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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) = pK(#S, T) 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 of the cevian triangle A'B'C' of T and the vertices of the anticevian triangle SaSbSc of S.


Conversely, if S = p : q : r and T = u : v : w are two given points and if (K) = pK(#S, T), the construction of X, Y and therefore that of C(X), C(Y) can be realized as follows. See figure below.

1. The polar conic C(S) of S in (K) passes through the harmonic conjugates of S with respect to A and Sa, B and Sb, C and Sc and also through S with tangent the line ST.

2. C(S) is actually the bicevian conic C(P1, P2) where P1, P2 are the common points of the line ST and the circum-conic Γ(S) with perspector S.

These two points are real and distinct if and only if : (q r u + r p v + p q w)^2 - 3 p q r (r u v + q u w + p v w) > 0.

3. Let L(T) be the polar line of T in Γ(S) which is also the trilinear polar of the cevapoint of S and T. L(T) meets Γ(S) at X and Y. These points X, Y are real and distinct if and only if : p q r (p v w + q w u + r u v) < 0.

4. C(X), C(Y) and Γ(S) meet at four points which lie on (K). The diagonal triangle of these four points is SaSbSc.

5. (K) meets C(X), C(Y) again at two pairs of points lying on the lines TX, TY respectively.

6. Note that the isoconjugation in ABC with fixed point S maps X, Y to X*, Y* which are the contacts of C(X), C(Y) with their fourth common tangent, namely the trilinear polar L(S) of S.

7. (K) is also a pivotal conic with respect to SaSbSc with pivot S and isopivot T.