The word "constructive" in "constructive geometry" might mean intuitionistic logic, or it might refer to ruler and compass constructions. In my first paper on the subject, Constructive Geometry, I showed that there is a connection: Things you can prove to exist in constructive geometry can be constructed with ruler and compass.
Within constructive geometry, different versions of the parallel axiom that are provably equivalent (using non-constructive, or "classical" logic) are no longer equivalent. Using certain metamathematical techniques, I was able to show that they really are not equivalent in constructive geometry. I reported on this work in my 2012 paper,Logic of Ruler and Compass Constructions.
In January 2014, I completed two papers presenting all my work on constructive geometry. New readers should ignore the early conference presentations mentioned above, and just read these two papers: