Building upon the projective framework established in Part I, this article investigates the Möbius isogonal cubic and its inherent symmetries. We define this class of cubics using Möbius transforms and prove their invariance under polar circle inversions and isogonal conjugacy relative to the orthic triangle of an associated orthocentric system. The article characterizes general cubic curves within a projective framework based on the interaction between involutions, perspectivity, and conjugacy. By generalizing the properties of the Möbius isogonal cubic, we demonstrate via projective transformations that any cubic can be represented as a locus of points defined by involutions. Furthermore, we establish an alternative characterization of cubics as the locus of conjugate pairs viewed through a central perspectivity, relative to a triangle system and an anticevian configuration of four points on the curve. Finally, we explore the concept of triangle polarity.
Hussein Khayou (Mon,) studied this question.