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X(6), X(44), X(187), X(524), X(1990), X(3284)

Sa, Sb, Sc extraversions of X(44)

midpoints of ABC

other intersections of the Steiner inellipse and the Thomson cubic K002

more details below

For any root P ≠ G on the Thomson cubic K002, there is a nK0 having a pencil of circular polar conics. Its pole Ω must the barycentric product of P and the infinite point of the line GP. In other words, for any nK0(Ω, P) of this kind, there is a line of points whose polar conics are circles which obviously form a pencil. This line is called the circular line of the cubic. When Ω is X(187), X(1990) we obtain K604, K393 respectively.

The transformation P -> Ω maps the Thomson cubic to Q087.

If P and P* are two isogonal conjugates on K002 (hence collinear with G), the corresponding poles Ω and Ω' have the following properties :

  1. K, P, Ω and K, P*, Ω' are collinear,
  2. Ω and Ω' lie on the trilinear polars of P and P* respectively,
  3. the lines GPP* and ΩΩ' are parallel,
  4. the points K, Ω, Ω' lie on a same circum-conic whose isogonal transform is a line through G meeting the line ΩΩ' on the Steiner inellipse,
  5. the enveloppe of ΩΩ' is a tricuspidal quartic tritangent to the sides of ABC at their midpoints, very similar to the Steiner deltoid. See below.

Q087 passes through K which is a triple point with tangents passing through the points Ta, Tb, Tc of K002 on the circumcircle. Note that the tangents at A, B, C are the symmedians.
Q087a

This tricuspidal quartic is tritangent at the midpoints A', B', C' to the sidelines of ABC.

The cusps are the images of A, B, C under the homothety with center G, ratio 3/2.

The cuspidal tangents are the medians of ABC.
Q087b

Construction

Let T be a point on the Steiner inellipse and (T) the tangent at this point.

S is the isotomic conjugate of the infinite point of (T) and aT is the anticomplement of T.

The parallel at G to the line S-aT meets the Thomson cubic at P, P*.

The parallel at T to this same line contains Ω and Ω' which also lie on (C), the isogonal transform of the line TG.

This latter parallel envelopes the quartic, the point of tangency Q being the reflection about T of its second intersection T' with the Steiner inellipse.

The same construction applied to the reflection T1 of T about G gives another point Q1 on the quartic and the line QQ1 is also a tangent to the quartic.