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X(2), X(4), X(278), X(281), X(393), X(459), X(1249), X(3346), X(6523), X(7952), X(40836), X(40837), X(40838), X(40839)

MaMbMc : medial triangle

HaHbHc : anticevian triangle of H

infinite points of pK(X2, X6527)

points of pK(X2, X393) on the Steiner ellipse

other points below

For any point M on the Lucas cubic K007, the barycentric product H x M and quotient H ÷ M lie on K879 = pK(X393, X2). These two points are X393-isoconjugates hence collinear with X(2).

It follows that K879 is anharmonically equivalent to the Thomson cubic K002. See Table 21.

K879 is :

• the isogonal transform of pK(X577, X394), where X(577) and X(394) are the barycentric squares of X(3) and X(63) respectively,

• the isotomic transform of pK(X3926, X3926), where X(3926) is the barycentric square of X(69),

• the complement of pK(X2, X6527). This latter cubic is the image of the Lucas cubic K007 under the symbolic substitution SS{a –> SB SC}.

K879 is also :

• the pK with pivot X(4), isopivot X(2) with respect to HaHbHc,

• the pK with pivot X(393), isopivot X(6523) with respect to MaMbMc.

K879 contains the following centers now added to ETC (2020-12-30) :

• P1 = X(40836) = X(4) x X(189), isogonal conjugate of X(7078),

• P2 = X(40837) = X(4) x X(5932),

• P3 = X(40838) = X(4) x X(1034),

• P4 = X(40839) = X(4) x X(14362).

Note that the line P2P3 passes through X(2) and the line P1P3 passes through X(4).