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X(2), X(3), X(20), X(63), X(69), X(77), X(78), X(271), X(394)
extraversions of X(63), X(77), X(78), X(271)
Denote by A', B', C' the antipodes of A, B, C on the circumcircle. For any point P, Pa is the trace of PA' on BC, Pb, Pc similarly.
The triangles ABC and PaPbPc are perspective if and only if P lies on the Darboux cubic K004.
The locus of the perspector is the Darboux perspector cubic i.e. K099 = pK(X394, X69).
K099 is anharmonically equivalent to the Thomson cubic K002. See Table 21.
This cubic is a member of the class CL042.
The isogonal transform of K099 is K445 = pK(X2207, X4) and its isotomic transform is K647 = pK(X2052, X264).
A locus property (César Lozada, 2022-06-06)
Let ABC be a triangle, P a point and A’B’C’ the cevian triangle of P. Let A” the point, other than A, at which circle with diameter AA’ cuts the circumcircle of ABC, and denote B”, C” cyclically. Then A’A”, B’B”, C’C” concur iff P lies on the cubic K099 and their point of intersection Q lies on K004.
ETC pairs (P, Q) : (2, 4), (3, 3), (20, 3183), (63, 40), (69, 20), (77, 1), (78, 1490), (271, 84), (394, 1498), (7013, 3182), (15394, 64), (46351, 2130).
K576 is the locus of the pseudo-poles of central cubics psK with center O such as K004, K426, K443.
The loci of the pseudo-pivots and the pseudo-isopivots are K099 and K172 respectively.
See Pseudo-Pivotal Cubics and Poristic Triangles and Central cubics.