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K659

∑ a^2 x (y^2 - z^2) = 0 or ∑ (c^2 y - b^2 z) y z = 0

X(2), X(6), X(76), X(194), X(2998)

vertices of the antimedial triangle

Consider a point P = u : v : w and its two Brocardians P1 = 1/w : 1/u : 1/v, P2 = 1/v : 1/w : 1/u. See Tucker cubics for more details.

The cevian triangles of P1, P2 are (P1a, P1b, P1c) and (P2a, P2b, P2c). The perpendicular bisectors of the three segments P1x-P2x are concurrent (at Q) if and only if P lies on K659. (Angel Montesdeoca, private message, 2014-01-24)

The locus of Q is a central cubic with center O (dashed blue curve on the figure).

K659 is the anticomplement of pK(X141, X2) and the isogonal transform of pK(X32, X32).