To make an interactive geometry application, maybe when a point or line or arc or whatever is selected, then the things you can construct from there should be immediately visible and selectable, with a little pushbutton to create each one. Maybe things like midpoints and intersections should be created as soon as segments and lines are, as in Laszlo Pandy’s (and Bret Victor’s?) dynamic drawing application.
To clean up clutter, you might want to have some constructions that are a little more at arm’s length; given a library of them, you could choose one to instantiate a cartoon of it with a little pushbutton on each of its inputs. Then you can constrain it to objects in your scene one by one.
Additionally, you may want to put some things in another layer, then hide that layer. In multitouch, this can be done with a layer quasimode.
(See Two-thumb quasimodal multitouch interaction techniques for more details.)