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A cubic is a member of the Steiner net when it is a circum-cubic passing through the (four) foci of the Steiner inscribed ellipse, the in-conic with center the centroid G of ABC. These foci are X{39162, 39163, 39164, 39165} in ETC.

Any such cubic is a spK(P, G) for some P. See CL055 for general properties of these cubics and their construction.

The following table gives a selection of spK(P, G) according to P.

P

spK(P, G)

type

centers and/or remark

X(2)

K002

pK

the only pK of the net

X(524)

K018

nK0

focal orthopivotal cubic

X(519)

K086

nK

strophoid

X(30)

K187

nK

focal cubic

X(538)

K248

nK

focal cubic

X(6)

K287

K0

central cubic, center G, isogonal transform of K729

X(527)

K352

nK

focal cubic

X(385)

K353

K0

isogonal transform of K705

X(3)

K358

-

stelloid, the only K60 of the net

X(7840)

K705

K0

isogonal transform of K353

X(3543)

K706

-

central cubic, center O, isogonal transform of K847

X(599)

K729

K0

isogonal transform of K287

X(3524)

K812

-

X(3), X(4), X(3531), X(3545)

X(20)

K847

-

isogonal transform of K706

X(323)

 

K0

X(2), X(6), X(186), X(381), X(1989)

X(381)

 

-

isogonal transform of K358, a circumnormal cubic

X(671)

 

nK

X(99), X(187), X(1379), X(1380)

X(1121)

 

nK

X(101), X(664), X(1055), X(1323)

 

 

 

 

Notes :

• spK(P, G) is circular if and only if P lies at infinity, giving a pencil of focal nKs with root G, focus F on (O), see yellow lines.

• spK(P, G) is also a nK when P lies on the Steiner (circum) ellipse in which case its root lies on this same ellipse and the pole on K229. Two examples are given in the pink lines.

• spK(P, G) is a K0 if and only if P lies on the line GK in which case it contains G and K, see red points P in the table. These cubics are in a same pencil of K0s, stable under isogonal conjugation, generated by K002 and K018 which are the only two self-isogonal cubics.

• if P1 and P2 are symmetric about G, then the cubics spK(P1, G) and spK(P2, G) are swapped under isogonal conjugation. They meet again at two isogonal conjugate points on K002 hence collinear with G.

 

Other cubics passing through the foci of the Steiner inscribed ellipse

These cubics are not circum-cubics of ABC but, in some cases, they are circum-cubics of some other triangle.

Those in the orange cells are circum-cubics of the Thomson triangle (T) and pass through G and K. They form a pencil which contains the Thomson cubic K002. The remaining points on (O) lie on a nK0(X6, R on GK) and the tangents at these points concur on a hyperbola passing through {2, 6, 110}.

Those in the blue cells are focal cubics with singular focus F on the circle passing through {2, 3, 110, 842, 8724, 14649, 14685, 34291, 35911}, with center X(44814), which is the (O)-inverse of the line {23,110} ans the Psi-image of the line {3,74}. These cubics are in a same pencil of circular cubics passing through X(2), X(3), X(110). They are Psi-invariant, see at the bottom of Table 60. Every such focal cubic meets the circumcircle (O) again at three points on an isogonal pK with pivot on the line GK.

Those in the green cells are focal cubics with singular focus F on the Fermat axis and same orthic line, namely the Euler line. These cubics are in a same pencil of circular cubics passing through X(30), X(31862), X(31863). The latter two points lie on the Fermat axis.

The red cubics are also in a same pencil of focal cubics with singular focus F on the line {3, 74, 110, 156, 246, 399, 1511, 1614, 2972, 3470, 5191, 5609, etc} which contains the circum-cubic K187 and the decomposed cubic, union of the line at infinity and the rectangular hyperbola with center G, passing through {3, 6, 381, 599, 2574, 2575} and the four foci of inSteiner. This hyperbola is the polar conic of X(30) in K003, of X(3) in K005 , of X(6) in K018. Every cubic of the pencil passes through X(30) and O, with tangent the line {3, 74, etc}, and orthic line the Euler line, except K038 which has a node at O.

The blue cubics are in a same pencil of stelloids with radial center X on the Euler line, which also contains the decomposed cubic above and the circum-cubic K358. Every cubic contains O, X(381) and the infinite points of the McCay cubic K003.

cubic

type

centers 39162, 39163, 39164, 39165 and ... / remark

K038

strophoid, F = X(14685)

3, 30, 36, 131, 187, 1511, 2482, 3184, 6150, 6592, 12095, 12096, 17729, 35204, 39987, 47077, 47078, 47079, 47080, 47081, 47082, 47083, 47084, 47085, 47086, 47087, 47088, 47089, 50388, 50389, midpoints of ABC

K463

focal, F = X(110)

2, 3, 15, 16, 30, 110, 5463, 5464 / K187 of (T) / (O) ∩ K002

K727

pK in CircumTangential triangle

2, 3, 7712, antipoints see Table 77 / isogonal transform of K833 wrt (T)

K758

central, center O

2, 3, 154, 165, 376, 3576, 10606, midpoints of (T) / isogonal transform of K002 wrt (T)

K800

focal, F = X(399)

1, 3, 5, 30, 191, 399, 5127, 5535, 6326, 7701, 13513, 18865, 31862, 31863, 39229, 39230, 40851, 40852, excenters of ABC

K816

focal, F = X(14685)

2, 3, 4, 74, 110, 125, 131, 541, 11472, 14685 / (O) ∩ K1169

K833

central stelloid, X = X(2)

2, 3, 381, 39641, 39642 / isogonal transform of K727 wrt (T) / (L∞) ∩ K003

K893

focal, F = X(3)

2, 3, 6, 110, 111, 542, 2482, 14916 / (O) ∩ K1156

K911

stelloid, X = X(5054)

1, 3, 381, 1340, 1341, 3576, excenters of ABC / (L∞) ∩ K003

K1092

focal, F = X(6)

6, 30, 1379, 1380, 5000, 5001, 11472, 13872, 16303, 31862, 31863

K1259

K006 of (T)

2, 3, 5373, 11472

K1275

strophoid, F = X(28662)

6, 115, 187, 524, 5526, 9721, 16303, 28662, 44909, 48653, 48654, 48721, 50536

K1286

focal, F = X(51800)

2, 3, 23, 110, 182, 187, 353, 9829, 11645, 15080, 51797, 51798, 51799, 51800 / (O) ∩ K102

K1288

focal, F = X(113)

30, 113, 2043, 2044, 31862, 31863, 47084

K1289

central focal, F = X(381)

30, 381, 31862, 31863, 51825, 51826

K1293

central, center O

1, 3, 40, 2043, 2044, 11472, 35237, 45633, 52050, 52051, 52052, 52053, 52054, excenters of ABC / (L∞) ∩ K243

K1294

central, center O

3, 30, 1379, 1380 / (L∞) ∩ K001

K1295

focal, F = X(8724)

2, 3, 110, 352, 511, 574, 599, 7998, 8724, 10717, 10989, 11178 / (O) ∩ pK(X6, X599)

K1296

focal, F = Psi(X399)

2, 3, 5, 13, 14, 110, 399, 3448, 6592 / (O) ∩ pK(X6, X37779)