It is a well-known fact (sometimes called the fundamental theorem of projective geometry) that given two lines and three points on each of the two lines, there is a unique projectivity between the two lines. It also appears to be a known fact that this projectivity can be extended to a collineation of the entire real projective plane.
I'm looking for a specific recipe (draw these lines here, construct the intersection, draw these other lines, etc.) for how to actually implement the collineation, without needing to pass to coordinates. It feels like there's almost such a recipe in various sources, (e.g., Coxeter, partly Richter-Gebert) but I haven't been able to quite make it work.
(Specifically, I want to be able to select a line, three points on the line, another line, three points on that line, and then any other 7th point, and be able to construct the image point via a tool in GeoGebra.)