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too complicated to be written here. Click on the link to download a text file. 

X(2), X(6) P1, P2, P1*, P2* defined at K231 

The trilinear polar of P meets the sidelines of ABC at U, V, W. Ub, Uc are the projections of U on AC, AB and Vc, Va, Wa, Wb are defined similarly. These triangles UbVcWa and UcVaWb are parallelogic if and only if P lies on the Thomson cubic or on the cubic K232 which is in fact the locus of P such that Uc, Va, Wb are collinear. Similarly, Ub, Vc, Wa are collinear if and only if P lies on K234. 
