Two Sweeps related to the incenter

This is one of my favorite pictures. All these points came from studying the Shiffler and co-Spieker points. The pattern of points inside the triangle follows a distinct path, which is the invariant curve of the projective transformation that takes ABCG to ABCIo. The point at infinity on the dual of Io has coordinates : c – a : Its isotomic conjugate is on the Steiner ellipse and causes the external sweep of points.