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X(99), X(385), X(512), X(694)

infinite points of the Steiner ellipse and their isogonal conjugates

other points below

The vertices A1', B1', C1' of the reflected first Brocard triangle are the reflections of those of the first Brocard triangle about the sidelines of ABC. See Table 32 for further properties. The isogonal conjugates of A1', B1', C1' are denoted A1", B1", C1".

The three triangles ABC, A1'B1'C1', A1"B1"C1" are two by two (simply) perspective with the same perspectrix, namely the trilinear polar (L) of X(3329), and perspectors X(83), X(39), X(384).

K020 is an isogonal pK passing through the vertices of these three triangles (and many other points). Similarly, there is an isogonal nK with the same property and this nK is K797.

K797 passes through the traces U, V, W of (L) on the sidelines of ABC, A1'B1'C1', A1"B1"C1". This gives a lot of triads of collinear points on K797 namely A B1' C1", A B1" C1', U B1' C1', U B1" C1", etc.

K797 has one real asymptote perpendicular to the Brocard axis and meeting K797 again at X whose isogonal conjugate X* is the intersection of the lines X(99)X(512), X(385)X(694). The two imaginary asymptotes meet at X(694) hence the polar conic of X(694) is an ellipse homothetic to the Steiner ellipse.

The polar conic (C) of X passes through X(99), X(385), X(512), X(694) hence K797 also contains the vertices X*, M1, M2 of the diagonal triangle of the quadrilateral. M1 = X(99)X(385) /\ X(512)X(694) and M2 = X(99)X(694) /\ X(512)X(385) are two isogonal conjugate points.

These points X, X*, M1, M2 are unlisted in ETC with SEARCH = -6.233881721061544, 0.3486465075577163, 2.759132445811658, -52.0425226179561 respectively.

It follows that K797 is a pK with respect to X*M1M2 whose pivot is X. Hence K797 also passes through XX* /\ M1M2, XM1 /\ X*M2, XM2 /\ X*M1.

K020 and K797 generate a pencil stable under isogonal conjugation. This pencil contains no other real psK nor nK.


K797 is also a member of the pencil of isogonal nKs containing K017 which is generated by two simple decomposed cubics namely :

• the union of the circumcircle (O) and the line at infinity L∞,

• the union of the Steiner ellipse (S) and the Lemoine axis.

Each cubic of this pencil passes through A, B, C, X(99), X(512), the infinite points of the Steiner ellipse and their isogonal conjugates on the Lemoine axis. Its root lies on the line X(2), X(6).

There is one fixed point Z whose polar conic in every cubic is a (possibly degenerated) circle.

Z = a^2 (-a^2 b^6+3 a^4 b^2 c^2-2 a^2 b^4 c^2+2 b^6 c^2-2 a^2 b^2 c^4-b^4 c^4-a^2 c^6+2 b^2 c^6) : : , SEARCH = 7.71122335962109.

Z lies on the lines {2,694}, {3,5106}, {6,6786}, {110,1691}, {111,5888}, {695,7807}, etc, and on the Jerabek hyperbola of the Thomson triangle.

Special cubics of the pencil :

K017 is the only nK0.

• four nodal cubics cKs with node the in/excenters of ABC.

• one K+ with asymptotes concurring at Z.

• one cubic (figure below) invariant under an oblique axial symmetry with root the barycentric product X(385) x X(5969), SEARCH = -11.6884825066246.

X(512) is a flex at infinity and the real asymptote passes through X(39).

The axis of symmetry is the line X(99), X(187).

Any parallel to the real asymptote meets the axis at M, the cubic at X(512), M1, M2 such that M is the midpoint of M1M2.

The axis meets the cubic at X(99) and two isogonal conjugate points P1, P2 hence also lying on the circum-hyperbola (H) passing through X(512), X(671), X(694), X(729). The tangents to the cubic at these three points are obviously parallel to the asymptote.