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X(2), X(4), X(278), X(281), X(393), X(459), X(1249), X(3346), X(6523), X(7952)

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 X2.

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 X577 and X394 are the barycentric squares of X3 and X63 respectively,

• the isotomic transform of pK(X3926, X3926), where X3926 is the barycentric square of X69,

• 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 X4, isopivot X2 with respect to HaHbHc,

• the pK with pivot X393, isopivot X6523 with respect to MaMbMc.

K879 contains the following non ETC centers :

• P1 = X4 x X189, isogonal conjugate of X7078, on the lines X1-X281, X2-X280, X4-X57, etc, SEARCH = 0.921269793787530,

• P2 = X4 x X5932, on the lines X2-X342, X4-X57, X9-X1767, etc, SEARCH = 0.184167170487233,

• P3 = X4 x X1034, on the lines X2-X342, X4-X282, X9-X1249, etc, SEARCH = 3.27842948799762.

Note that the line P2P3 passes through X2 and the line P1P3 passes through X4.