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X(2), X(4), X(67), X(69), X(316), X(524), X(671), X(858), X(2373), X(11061), X(13574), X(14360), X(14364) Ga, Gb, Gc vertices of the antimedial triangle Other points below 

The Droussent cubic is basically the only isotomic circular pK. Its pivot is X(316) reflection of X(99) in the de Longchamps axis and isotomic conjugate of X(67). Its singular focus F = X(10748) is the midpoint of X(4)X(14360) and the intersection of the lines X(3)X(126), X(5)X(111), X(30)X(1296). K008 meets : – its real asymptote (the line X(111)X(524)) at X on the line X(4)X(14360). – the circumcircle at A, B, C, X(2373) and the circular points at infinity. – the Steiner ellipse at A, B, C, X671 and two imaginary points on the de Longchamps axis (which contains X(858) as well). K008 is welldocumented in Droussent's original paper. See the bibliography. See also Droussent central cubic, Droussent medial cubic and K007, property 8. The isogonal transform of the Droussent cubic is K108 = pK(X32, X23) and its antigonal transform is K273. Locus properties :




K008 has always three real prehessians P_{1}, P_{2}, P_{3}. The centers of the polar conics of X(524) with respect to these prehessians are M1, M2, M3 where M1 = A X(671) /\ Ga X(69), M2 and M3 likewise. Furthermore the polar conic of X(524) in K008 is a rectangular hyperbola. It follows that K008 is the isogonal pK with pivot X(524) with respect to the triangle M1M2M3. Thus it must pass through the in/excenters of this latter triangle with tangents parallel to the real asymptote. X is then the isogonal conjugate of X(524) in M1M2M3 and the singular focus F is its antipode on the circumcircle of M1M2M3. 

This can easily be generalized for any point M on K008 as far as M is not a flex. If M1, M2, M3 are the centers of the polar conics of M with respect to the prehessians then K008 is a pivotal cubic with pivot M in the triangle M1M2M3. The tangents at M1, M2, M3 (and M) concur at the isopivot M' and the polar conic of M' contains these five points. Examples : • when M = X2, M1M2M3 is the antimedial triangle and M' is X(316). • when M = X(67), M1M2M3 is the cevian triangle of X(316) and M' is the tangential of X(67). 
