This "comment" is a little too long for the comments.
I think, without diving too deep in to the link that you posted, that what you're lamenting is the lack of what I would call synthetic geometry, in particular as opposed to analytic geometry or other approaches to geometry ( eg algebraic geometry ). This would be things like ruler/compass constructions. To this complaint, I think many many mathematicians consider synthetic geometry very beautiful and it has a special place in their hearts, but it's kind of like owning a horse that you ride for pleasure and exercise, but you still drive a car to get places; we typically have better tools now. Synthetic geometric intuitions still play a big part and occasionally I've seen proofs that were more enlightening when done synthetically, but for the most part, practicing mathematicians use the more modern tools.
If it's synthetic geometry you're actually interested in, you shouldn't restrict yourself to Euclidean geometry! There's actually a lot of fun to be had trying to do some of the same things in non-euclidean geometry that were done for centuries in Euclidean geometry. I recommend the book "Euclidean and non-Euclidean Geometry". For more of a similar flavor, you could study projective geometry. I've never read it, but Robin Hartshorne has a very popular book that is of a synthetic flavor ( Note: His more famous algebraic geometry books are decidedly not of this flavor ).
As far as "Euclidean" geometry goes, though, that is still quite actively studied. It may not be the hottest topic, and the tools used are often pretty far removed the days of the Greeks or even Descartes, but there is plenty of geometry still done that studies good old Euclidean space.