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the cusp X = X(15724) described below

Let (C) be a circle with center O and radius k R, where R is the circum-radius. The locus of M such that M and its isogonal conjugate M* are conjugated with respect to (C) – i.e. such that M* lies on the polar line of M with respect to (C) – is an equilateral isogonal nK with root R on the line X(2)X(6).

When the real number k varies, all these cubics belong to a same pencil generated by K024 (obtained when k = 1) and the union of the line at infinity and the circumcircle (obtained when k = ∞). When (C) passes through the incenter X(1), we find the nodal cubic K085. See also K098, K105, K686.

Each (proper) cubic has three real asymptotes parallel to those of K024 and forming an equilateral triangle with center G. These asymptotes meet the cubic at three finite collinear points lying on the satellite line (S) of the line at infinity.

When k varies, (S) envelopes the cuspidal cubic K687.


The cusp X is the intersection of (L) = X(351)X(523) and the cuspidal tangent (T) which is the homothetic of the Lemoine axis under h(X2, 2/3).

(T) is actually the satellite line (S) of the line at infinity for the cubic K024 itself.

X(523) is a point of inflexion on K687 and the inflexional tangent is the line at infinity.

The Hessian of K687 splits into three lines namely (L) and (T) counted twice.

The first barycentric of X is :

(b^2-c^2) [6 a^4+b^2 c^2-2 a^2 (b^2+c^2)]

with SEARCH = 0.507422047673261.