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X(39), X(512), X(1015), X(1500), X(2028), X(2029), X(14090)

A1, B1, C1 : extraversions of X(1015)

A2, B2, C2 : extraversions of X(1500)

U, V, W : vertices of the cevian triangle of X(39)

A', B', C' : vertices of the anticevian triangle of X(512)

other points below

The Steiner and Brocard inscribed ellipses meet at four points namely X(1015) and its three extraversions A1, B1, C1.

The triangle A1B1C1 is perspective

  • to ABC at X(1500),
  • to the cevian triangle UVW of X(39) at P2 = X(14991) = X(39)/X(1015) = X(39)-Ceva conjugate of X(1015),
  • to the anticevian triangle A'B'C' of X(512) at X(1015),
  • to A2B2C2 at X(39).

K554 is the pivotal cubic that contains all these points and also :

  • P1 = X(14990) = X(39)/X(512), perspector of the triangles UVW, A'B'C',
  • P3 = X(14992) = X(39)/X(1500), perspector of the triangles UVW, A2B2C2,
  • P4 = X(39)/X(2028) and P5 = X(39)/X(2029). The tangents at the two points X(2028), X(2029) to the Brocard ellipse are perpendicular to the Brocard axis.
  • P6 and P7 intersections of the Steiner in-ellipse and the line X(2)X(39). The tangents at these two points to this ellipse are perpendicular to the Brocard axis.

One of the asymptotes of K554 is the perpendicular at X(39) to the Brocard axis.

See the related K925, K927.


More generally, two inconics C(X), C(Y) with perspectors X, Y meet at four (all real or all imaginary) points lying on the pivotal cubic K(X, Y) whose pivot T is the crosspoint of X and Y, which is invariant under the isoconjugation with fixed point the intersection S of the trilinear polars of X and Y.

Recall that the trilinear polar L(S) of S is the fourth common tangent to C(X), C(Y).

K(X, Y) contains the vertices A'B'C' of the cevian triangle of T, the vertices QaQbQc of the anticevian triangle of S.