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X(1), X(3), X(4), X(10996)

excenters

Let P be a point and A0B0C0 its pedal triangle. Consider the point A1 defined as follows :

• A0b is the orthogonal projection of A0 on PB0,
• A0bc is the orthogonal projection of A0b on PC0,
• A1 is the orthogonal projection of A0bc on PA0.

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.