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The Neuberg cubic K001 and the Lemoine cubic K009 generate a pencil of cubics passing through A, B, C, O, H and the three points Ua, Ub, Uc mentioned in the Neuberg cubic page. These cubics have the same tangent at O which is the line OX(54). See a generalization in table 18. Each cubic of this pencil meets the Euler line at O, H and a third point X such that OX = t OH (vectors), in which case the cubic is denoted by F(t). Special cases :
Properties :


Connection with CL055 : Any cubic F(t) is spK(Y, M) as in CL055 with X, Y as above and M midpoint of XY. The isogonal transform F(t)* of F(t) is then spK(X, M) passing through O, H, Ua*, Ub*, Uc*and meeting the line at infinity like pK(X6, X), the circumcircle like pK(X6, Y). See also Table 54. *** The following table gives a selection of cubics from the NeubergLemoine pencil. 



The figure below shows several interesting cubics. 
