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K1238

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X(3), X(30), X(599), X(2574), X(2575), X(5895), X(10510), X(10602), X(21230), X(22115), X(41198), X(41199), X(44782)

X(44747) → X(44758), X(44786) → X(44791)

other points below

Geometric properties :

K1238 is defined in ETC, article X(44747).

K1238 is a crunodal cubic with node X(3) and nodal tangents passing through X(74), X(125).

K1238 has three real asymptotes namely those of the Jerabek hyperbola (J) and the parallel at X(110) to the Euler line. The tangential of X(30) is X(22115).

The satellite line of the line at infinity passes through X(3564), X(5622), X(10257) and obviously X(22115) and the two points of K1238 on the asymptotes of (J).

K1238 meets the sidelines of ABC at three real points A', B', C', on the trilinear polar (L) of X(525), and three pairs of always imaginary points. These points lie on the isotropic lines passing through X(110) and the midpoints of the three pairs are collinear on the parallel to the Euler line at X(113).

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Other analogous cubics

Recall that, if P is a point with pedal triangle A1B1C1 with respect to the antimedial triangle, then A2, B2, C2 are the reflections of A1, B1, C1 in A, B, C respectively. C(P) is the circumcircle of A2B2C2 with center Ω(P).

As mentioned in ETC, article X(44747), when P lies on the Euler line of ABC then Ω(P) lies on K1238. In this case, C(P) passes through X(110).

These properties also hold when P lies on the hyperbola which is the homothetic transform of the Jerabek hyperbola under h(X20, 2).

Now, when P lies on a line (∆) passing through H, we find an analogous unicursal cubic passing through the infinite point ∞(∆) of (∆), meeting the line at infinity again at two points of the rectangular circum-hyperbola (H) passing through the isogonal conjugate of ∞(∆). In other words, this hyperbola is the isogonal transform of the parallel at O to (∆).

The node N lies on (∆) and on the line passing through O and the isogonal conjugate of ∞(∆). In other words, N is the Cundy-Parry Psi image of ∞(∆), hence N lies on K028. See CL037.

K1238 is obtained when (∆) is the Euler line and (H) is the Jerabek hyperbola, so K1238 can be renamed the Jerabek HM (crunodal) cubic.

When (∆) passes through X(69) and X(511), (H) is the Kiepert hyperbola and K1239 is the Kiepert HM (acnodal) cubic with node X(76).

When (∆) passes through X(8) and X(517), (H) is the Feuerbach hyperbola and K1240 is the Feuerbach HM (cuspidal) cubic with node X(8).