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Contributed by Wilson Stothers (private message 20040510) with some additions by myself A Stothers cubic is a cubic of the form ST(F) = cK0(#F,TG(F)), where TG(F) denotes the tripolar centroid of F (see a definition at CL029). These cubics are always of type cK0 since TG(F) lies on the tripolar of F. These have all the properties of a cK0, but are unusual since we can identify the nodal tangents. Notations For any P,
*** General properties of cK0(#F,R)  so that the root R is on T(F). T(R*) is also the tangent to C(R) at F and the polar of R with respect to C(F). R* lies on C(F)
The pivotal conic and I(R*) touch at R, with common tangent T(F). For example, with F = I (incenter), we obtain isogonal cubics and T(F) is the antiorthic axis. In particular, when R = X(650), X(513), X(661) we find K040 (the Pelletier strophoid), K137 = Z+(IK), K221 respectively. *** The special case ST(F) = cK0(#F,TG(F)). Say F has barycentrics (f : g : h). Observe that the intersections in property 3 are the points P1, P2 with first barycentrics f/(gh) and f/(g+h2f). This gives the nodal tangents as
The pivotal conic is the inconic of the anticevian triangle of F touching (any two of) T(F), T1, T2. The general equation of ST(F) is : f(g  h)(g + h  2f)x(h^2y^2 + g^2z^2) + cyclic = 0. Example : ST(K) with K = X(6). Then
Observe that T1 and T2 are perpendicular. This is true for any ST(F) with F on the quintic Q012. We find several other interesting cubics with the same singularity K such as K222, K223, K224, K225 with roots X(512), X(647), X(649), X(663) respectively. 
