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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 :

• A, B, C with tangents the bisectors, for which we obtain three strophoids,
• G with tangents parallel to the asymptotes of the Kiepert hyperbola,
• the Fermat points X(13), X(14) for which we obtain the strophoids K061a and K061b,
• X(110),
• the circular points at infinity,
• Ae, Be, Ce, Ai, Bi, Ci : vertices of the Napoleon triangles,
• A3, B3, C3 : intersections of altitudes with the parallels at G.

Another remarkable example is K1051 = O(X11).