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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 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) 

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