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K635

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

X(1), X(876), X(885), X(2283)

centers of the Apollonian circles

points A', B', C' on (O) whose Simson lines pass through the center of (C) : these points lie on nK0(X6, X5275)

infinite points of nK0(X6, X5275)

K635 is an isogonal cK with root X(6) comparable to the Kjp cubic K024. See also Table 44.

K635 is a nodal cubic with node the incenter X(1).

Locus properties

  1. Locus of P whose pedal circle is orthogonal to the circle (C), the member of the pencil of circles generated by the circumcircle and the nine point circle that is also orthogonal to the incircle. The center of (C) is the common point of the Euler line and X(11)X(57).

 

K635abc

Since K635 is a weak cubic, it is associated by extraversion to three other nodal cubics namely :

K635A = cK(#Ia, X6), K635B = cK(#Ib, X6), K635C = cK(#Ic, X6),

where Ia, Ib, Ic are the excenters of ABC.

Naturally, each cubic contains the corresponding extraversions of the weak points X(876), X(885), X(2283).