X(2), X(11), X(13), X(14), X(110),
extraversions of X(11)
Ae, Be, Ce, Ai, Bi, Ci : vertices of the Napoleon triangles
A3, B3, C3 : intersections of altitudes with the parallel at G to the relative sideline of ABC.
(these 9 latter points on the Napoleon cubic)
Q015 is the locus of orthopivots of singular orthopivotal cubics. See the FG paper "Orthocorrespondence and orthopivotal cubics" in the Downloads page.
It is a 12th degree bicircular curve I call the Lang curve in honour of Fred Lang who kindly gave me the discriminant which allows computation of the equation.
For each point P on Q015, the orthopivotal cubic O(P) is either degenerate or has a singularity. See in particular §§6.1, 6.5.1, 6.5.2 in the paper mentioned above.
Q015 contains a large number of points on the Napoleon cubic and a total of 18 double points :
Another remarkable example is K1051 = O(X11).