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X(69), X(95), X(298), X(299), X(319), X(320), X(340), X(1273), X(1494), X(2992), X(2993)

K636 is the isotomic transform of K060. It is the pivotal cubic with pivot X(69) and pole the isotomic conjugate X(7799) of X(1989) which is also the barycentric product X(340) x X(69). This pole lies on the lines X(2)X(39), X(15)X(298), X(16)X(299). Its first barycentric coordinate is : 4 SA^2 - b^2c^2.

It is related to the Neuberg cubic K001 as follows. For any X on K001, let A'B'C' be the cevian triangle of a variable point P. The parallels at A', B', C' to the Euler lines of triangles XBC, XCA, XAB concur if and only if P lies on an isotomic pK whose pivot Q is a point on K636. Recall that these Euler lines also concur, see K001 property 4. The mapping X -> Q can be defined in this way : denote by Xa, Xb, Xc the isogonal conjugates of A, B, C in the triangles XBC, XCA, XAB respectively. ABC and XaXbXc are perspective at the isotomic conjugate of Q.

The table gives a selection of catalogued cubics (or a list of centers on the cubic) according to X on K001.

note : this cubic is the anticomplement of pK(X233, X2) and contains X(2), X(3), X(5), X(95), X(264).

More generally, if A'B'C' is replaced by the anticevian of P, the analogous parallels as those above also concur but when P lies on a pK with pivot X(2) and pole the complement of Q which appears to be the complement of the corresponding previous cubic. For example, when X = X(3), the two cubics are the Lucas cubic K007 and the Thomson cubic K002 and when X = X(3065), they are the Parry cubic K311 and its complement K453.

When X = X(1), we find K455 and K637 = pK(X1100, X2) cited in ETC, preamble of X(3647).

The complement of K636 is K752.