We use the notation of the Simson and Hirst pages.
K137 : cK(#I,X(513)
R = [a(b-c)] = X(513), F = I, P = R, U = [a(b-c)2] = X(244).
This is of Hirst type with R the circumconic with perspector R (in TCCT) and F is the
circumconic with perspector I, centre X(9).
New facts:
The nodal tangents are FT1, FT2 where T1, T2 are the intersections of R and
the antiorthic axis - the tripolar of F.
The line of F-harmonics is the tangent to R at I - the line IK.
K147 : cK(#X(110),K)
R = K, F = [a2/(b2-c2)] =X(110), P = R, U = [(a2(b2-c2)] = X(512).
This is of Hirst type with R the Circumcircle, F the circumconic with perspector X(110).
New facts:
The nodal tangents are FT1, FT2 where T1, T2 are the intersections of R and
the tripolar of F - the Brocard axis. These are X(1379), X(1380).
Note that U is at infinity. Therefore the tangents to R are parallel so T1,T2 are antipodes.
F is also on R - the Circumcircle - so the nodal tangents are perpendicular (well-known).
Of course the Hirst inverses are F.
The circumconics F and R meet at X(691), so X(691) and its isoconjugate lie
on the cubic. (Known - misprint in Gibert says X(651)). The isoconjugate is unlisted, but
lies on the tripolar of F.
The line of F-harmonics is the tangent to the Circumcircle at X(110) - X(110)X(351).
K162 : cK(#K,O)
R = O, F = K, P = X(32), U = X(184).
Not of Hirst type as R ≠ P.
L002 : cK(#G,X(512)
R = X(512), F = G, P = X(512), U = X(1084) - the centre of R, and on the Steiner Inellipse.
This is of Hirst type with R with perspector X(512) (in TCCT), F the Steiner Ellipse.
Since F = G, it is also of mid-point type, with line L = GX(39).
The cubic contains X(538), the infinite point of L, and X(385) the G-Hirst inverse of K on R,
and hence also its isotomic conjugate X(1916).
The nodal tangents are FT1, FT2 where T1, T2 are the intersections of R and
the line at infinity - the tripolar of F - i.e. are parallel to the asymptotes of R.
L003 : cK(#G,X(514)
R = X(514), F = G, P = X(514), U = X(1086) - the centre of R, and on the Steiner Inellipse.
Here R is the Yiu hyperbola - through G and X(7).
This is of Hirst type, with F the Steiner ellipse.
Since F = G, it is also of mid-point type with line L = GI.
It contains X(519), the infinite point of GI, and hence its isotome X(903) (on the Steiner inellipse).
Note that X(903) is the fourth intersection of R and F, so is fixed by the inversion.
The cubic also contains X(447), X(350), X(291) as the F-Hirst inverses of X(27), X(75), X(335).
Now, X(350),X(291) are isotomic conjugates, so X(75), X(335) are antipodes, as are X(2),X(903).
The nodal tangents are FT1, FT2 where T1, T2 are the intersections of R and
the line at infinity - the tripolar of F - i.e. are parallel to the asymptotes of R.
L004 : cK(#K,X(512)
R = X(512), F = K, P = X(512), U = [a2(b2-c2)2]
R is listed in TCCT. It contains, among others, G,K,X(25),X(111).
This is of Hirst type, with F the Circumcircle.
The line of F-harmonics is the Brocard axis. The tripolar of F is the Lemoine axis.
Taking F-Hirst inverses, the cubic contains X(385),K,X(232),X(111).
Taking isoconjugates, we get further points X(248) and X(187) - the intersection of the above axes.
The point X(111) is the F2 for the cubic.