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The Darboux cubic K004 and the Lucas cubic K007 generate a pencil of cubics passing through A, B, C, H, L = X(20) and the four CPCC points, see table11.

This pencil contains several remarkable cubics mentioned in the following table.

cubic

other centers Xi for i =

remarks

K004

see the page

one of the three pK of the pencil, a pK++

K007

see the page

one of the three pK of the pencil

K032

see the page

 

K122

see the page

also member of the pencil K005 - K060

K268

see the page

the only equilateral cubic of the pencil, a K60+

K313

see the page

the only circular cubic of the pencil

K329

see the page

one of the three pK of the pencil, a pK+