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X(370) and its five mates, see Table 10

vertices of the antimedial triangle.

see below for other points

A point P is an equilateral cevian point if the cevian triangle of P is equilateral (TCCT, p.267).

P must lie on the three cubics K143-A, K143-B, K143-C.

It also lies on the Neuberg cubic (conjectured by Edward Brisse, proven by Gilles Boutte) and on K144 (X(370) antimedial cubic). See also Q033.

There are at least four (see the figure above), at most six (see K144) real such points, one of them being X(370) always inside ABC, the others outside. The centers of the corresponding equilateral triangles lie on the Napoleon cubic and on the perpendicular bisectors of X(13)X(15) and X(14)X(16). (Jean-Pierre Ehrmann)

K143-A contains the six mentioned points, A, the midpoint of BC, the vertices of the antimedial triangle.

Hence, K143-A, K143-B, K143-C belong to the same pencil of cubics passing through the six points and the vertices of the antimedial triangle. This pencil also contains K144.

See table 10 : X(370) related curves.