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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.

Tucker circle


ABC (trivial case)


second Lemoine circle


with center X(61) or X(62)


degenerate into the line at infinity and the Lemoine axis




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).