I have only a rudimentary knowledge of Geogebra, but I think you would find that anything you can do in Geogebra you can duplicate in Mathematica, but Mathematica is immensely more powerful.

In Mathematica you don't "create a 3D axis (itself a problematic idea) and a plane" as much as define geometric objects in relation to a coordinate system. All your geometric and display operations are carried out on that data set.

With Geogebra, as with other drag-n-drop programming systems, you can produce dazzling, expressive, illuminating results very quickly, but you're ultimately limited to the scope the package authors thought of.

It's going to be a universal problem, figuring out how to make the transition between the drag-n-drop sensibilities and the bits-n-bytes approach. Maybe, the drag-n-drop systems will advance to the point that they'll be capable of doing everything.

I think you must be on the leading edge of this, so you're going to have to tell us.

Hth,

Fred Klingener