Rename $C \equiv O.$ Isogonal conjugation with respect to $\triangle ABC$ takes its circumcircle $\omega$ into the line at infinity, takes centroid $G$ into the symmedian point $K$ of $\triangle ABC$ and circumconics $\mathcal{K} \equiv ABCG$ into a pencil $\ell$ of lines passing though $K.$ Since $K$ is inside $\omega,$ then each $\ell$ cut $\omega$ at two points $\Longrightarrow$ each $\mathcal{K}$ has two points at infinite, i.e $\mathcal{K}$ are all hyperbolas. According to About Nine-point conic, locus of centers of $\mathcal{K}$ is the conic passing through midpoints of $BC,CA,AB$ and midpoints of $GA,GB,GB,$ i.e. the Steiner inellipse of ABC.