They involve simple mechanisms that turn, slide, and rotate to make things go. Anyone who has tried to clear a paper jam from the inside of a Xerox machine has seen multiple examples of mechanisms at work. Anyone who tries to fix such a machine needs to understand how geometry serves the cause of making things move. Problems of this type exercise geometric thinking in powerful ways. They cry out for physical models, which are easy to build.
Many variations are possible, and all have good physical applications.
The mathematics involves applications of coordinate geometry as well as trigonometry and simple algebra. Much of the analysis can be carried out either by geometry, analytic geometry, algebra, or spreadsheets.