Lines formed from the sweep of the incenter

The "sweep" of a point is a pattern formed by the direct product of the orbit of a point with the orbit of the incenter. It is an organizing scheme involving many points. Often a set of these points is colinear. If so, the direct product pattern in the point is repeated for lines. When points have patterns, lines do too. This page shows the repeated application of a projective transformation on lines.

These four points are colinear and are part of the sweep of the Nagel point. The point is given followed by its Kimberling X-number. We begin with the G—Io line and use four point on it. There are of course more which we can include at will. Geometrically what happens under this transformation is that the line rotates around one of the points on the curve enveloping a curve discussed much on these web pages.

Io (1) — G (2) — No (8) — :b s_b² : (200)

So are these

K (6) — Io (1) — Mo (9) — :b² s_b² : (220)

and

pIo (31) — K (6) — pNo (9) — :b³ s_b² : (1253)

and

pK (31) — pIo (6) — pMo (9) — :b⁴ s_b² : (?)

The coordinates of the points in each row are multiplied by b, representing the projective transformation that preserves A, B, and C while taking G to Io. The lines shown in red) envelope a curve very much like the orbit of the incenter itself.