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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.

k

P

cubic / other centers X(i) on the cubic for i =

type

remarks/ notes

X(6)

K102 / 1, 43, 87, 194, 3224

pK

(7)

- k1

?

K757 / 182, 264, 287, 511, 694

 

k1 = 1 - 4 sin2ω

-3 GK/R

?

- / 1114, 1344, 2574

nK0

decomposed

-3

X(2)

isogonal transform of K314 /

 

(1)

-2

X(141)

K754 / 17, 18, 141, 9481

spK(X3629, X6)

(3)

-1

X(69)

K168 / 3, 69, 485, 486, 5374, 5408, 5409, 6337, 8770

pK

(4)

0

X(524)

K018 / 13, 14, 15, 16, 111, 368, 524, 5000, 5001

nK0

focal cubic, (6)

1

X(193)

K233 / 4, 25, 193, 371, 372, 2362, 7133

pK

(5)

2

X(3629)

K755 / 61, 62, 251, 3629, 9484

spK(X141, X6)

(2)

3

X(1992)

K314 / 1992

K++, spK(X2, X6)

central cubic

3 GK/R

?

- / 1113, 1345, 2575

nK0

decomposed

k1

X(385)

K756 / 98, 184, 232, 262, 385, 7709

 

k1 = 1 - 4 sin2ω

(3 - k2)/2

?

- /

K+

k2 below

(3 + k2)/2

?

- /

K+

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