Isogonal Conjugate of line through Centroid

This sketch shows the isogonal conjugate of line through centroid G is the hyperbola that passes through vertices of triangle. When line is a median the conjugate is the symmedian. K is the symmedian point. P is a point on the line and P' its isogonal conjugate. Putting P at a vertex gives you the symmedian.
Inspired by the work of Steve Phelps @giohio