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X(2), X(6), X(30), X(698), X(3070), X(3071), X(11542), X(11543), X(39641), X(39642)

X(39641), X(39642) : imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola

X(42413) → X(42421), X(42576) → X(42579)

Reference : J. Cullen, Mathematical Questions and Solutions from the Educational Times 7 (1905) 24, #14754.

A Tucker circle meets the sidelines of ABC at six points forming the two Tucker triangles AbBcCa and AcBaCb. Note that any line Xy-Yx is parallel to a sideline of ABC and that any line Yx-Zx is antiparallel to a sideline of ABC. For any Tucker circle, there is a conic with center W inscribed in both Tucker triangles and the locus of its center is the Cullen cubic K369.

K369 is a nodal cubic with node X(30), the infinite point of the Euler line. The nodal tangents (asymptotes) are the parallels at X(61), X(62) to the Euler line.

K369 has a third real asymptote with infinite point X(698), the even (- 4, - 2) infinity point. Thus, this asymptote is parallel to the lines X(6)X(194), X(75)X(257), X(76)X(141).

K369 contains G (with tangent the line GK) and K (with tangent the line KX(20)).

The following table gives the correspondence between a Tucker circle and the point W on K369.

The other "classical" Tucker circles give very complicated points W.

Note that the line passing through W and the center T of the corresponding Tucker circle is parallel to the Euler line.

***

K369 is a KHO-curve, see K1191 and CL075 for a generalization. Analogous cubics are K1197, K1198 and K1199, the Cullen (second, third, fourth) cubics.

The KHO-eaquation of K369 is : x^2 (2y + z) - 3 (2y - z) (y + z)^2 = 0

and that of its Hessian is : x^2 (2y + 3z) + 9 (2y + z) (y + z)^2 = 0.

This latter curve passes through X(6), X(20), X(30) and X(42584) → X(42589).