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on K1384-a :

X(2), X(6), X(1380), X(6189)

on K1384-b :

X(2), X(6), X(1379), X(6190)

other points below

Geometric properties :

K1384-a and K1384-b are both Thomson centroidal cubics as in CL040, namely TC(X1380) and TC(X1379) respectively.

They are also the non-pivotal cubics nK0(Ω1 = X1380, R1 = X30509) and nK0(Ω2 = X1379, R2 = X30508).

They are also the sympivotal cubics spK(X2, Q1 = X39023) and spK(X2, Q2 = X39022) as in CL055.

They belong to the pencil of spK(X2, Q on X2,X6) which are also the Thomson centroidal cubics TC(S) where S = X3,X6 /\ X30,Q. Hence, every cubic is a K0 and passes through the vertices of the pedal triangle of S. When Q is on (O), this pedal triangle becomes a Simson line and the cubic becomes a nK0 and one of the cubics K1384-a and K1384-b.

All these cubics pass through A, B, C, G, K (double) and the infinite points of K002 which is the only pK of the pencil.

This pencil also contains :

• the union of the line at infinity and the circum-conic passing through G and K, which can be seen as a degenerate nK0.

• K281 = spK(X2, X597) = TC(X182), nodal cubic with node X6.

• K314 = spK(X2, X6) = TC(X6), central cubic with center X6, the only K+ of the pencil.

***

Other points on K1384-a and K1384-b

• A1, B1, C1 and A2, B2, C2 , on the trilinear polar of the roots R1, R2 which are the asymptotes of the Kiepert hyperbola and also the Simson lines of Ω1, Ω2.

• points on (O) and the parallels at G to these asymptotes which are the axes of the Steiner ellipses. These points lie on the respective polar conics of Ω1 = X(1380), Ω2 = X(1379) which are rectangular hyperbolas with common points X(6), X(110), X(2574), X(2575), passing through X(1380), X(6189) and X(1379), X(6190) respectively.

• points on the Steiner ellipse and the parallels at K to these asymptotes.

• the points on the Kiepert hyperbola – apart A, B, C, G – lie on the parallels to the Euler line passing through Ω1, Ω2 respectively.