I work as a teaching assistant in a high school and geometry gives most students headaches.

I emphasize understanding the problem by constructing lines and using elementary properties to arrive at the solution instead of merely pulling a formula and plugging numbers. However, even when I've presented multiple ways to think about a problem they still do not attempt constructions to understand and/or solve it or similar ones.

What are good ways to foster this kind of initiative in students dealing with elementary geometry?

I mean constructions in the sense of drawing diagrams, constructing (by hand) auxiliary lines that shape the problem into something known (perpendiculars, etc).