X(1), X(3), X(4)
Let P be a point and A0B0C0 its pedal triangle. Consider the point A1 defined as follows :
B1 and C1 are defined similarly. The triangles ABC and A1B1C1 are perspective if and only if P lies on the (first) Ariadne's cubic K152.
K152 is an isogonal pK with pivot X(10996) on the Euler line, hence it is a member of the Euler pencil of cubics (Hyacinthos #8012 & sq.). See Table 27.
You will find other Ariadne's cubics in Antreas Hatzipolakis' Anopolis.