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

X(2), X(3) vertices of the CircumTangential triangle and its midpoints vertices of the Thomson triangle foci of the Steiner inellipse fixed points of the involution f in page K024 infinite points of pK(X6, X3534) antipoints, see Table 77 

K727 is pkT(X3) as defined in the page K024. The Thomson isogonal transform of K727 is K833. See there for further properties. K1249 is an analogous cubic. 
