Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

X(4), X(6243)

reflections A', B', C' of A, B, C in the sidelines of ABC

points at infinity of the McCay cubic

imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola

"golden points" 𝚽1 and 𝚽2 defined by

The golden cubic belongs to the pencil of cubics containing the Neuberg cubic, the Soddy cubic and the union of the three altitudes. See "Two Remarkable Pencils..." in the Downloads page.

It is also a member of the pencil of stelloids generated by the McCay cubic and the union of the line at infinity with the Kiepert hyperbola. See Table 51.

It is a K60+ with asymptotes parallel to those of the McCay cubic and concurring at Z = X(3060), the intersection of the parallel at G to the Brocard axis and the line X(6)X(22).

It meets the Euler line at H and the two "golden points". These points are inverses in the circle with center O passing through H.


K115 is spK(X3, X52) as in CL055 and it is the only cubic C(k) which is a spK.

It follows that K115 passes through :

• the foci of the inconic with center X(52).

• the points U, V, W on the sidelines of ABC.

U lies on the line A' X(6243) which is parallel to the A-cevian line of O, V and W similarly.