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X(2), X(4), X(92), X(253), X(264), X(273), X(318), X(342), X(2052)

When P lies on the Lucas cubic K007, the point H÷cP lies on K647. See details at the page K645.

K647 is the isogonal transform of K576 and the isotomic transform of K099.

It is also the barycentric products X(4) x K184, X(76) x K445, X(92) x K034, X(264) x K002, X(1969) x K175, X(2052) x K099, X(18022) x K346, X(18027) x K576 hence, in particular, the barycentric quotient K002 ÷ X(3).

K647 meets the Steiner ellipse at the same points as pK(X2, X317).

Locus property

Let HaHbHc be the orthic triangle of ABC and a point P. Bap and Cap are the perpendicular feet of B on AP and C on AP respectively.

The triangles BHbBap and HcCCap are perspective at A (by construction) and the perpectrix meets BC at A'. The points B', C' are analogously defined.

The triangles ABC, A'B'C' are perspective if and only if P lies on the Darboux cubic K004. The perspector lies on K647. (Angel Montesdeoca, 2018-08-23, see here).