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X(2), X(4), X(13), X(14), X(542), X(1316)

two imaginary points S1, S2 on the Kiepert hyperbola, the orthocentroidal circle, the line X(115)X(125)

X = X(16279) = X(6)X(30) /\ X(542)X(2452)

Y = X(16280) = X(542)X(1316) /\ X(115)X(125)

Q₁ = X(10653) = X(2)X(13) /\ X(4)X(14)

Q₂ = X(10654) = X(2)X(14) /\ X(4)X(13)

K876 and the axial cubic K873 are the two members of the pencil described in Table 59 having the same circular polar conic (C), namely the circle passing through X13, X14, X868, X1316, X1640.

The real asymptote is the image of the Fermat axis under the homothety with center X(1316), ratio 2, a line passing through X(542), X(2452). K876 meets this asymptote at X = X(16279).

Note that K876 has two very simple real centers of anallagmaty which are X(2), X(4). It follows that :

• the tangents to K876 at these points are parallel to the asymptote and to the Fermat axis.

• K876 contains two imaginary points on the line X(523)X(1316). These are the other centers of anallagmaty.

• K876 is invariant under the two inversions i_G, i_H with pole X(2) – resp. X(4) – that swap X(1316) and X(4) – resp. X(2).

In particular, i_G (X13) = i_H (X14) = Q₁ and i_G (X14) = i_H (X13) = Q₂.

Locus properties

K876 is the locus of contacts of tangents drawn through X(1316) to the circles passing through the Fermat points X(13), X(14).

K876 is the locus of M such that the sum of directed angles (MG, MQ₁) + (MH, MQ₂) = 0 (mod. π).