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P = X(4), P' = X(69), X(877), X(879)

ten points obtained from all the permutations of the coordinates of P and P'.

points at infinity of ABC sidelines (inflexion points)

This is the "original" Tucker cubic. It is the locus of point M such that the orthic triangle and the cevian triangle of M have the same area. See "Tucker cubics" in the Downloads page.