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X(4), X(107), X(523), X(7471), X(14220), X(14221), X(14222), X(14223), X(14224), X(14225)

The antigonal image of a line is, in general, a bicircular quintic passing through A, B, C, H (which are singular) and the point at infinity of the line. More precisely, if the line is L = px + qy + rz = 0, the barycentric equation of this quintic can be written under the form :

COR^2 L - 2 COR Linf [px(SB y + SC z) + cyclic] + 4 Linf^2 (p SB SC + cyclic) xyz = 0, where COR = a^2 yz + cyclic = 0 is the circumcircle and Linf = x+y+z = 0 is the line at infinity.

This quintic decomposes into the circumcircle and a circular nodal circum-cubic with node H if and only if L passes through H. In this case, the cubic is the orthoassociate (inversive image in the polar circle) of a conic passing through H, the vertices of the orthic triangle and those of the cevian triangle of the H-isoconjugate of the infinite point of the direction perpendicular to L.

When L is the Euler line, this cubic is the Ehrmann strophoid K025.

The term in xyz vanishes if and only if L is the line HK and the corresponding cubic is the Gigha cubic K288. See also K289, the Canna cubic obtained when L passes through X(69).

The cubic is a nK if and only if L is the perpendicular at H to the Euler line : it is the Iona cubic denoted by K186 (see figure above). Its pole is X393 (barycentric square of the orthocenter) and its root is R = (4SA^2 - b^2c^2)/SA^2 : : = 3 – tan²A : : . R is the trilinear pole of the line X(526)-X(1986), now X(14165) in ETC (2017-09-01).

The tangent at X(107) meets the real asymptote at X on the curve.

K186 is the orthoassociate of the conic passing through H, X(125), the vertices of the orthic triangle.

See a generalization and other related cubics in Table 43.