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S^2 xyz + ∑ a^2 y z (c^2 y + b^2 z) = 0,

or equivalently

S^2 xyz + ∑ a^2 x (c^2 y^2 + b^2 z^2) = 0,

with S = 2*area(ABC)

centers of the Apollonian circles

points A', B', C' on (O) whose Simson lines S(A'), S(B'), S(C') pass through G : these points lie on nK0(X6, X5304)

infinite points of nK0(X6, X5304)

K634 is an isogonal nK with root X(6) comparable to the Kjp cubic K024. See also Table 44.

Locus properties

1. Locus of P whose pedal circle is orthogonal to the orthoptic circle of the Steiner in-ellipse.
2. Let P be a point, A*B*C* its circumcevian triangle, A'B'C' its pedal triangle and let A"B"C" be the antipodal triangle of A'B'C' in the pedal circle of P. A*B*C*and A"B"C" are perspective if P is on the circumcircle (the circumcevian triangle degenerates) or on K003 or on K634. If P is on K634, the perspector of the triangles A*B*C* and A"B"C" is a point in the circumcircle (Antreas P. Hatzipolakis, Hyacinthos #21738, Angel Montesdeoca, Hyacinthos #21743). See figures below (Angel Montesdeoca's work)
3. If P lies on the cubic K634 (together with a quadricircular octic), then the pedal and antipedal circles of P are tangent. The point of tangency of the two circles is on the circumcircle (Antreas P. Hatzipolakis, Angel Montesdeoca, Francisco Javier Garcia Capitan, Hyacinthos #21746 and sq.). See figures below.
4. Let ABC be a triangle and P a point with circumcevian triangle XY Z. The lines XY, XZ intersect BC at Ac, Ab respectively. We define Bc, Ba and Ca, Cb cyclically. These six points Ab, Ac, Bc, Ba, Ca, Cb lie on a conic C(P). This conic is a hyperbola, an ellipse or a parabola according to P is interior, exterior or lies on K634 (Francisco Javier Garcia Capitan, private message, 2020-10-17). Furthermore, C(P) is a circle if and only if P is one of the seven Rescassol-Skoubidou points mentioned in page K003. C(P) is called perspeconic in ETC, see preamble before X(34807).