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X(1) singular

vertices of the circum-tangential triangle

infinite points of K024

There is only one isogonal cK60 with node I = X(1). See Special isocubics ยง7.2.3. It is K085 a member of the class CL004 of cubics.

Its root is X(4383) intersection of the lines X(2)X(6), X(43)X(55) and many others.

It is the locus of point M such that the circle with diameter MM* (isogonal conjugate) is orthogonal to the circle centered at O passing through I or, equivalently, the locus of M such that M and its isogonal conjugate M* are conjugated with respect to this same circle.

It is an equilateral unicursal cubic with three real asymptotes forming an equilateral triangle with center G whose circumradius is 4/3 r where r is the radius of the incircle. The incenter X(1) is the node.

It is a member of the pencil of cubics generated by K024 and the union of the line at infinity and the circumcircle.

See a generalization in the page K687.