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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 XyYx is parallel to a sideline of ABC and that any line YxZx 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 KHOcurve, see K1191 and CL075 for a generalization. Analogous cubics are K1197, K1198 and K1199, the Cullen (second, third, fourth) cubics. The KHOeaquation 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). 
