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The second Brocard cubic K018 and the Grebe cubic K102 are both isogonal cubics, the former being a nK0 and the latter a pK.

They generate a pencil of K0s passing through the base points A, B, C, G = X(2), K = X(6) and the four foci of the inconic with center X(6).

Any such cubic Γ(k) of the pencil has an equation of the form :

∑ (y - z) [a^4 y z + b^2 c^2 x (k x + y + z)] = 0, where k is a real number or infinity.

Properties

• the pencil is stable under isogonal conjugation and the isogonal transform of Γ(k) is Γ(k)* = Γ(–k).

It follows that K018 = Γ(0) and K102 = Γ(∞) are the only self-isogonal cubics of the pencil.

• Γ(k) meets the line GK = X(2)X(6) again at P(k) = b^2 + c^2 - (2 + k) a^2 : : , the homothetic of G under h(K, - 3/k).

• Γ(k) meets the line at infinity at the same points as pK(X6, P(-k)) and the circumcircle at the same points as pK(X6, P(k)). Obviously, P(0) = X(524) and P(∞) = X(6).

• Γ(k) is spK(P(-k), X6) in CL055.

• the pencil contains two other pKs, two other (decomposed) nK0s, two K⁺ and one K⁺⁺ but no K60. These are shown in the table below.

k2 is the square root of : 25 + 16 cosA cosB cosC

Special points on these cubics

(1) : vertices of the Thomson triangle

(2) : Ix-anticevian points, see Table 23

(3) : isogonal conjugates of the Ix-anticevian points, see Table 23

(4) : H-cevian points, see Table 11

(5) : isogonal conjugates of the H-cevian points, see Table 11

(6) : four foci of the Steiner inellipse

(7) : imaginary foci of the Brocard inellipse