(115) X(2) X(2)-X(1637) see CL009 K393 nK0 -- X(1637) orthic axis see CL044 K601 pK X(476), X(3258) X(2) X(2)-? see CL009 K602 pK X(99), X(620) P602 X(351)-X(2799) see CL049 K603 pK X(100), X(3035) P603 X(2804)-P603 see CL049 K604 nK0 -- X(351) X(351)-X(523) see CL044 K628 nK X(1), X(1784), X(2588), X(2589) P628 X(523)-P628 nK0(X468, X468) nK0 -- X(2) X(2)-X(6) see CL026 nK0(X523, X523) nK0 -- X(2) X(2)-X(525) see CL026 nK0(X1990, X1990) nK0 -- X(2) Euler line see CL026 nK0(X57 x X527, X57) nK0 X(1), X(1323) X513 x X527 X(514)-? see CL044 psK(cX113, X3260, X74) psK X(74), X(146), X(265) ? ? see CL063 psK(X620, X523, X99) psK X(99), X(148), X(620) ? X(2799)-? see CL063 psK(X3035, X693, X100) psK X(100), X(149), X(3035) ? X(2804)-? see CL063  Notes : P602 =(b-c) (b+c) (2 a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) : : , SEARCH=8.660881599853399 P603 = a (a-b-c) (b-c) (2 a^3-2 a^2 b-a b^2+b^3-2 a^2 c+4 a b c-b^2 c-a c^2-b c^2+c^3) : : , SEARCH=19.08170495120384 P628 = (b-c) (-2 a^4+a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4) : : , SEARCH=-2.260132844466736 These points are now X(11123), X(11124), X(11125) in ETC (2016-12-07).  