Reddit mentions of Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics)

I'm not sure I know of anything exactly like what you're asking for, but a lot of college axiomatic geometry textbooks will have similar proofs to Euclid, or at least prove the same theorems as Euclid, and be in more modern languages. In my geometry class we used Kay's Geometry Book, and I found it pretty readable. I guess some of the reviewers on Amazon don't agree, though. But you could look for another textbook and it would probably have the proofs.

Edit: A Google Search gave me This Book which seems highly praised. I might pick it up myself. I haven't done any Euclidean geometry in half a year and I should brush up.

I took a course in geometry recently. We used http://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/1441931457 The first chapter is the only one to cover Euclid, and it only reviews books 1 - 4 but if you read it and work through the problems it'll give a good foundation to cover the rest of Euclid as you see fit. The real reason I mention this book is that almost all the problems are constructions with straightedge/compass. They give a par (the minimum number of steps a construction can be completed in with a reasonable amount of time spent thinking). You could give out the problems you are interested in without the par so that your students could compare construction methods. When I began the course I had no geometric intuition. I spent a lot of time trying to find the best possible constructions and felt my (euclidean) geometric intuition bloom.

Finally, after chapter 1 the book goes into Hilbert's Axioms to show how we develop modern geometry and develops a number of interesting geometries. I can't speak of most of this as we only covered Euclid, Hilbert's Axioms, and a quick bit of non-Euclidean geometry. But I found it very interesting and think the book can be used to study geometry in a number of ways. Either way good luck.

Geometry: Euclid and Beyond by Hartshorne is a really good book. It starts out by walking you through Euclid's elements (you need a separate copy of that). Then it goes on to Hilberts's axiomatization of geometry, while discussing the ways in which Euclid's was lacking. Then it moves on to other things, like the relationship between algebra and geometry, and non-Euclidean geometries. It's not an easy book, but it does not really have any prerequisites, and it's a lot of fun. It makes Euclid a lot more clear.

This has turned out to be a much more interesting question than I had thought it would be. It seems to be unexpectedly hard to find a good, short book on Euclidean geometry. Most of the really good books are advanced treatments that have a lot more to say than what you probably want. Anyway, there is a good discussion of this question on mathoverflow. It appears that Kiselev is a pretty good choice. Hartshorne might be good as a guide to learning straight from Euclid (and lots more besides). I don't know how far you really want to go with this project. It might be enough to just get a taste of how the whole synthetic geometry program is organized.

By the way, you know about libary.nu, right?