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X(2), X(3), X(54), X(69), X(95), X(96), X(97)

Let A'B'C' the cevian triangle of a point P. Let Ab, Ac be the reflections of C, B in the perpendiculars dropped from A' onto AC, AB. Let Oa be the circumcenter of triangle AAbAc and define Ob, Oc likewise. Then

• OaObOc and A'B'C' are perspective if and only if P lies on K045 in which case the perspector lies on K044.

• OaObOc and ABC are perspective if and only if P lies on K646 in which case the perspector also lies on K044. (Angel Montesdeoca, Anopolis #845)

K646 is the pivotal cubic pK(X97, X95). It is the isogonal transform of K350 and the isotomic transform of K674 = pK(X324, X264).

K646 meets the circumcircle at the same points as K007 and pK(X6, X2979). These are the vertices of the Lucas triangle.