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X(98), X(511), X(10221)

Brocard points Ω1 and Ω2

foci of the K-ellipse (inellipse with center K when the triangle ABC is acutangle)

foci of the inconic with center O, perspector X(69)

more points and other figures below

K019 = nK0(X6, X647) is the circular isogonal focal nK0 with root X(647) and singular focus the Tarry point X(98). See the related cubics K048, K417, K418. It is also spK(X511, X6) in CL055.

K019 is the locus of :

  1. foci of inscribed conics in ABC whose center is a point on OK. See also Z+(O) = CL025.
  2. point M such that M and M* are conjugated with respect to the Jerabek hyperbola. See Table 4. Hence, K019 is invariant in the transformation denoted JS which maps a point M to the intersections of the polar lines of M in the Jerabek and Stammler rectangular hyperbolas. See also Table 62.
  3. contacts of tangents drawn from the Tarry point X(98) to the circles centered on the line through the Brocard points and orthogonal to the Brocard circle.
  4. intersections of the circles centered at M on the Brocard axis passing through the Brocard points and the line MX(98).


Points on K019

K019 contains :

• A, B, C.

• the intersections of the sidelines of ABC and the trilinear polar of the root X(647).

• X(511), the infinite point of the Brocard axis OK. The real asymptote is the parallel at the Steiner point X(99) to the Brocard axis.

• two other points on this line OK which are imaginary and also lie on pK(X6, X511).

• the circular points at infinity.

• the Brocard points Ω1 and Ω2.

• X(98), the singular focus. The tangent at this point meets the real asymptote at X on the cubic and on the circle through X(98), Ω1 and Ω2.

• the orthogonal projection E1 of X(98) on the line through Ω1 and Ω2.

• its isogonal conjugate E1*, common tangential of the Brocard points.

• the four foci of the "K-ellipse", the inconic with center K.

• the four foci of the inconic with center O.

• three points on the Euler line which also lie on Q002, the Euler-Morley quartic, one of them E2 = X(10221) is always real.

• three points on the Jerabek hyperbola (apart A, B, C) which also lie on Q003, the Euler-Morley quintic, one of them E3 is always real. E3 is the isogonal conjugate of E2 = X(10221).

• the six intersections of the altitudes and the cevian lines of O (Wilson Stothers).

• two points S1, S2 on the line X(2)X(98) which are now X(13414), X(13415) in ETC (2017-05-22). See also Table 62.

These points are always real and lie on

  • the Brocard circle,
  • the parallels at K and O to the asymptotes of the Jerabek hyperbola or to the axes of the K-ellipse,
  • the parallels at X(1113), X(1114) to the Brocard axis.
  • the McCay hessian cubic K048.

• their isogonal conjugates S1*, S2* which lie on

  • the line through X(98) and X(648),
  • the circum-hyperbola through X(6), X(232), X(250), X(262), X(264), X(325), X(511), X(523), X(842), X(1485), X(2065).
  • the circle through H, Ω1 and Ω2.
  • the parallels at S2, S1 to the Brocard axis.


Hessian and prehessians of K019


The hessian (H) of K019 contains O, H, X(648), the intersection of the Euler line with the trilinear polar of the root X(647). Note that the polar conic of X(648) in K019 is the union of these two latter lines.

K019 has always three real prehessians (H1), (H2), (H3) and one of them – say (H1) – is a stelloid with radial center X(98), the singular focus of K019.

The cubics – apart (H1) – share the same orthic line, namely the Brocard axis.

Each prehessian is associated with a natural conjugation Fi on K019 which swaps a point and the center of its polar conic in (Hi).


Conjugations on K019


K019 is the isogonal pK with pivot X(511) with respect to the triangle Ω1 Ω2 X(98).

Hence, K019 contains the in/excenters of this latter triangle which are four centers of anallagmaty on the cubic. These points lie on the polar conic of X(511), a rectangular hyperbola with center X(2968), passing through X(39) and X(512).

It follows that K019 is now invariant under ten conjugations namely :

• isogonal conjugation in ABC,

• isogonal conjugation in Ω1 Ω2 X(98),

• transformation JS (see Table 62) which maps a point M to the intersections of the polar lines of M in the Jerabek and Stammler rectangular hyperbolas,

• three involutions F1, F2, F3 above,

• four inversions with center one of the in/excenters of triangle Ω1 Ω2 X(98) that swap another in/excenter and the corresponding vertex of triangle Ω1 Ω2 X(98).