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X(13), X(14), X(671), X(56395)

S1, S2 on the Steiner ellipse and on the line {51,512}

T1 = X(13)S1 /\ X(14)S2

T2 = X(13)S2 /\ X(14)S1

The barycentric products S1 x S2, T1 x T2 are X(1989)

Geometric properties :

For every point P on the line {5,523}, –the perpendicular bisector of OH – , one can find a stelloidal nK with pole X(1989), root P.

These cubics are in a same pencil which contains :

• K139, a central cubic with P = X(14566),

• K205, a nK0 with P = X(523),

• K669, the McCay stelloid spK(X3, X125) with P = X(14592).

K1383 is obtained with P = X(15475), on the line {51,512} and on the Lester circle.

The radial centers of all these stelloids lie on the circle with center X(38724), radius R/3, passing through X(5627), X(14846), X(14644) corresponding to the three cubics mentioned above. This circle is the homothetic of (O) under h(X125,-1/3) or h(X265,1/3).

That of K1383 is X = a^10-2 a^8 b^2+4 a^6 b^4-2 a^4 b^6+7 a^2 b^8-2 b^10-2 a^8 c^2-2 a^6 b^2 c^2-29 a^2 b^6 c^2+7 b^8 c^2+4 a^6 c^4+45 a^2 b^4 c^4-5 b^6 c^4-2 a^4 c^6-29 a^2 b^2 c^6-5 b^4 c^6+7 a^2 c^8+7 b^2 c^8-2 c^10 : : , SEARCH = 0.442902848071454. X is the homothetic of X(39446) under h(X125,-1/3).

The polar conic of X(671) is the rectangular hyperbola (H) passing through X(13), X(14), X(671), X(892), X(5466), S1, S2 hence the tangents to K1383 at X(13), X(14), S1, S2 concur at X(671).

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Remark : with P = X(43084), the stelloid has its asymptotes parallel to those of K024 and these concur at X(9140), the reflection of X(2) in X(125). These two cubics have six other common finite points which lie on the rectangular circum-hyperbola passing through X(691).