The "pro" operation

This operation is a projective transformation that is the isogonal conjugate of the isotomic conjugate. It also serves as an example of the barycentric product K•Q, where K is the symmedian point. Conway has names this the "pro" operation, which is short for "projective." This operation takes G to K, medians to symmedians, the Steiner circumellipse to the Circumcircle, and the Seiner inellipse to the Brocard inellipse. It also preserves the sweep of the incenter, which is the curve supporting all barycentric powers of the incenter. It also takes the Steiner point to the Focus of the Kiepert parabola.

The arrows in this picture show the direction and distance the "pro" operation takes points in various regions around ABC. The blue points on the black curve are powers of Io and Ia. The pro operation preserves this structure.

For this shape triangle, the pro operation accumulates at B (an attractor) and is repelled from the other two vertices. The 3 vertices are the fixed points. This terminology indicates that this operation can be thought of as a dynamical system.

The pro operation is strength preserving.

This picture is a good illustration of a projective transformation that preserves the three vertices of the triangle.

The pro operation takes ABCG to ABCK.
It also takes the Cevians (internal and extranal) of G to those of K.
It also takes G—K, the symmedian track into O—K—pK, the Brocard Meridian Line.
It also takes the conics with perspector Q to those with perspector pQ. In particular it takes the Steiner ellipse into the circumcircle.
It also preserves the orbit of the incenter, the black line in the picture.