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X(1), X(10), X(58), X(1794), X(1838)

imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola

The three internal bisectors AI, BI, CI of ABC are rotated about each corresponding vertex of ABC of a same angle t, all outwardly or all inwardly. The six rotated bisectors define a triangle NaNbNc which is perspective to ABC at P. See figure below.

When t varies, the locus of P is the nodal cubic cK(#X1, X101) = K588.

K588 is a nodal isogonal non-pivotal cubic with node I = X(1). See Table 69.