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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.

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

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

X(3524)

K812

-

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

X(20)

 

-

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.

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