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X(8), X(10), X(80), X(369), X(519), X(6740), X(7026), X(7043), X(36590), X(36926), X(36927), X(36928), X(36929), X(36930), X(36931), X(36932), X(36933), X(36934), X(36935), X(36936), X(36937), X(36938)

Geometric properties :

K311 and K1150 generate a pencil of cubics passing through A, B, C, X(8), X(80), X(369), X(519), Z1, Z2 which contains a third pK namely K1151 and a circular cubic. See Table 42 for further details.

X(36910) is the barycentric product X(8) x X(80) hence X(8) is the isopivot. The tangents at A, B, C, X(80) pass through X(8) and the polar conic of X(8) in K1151 is the Feuerbach hyperbola.

Remarks :

• The circular cubic above also passes through X(4), X(1698).

Its singular focus F= X(36939) is on the lines {106, 9624}, {114, 2796}, {355, 2802}, {1054, 7988}, {2827, 4010}, {8055, 9519}, {10164, 11814}, {10283, 11717}.

• K1147 and K1151 generate a pencil of cubics passing through A, B, C, X(369), X(519) counted twice, X(36590), Z1, Z2 which contains a third pK namely pK(X4370, X6224) also passing through X(1145), X(6224) and a trident.