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All central nKs with center O = X(3) have their pole W on the trilinear polar of O and their root R on the trilinear polar of X(69).

W is the isogonal conjugate of the tripole of a line L passing through H. The asymptotes of the cubic are :

  • one (always real) which is perpendicular at O to L,
  • two (not always real) which are parallel at O to those of the conic isogonal transform of L.

For example, when the line is the Euler line, the cubic is K069 with three real points at infinity : X(523) and those of the Jerabek hyperbola.

See a generalization in the page P-conical cubics.

The figure 1 shows the cubic with pole X(684) and the circum-conic through O and X(76). The three points at infinity are X(511), X(525) and (b^2 - c^2)(b^4 + c^4 - a^2b^2 - a^2c^2) : : .

The figure 2 shows the cubic with pole X(520), root X(441) and the circum-conic through O and G. The three points at infinity are X(525) and those of the conic.