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

X(4), X(5), X(20), X(481), X(482), X(485), X(486) points at infinity of the orthocubic K006 points on the circumcircle and on pK(X6, X5562) midpoints of the altitudes CPCC or Hcevian points, see Table 11 Ixanticevian points, see Table 23 vertices of the diagonal triangle of these four points, see figure below 

This cubic belongs to the pencil of cubics containing the Napoleon cubic K005, the Kn cubic K060 and many other cubics. See "Two Remarkable Pencils..." in the Downloads page. It also belongs • to the pencil of cubics generated by the Darboux cubic K004 and the Lucas cubic K007. K122 is the Orion cubic K(X5). See figure below and Table 11, Table 15. • to the pencil of cubics generated by the Orthocubic K006 and the cubic decomposed into the line at infinity and the Kiepert hyperbola. It follows that its asymptotes are parallel to those of the Orthocubic. The isogonal transform of K122 is K633. 

K122 and the CPCC points 
K122 and the Ixanticevian points 

The green points are the Ixanticevian points, see table 23. (H) is the polar conic of X(5) in K122, a rectangular hyperbola passing through these points and X5, X6, X52, X195, X265, X382, X2574, X2575. The red points are the vertices of the diagonal triangle of these four points. K122 is a pK in this triangle with pivot X(5). The isopivot (tangential of X5) is unlisted in ETC with SEARCH = 0.449196530387458. 
