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vertices of the CircumTangential triangle and its midpoints
vertices of the Thomson triangle
foci of the Steiner in-ellipse
fixed points of the involution f in page K024
infinite points of pK(X6, X3534)
anti-points, 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.