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See the general equation in Table 32

K419a = Cev(–π/3) = pK(X13, Qi)

K419b = Cev(π/3) = pK(X14, Qe)

on both cubics :

X(2), X(13), X(14), X(30),

infinite points of the Steiner ellipse

K419a :

X(298), X(395), X(472), X(1081)

3 (always real) mates of X(370) (blue points), see Table 10

K419b :

X(299), X(370), X(396), X(473), X(554)

2 (not always real) mates of X(370) (red points), see Table 10

K419a and K419b are the Kiepert Cevian Mates of the Neuberg cubic K001. See explanations in Table 32. See also CL041.

K419a = pK(X13, Qi) where Qi = X(2)X(13) /\ X(30)X(74) = barycentric quotient X(13) ÷ X(14).

K419b = pK(X14, Qe) where Qe = X(2)X(14) /\ X(30)X(74) = barycentric quotient X(14) ÷ X(13).

These two points Qe, Qi are actually the G-Hirst inverses of X(13), X(14) respectively and lie on the real asymptote of the Neuberg cubic.

Qi, Qe are now X(11078), X(11092) in ETC (2016-12-03).

These two cubics have each only one real asymptote since they both pass through the infinite points of the Steiner ellipse and this asymptote is parallel to the Euler line.

That of K419a is the parallel at the point Si where K419b meets the Fermat line again. Si also lies on the line containing G and X(2993), the isotomic conjugate of the anticomplement of X(16). Similarly for Se and the asymptote of K419b. Note that the midpoint of Si, Se is X(3163), barycentric square of X(30), on the Steiner inellipse.

The tangentials of X(30) in K419a, K419b are Xi, Xe respectively and {X30, Xi, Se}, {X30, Xe, Si} are two triads of collinear points.

One remarkable thing to observe is that these two cubics are tangent to the tangents at the in/excenters to the Neuberg cubic. This is also true for the cubic Kn = K060.

The contacts with these tangents are :

  • X(1081) and extraversions for K419a, where X(1081) = X(1)X(30) /\ X(13)X(226),
  • X(544) and extraversions for K419b where X(554) = X(1)X(30) /\ X(14)X(226),
  • X(79) and extraversions for K060.

K419a and K419b generate a pencil of pKs containing K472 = pK(X30, X2) and also the decomposed cubic union of the line at infinity and the Kiepert hyperbola.

All these cubics share the same orthic line, namely the Brocard axis.

The isotomic transforms of K419a, K419b are K867a, K867b respectively.