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∑ (b^2 - c^2) x (c^2y^2 - b^2z^2) = 0

X(1), X(110), X(523), X(21381), X(39137), X(39138)

excenters

imaginary foci of the MacBeath inconic

Geometric properties :

K1155 is the isogonal circular pK with pivot X(523) and singular focus X(74).

Let P be a point and Pa, Pb, Pc its reflections in the sidelines BC, CA, AB of ABC respectively.

The lines APa, BPb, CPc concur (at Q) if and only if P and its isogonal conjugate P* lie on the Neuberg cubic K001 (see locus property 1). The locus of Q is K060 and P, P*, Q are collinear. Note that P* is the circum-center of triangle PaPbPc.

The circles with diameters APa, BPb, CPc concur (at X) if and only if P and P* lie on K1155 = pK(X6, X523). (Angel Montesdeoca, see here in Spanish). The locus of X is Q155 where further details are given.