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Consider a circum-conic (C) with center T and perspector S = G/T (Ceva conjugate).

A variable line (L) passing through T meets (C) at two points M, N. Let C(M, N) be the bicevian conic passing through the vertices of the cevian triangles of M and N. (C) and (L) meet at two points P, Q and, when (L) rotates around T, the locus of P, Q is a cubic K(T).

K(T) is a pivotal cubic with pivot T and pole the barycentric square Ω of G/T. Hence, K(T) = K(G/S) = pK(S^2, G/S) = pK((G/T)^2, T).

The isoconjugation with pole S transforms K(T) into the isotomic pivotal cubic pK(G, aS) with pivot aS, the anticomplement of S.

The most remarkable example is K172 = pK(X32, X3) since (C) is the circumcircle of ABC. The corresponding isotomic cubic is the Lucas cubic K007.

Here is a selection of some cubics K(T) and pK(G, aS) with a list of centers when the cubic is not listed.

T

S

Ω

K(T) = pK(Ω, T)

pK(G, aS)

X(1)

X(9)

X(220)

X(1), X(9), X(200), X(3158)

X(2), X(7), X(8), X(145)

X(3)

X(6)

X(32)

K172

K007

X(5)

X(216)

?

X(5), X(51), X(216), X(418)

K045

X(6)

X(3)

X(577)

X(3), X(6), X(394), X(493), X(494), X(1124), X(1335), X(3167)

K170

X(9)

X(1)

X(6)

K351

K200

X(10)

X(37)

X(1500)

X(10), X(37), X(42), X(65), X(71), X(210), X(227), X(1826)

K034

X(37)

X(10)

X(594)

X(10), X(37), X(321)

X(1), X(2), X(75), X(192), X(330), X(3223)

X(214)

X(44)

X(1017)

X(1), X(44), X(214), X(678), X(1319)

K311

 

 

 

 

 

 

 

 

 

 

Remark : when T = G, the cubic K(T) is the union of the medians of ABC.