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X(54), X(112), X(115), X(826)

Brocard points

A', B', C' : vertices of the cevian triangle of K

Consider two points X(t), Y(t) with trilinear coordinates X(t) = cos(A + t) : : and Y(t) = sin(A + t) : : . These points lie on the Brocard axis. See Table 38.

For any real number t,

• X(t) and Y(t) are conjugated with respect to

  1. the Kiepert hyperbola, the locus of the isogonal conjugates of X(t) and Y(t),
  2. the inscribed conic with perspector X(59), the isogonal conjugate of X(11). This conic is the locus of points with trilinear coordinates 1 ± cos(A ± t) : : or 1 ± sin(A ± t) : : ,
  3. any conic of the pencil generated by these two latter conics.

• the circle with diameter X(t)Y(t) contains the Brocard points,

• the circum-conic passing through X(t), Y(t) contains X(54), the isogonal conjugate of X(5).

These two curves meet at two other points collinear with X(115) that lie on the cubic K629.

K629 is a circular cubic with singular focus X(14675) whose isogonal transform is K630.

These two cubics generate a pencil of circular circum-cubics passing through the Brocard points and two other (not always real) points which are the common points of the line X(110)X(512) and its isogonal transform. See a figure at the page K630.

This pencil is invariant under isogonal conjugation and contains K021 = pK(X6, X512), nK(X111, X111, X523), nK(X187, X6, X110) and the isogonal focal cubic nK(X6, X187, X3906*) where X3906* is the isogonal conjugate of the infinite point X3906.