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X(6), X(523), X(2574), X(2575), X(5968), X(8105), X(8106), X(52038)

vertices of the Thomson triangle

M1, M2 on the Fermat axis and on the rectangular circum-hyperbola through X(477)

K624 is a nK0 passing through the vertices of the Thomson triangle and having three real asymptotes concurring at G.

Two asymptotes are parallel to those of the Jerabek hyperbola and one is perpendicular to the Euler line.

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More generally, a nK0 passing through the vertices of the Thomson triangle must have its root on the line at infinity and its pole on the Lemoine axis. It always contains the Lemoine point K = X(6). See K625 for another example.

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Every nK0(X512, R), with root R on the Euler line, is a circum-cubic of a pencil with base-points A, B, C, X(6), X(523), X(2574), X(2575), X(8105), X(8106) with can be generated by two decomposed cubics, namely :

• the union of the line at infinity and the circum-conic passing through G and K,

• the union of the orthic axis and the Jerabek hyperbola.

The remaining points on (O) of any cubic are those of an isogonal pK with pivot on the line GK.

The asymptotes at X(2574), X(2575) concur at X on the trilinear polar of X(2966), a line passing through {2, 98, 110, 114, 125, 147, 182, 184, 287, 542, 1352, 1899, 1976, 2001, 3047, 3410, 3448, etc}.

The tangential Y of X(523) lies on the cubic and on the circum-conic with perspector X(3569), passing through {6, 232, 250, 262, 264, 325, 511, 523, 842, 1485, 2065, etc}.

The X(512)-isoconjugate Z of Y is obviously also on the cubic and on the trilinear polar of X(98), a line passing through {6, 523, 879, 1316, 1640, 2395, 2422, 2451, 2452, 2453, 3049, 3050, 3287, 3288, etc}.

K624 is the only cubic which is a nK0+ with three asymptotes concurring at G.

K1356 = nK0(X512, X23) is the only cubic which is a spK, namely spK(X110, X6593) where X(6593) is the midpoint of X(6), X(110). It follows that the points on (O) are the vertices of the Grebe triangle. See also Table 57. In this case X = X(182) and Y = X(14356).

See Q187 where nK0(X512, X403) is mentioned with X = X(1899) and Y on the lines {6, 1112}, {22, 250}, {24, 842}, {25, 523}, {107, 264}.

The pencil contains two other decomposed cubics into a line through X(6) and one of the points X(2574), X(2575) and the circum-conic which is its X(512)-isoconjugate.