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When four tangents are drawn from a point M on the Neuberg cubic to the Neuberg cubic itself, their anharmonic ratio is constant (Salmon) and, up to a permutation of these tangents, is equal to :

which is obtained when the tangents are those drawn from X(74) to the cubic namely AX(74), BX(74), CX(74), X(30)X(74) in this order.

Now, given another pivotal cubic pK with pole Ω = p:q:r and pivot P = u:v:w, the tangents at A, B, C, P concur at P* and their anharmonic ratio is evaluated similarly.

The cubic pK is said to be equivalent to the Neuberg cubic if and only if these ratios are equal. This gives the condition :

a^2 (b^2 - c^2) (4SA^2 - b^2c^2) qr u^2 = 0

which can be construed as follows :

• Ω lies on the circum-conic C(P) with perspector the barycentric product X526 x P^2, passing through P^2 corresponding to a degenerate pK,

• P^2 lies on the trilinear polar of the barycentric quotient Ω ÷ X526 which means P lies on the diagonal conic D(Ω) with center Ω ÷ X526 passing through the four (real or not) square roots of Ω.

Special cases :

• with Ω = X(6), we obtain that an isogonal pK equivalent to the Neuberg cubic must have its pivot on the diagonal rectangular hyperbola passing through X(1), X(5), X(30), X(395), X(396), X(523), X(1749), etc, and whose center is the Tixier point X(476). This conic is the polar conic of the infinite point of the Euler line in the Neuberg cubic.

• with Ω = X(323), D(Ω) is the Wallace hyperbola with center X(99) passing through X(1), X(2), X(20), X(63), X(147), X(194), X(487), X(488), X(616), X(617), X(627), X(628), X(1670), X(1671), X(1764), X(2128), X(2582), X(2583), X(2896), etc.

• with Ω = X(50), D(Ω) is the Stammler hyperbola with center X(110) passing through X(1), X(3), X(6), X(155), X(159), X(195), X(399), X(610), X(1498), X(1740), X(2574), X(2575), X(2916), X(2917), X(2918), X(2929), X(2930), X(2931), X(2935), X(2948), X(3216), etc.

• with P = X(2), C(P) passes through X(2), X(15), X(16), X(186), X(249), X(323), X(842), X(1138), X(2411), etc, giving cubics passing through the vertices of the medial triangle and whose anticomplements denoted a(pK) are also pivotal cubics with pole equivalent to K001. See blue lines.

• with P = X(1), C(P) passes through X(6), X(50), X(323), X(2436), X(2623), etc.

• a pK with coincident Ω and P equivalent to K001 must have its pole/pivot on the trilinear polar of the isotomic conjugate of X526, the line passing through X(2), X(94), X(300), X(301), X(328), X(338), X(566), X(1272), X(1989), X(2410), X(2493), etc. See yellow lines. The isotomic transforms of these pKs have pivot X(2). See blue lines.

***

Any projection or linear transformation, any isoconjugation with pole Q transform the Neuberg cubic into an equivalent cubic. In particular, the barycentric product of K001 by a point gives a simple equivalent cubic.

The following table gives a selection of pivotal cubics equivalent to the Neuberg cubic. As usual, g, t, a, c, i denote isogonal, isotomic, anticomplement, complement, inverse in (O).

Ω

P

cubic or centers on the cubic

remark

X(6)

X(5)

K005 Napoleon cubic

 

X(1989)

X(265)

K060

circular, g(K073)

X(50)

X(3)

K073

circular, i(K001), g(K060)

X(6) x X(1989)

X(1989)

K095

 

X(6)

X(395)

K129a

 

X(6)

X(396)

K129b

 

X(13) x X(16)

X(13)

K261a

circular, X13 x pK(X323,X2)

X(14) x X(15)

X(14)

K261b

circular, X14 x pK(X323,X2)

X(76)

X(3260)

K276

Q = G, t(K001)

X(1989)

X(1989)

K278

 

X(15)

X(2)

K341a

 

X(16)

X(2)

K341b

 

X(300)

X(300)

K342a

t(K341a)

X(301)

X(301)

K342b

t(K341b)

X(50)

X(6)

K390

g(K278)

X(2)

X(621)

X(2), X(4), X(69), X(616), X(621), X(628), X(2992)

a(K341a)

X(2)

X(622)

X(2), X(4), X(69), X(617), X(622), X(627), X(2993)

a(K341b)

X(6)

X(523)

X(1), X(110), X(523)

circular

X(6)

X(1749)

X(1), X(484), X(1749)

 

X(32)

X(1953)

X(6), X(19), X(48), X(1953), X(2148)

 

X(32)

X(2173)

X(6), X(19), X(48), X(2151), X(2152), X(2153), X(2154), X(2159), X(2173)

Q = X(31), X1 x K001

X(50)

X(1)

X(1), X(35), X(36), X(1094), X(1095), X(2169)

 

X(50)

X(195)

X(54), X(195), X(1154), X(2914)

 

X(50)

X(399)

X(74), X(399), X(1511), X(2914)

 

X(323)

X(2)

X(2), X(97), X(323), X(2981)

g(K095)

X(323)

X(63)

X(63), X(2167), X(2349)

 

X(323)

X(616)

X(74), X(616), X(2992)

 

X(323)

X(617)

X(74), X(617), X(2993)

 

X(323)

X(627)

X(54), X(627), X(1273), X(2993)

 

X(323)

X(628)

X(54), X(628), X(1273), X(2992)

 

X(393)

X(1784)

X(1), X(4), X(158), X(1784)