Hidden geometrical gems in Euclid's Elements? I am teaching a course in Euclidean geometry at the University of South Carolina, and it seemed highly appropriate and interesting to read Euclid himself. (See here for a wonderful, and completely free, translation and guide.)
We are up through Proposition 17 now, and although this is very instructional for both the students and myself, I sense that before too long the novelty will wear off and it will be good to return to a modern treatment. (I will, though, want to say a little bit about his treatment of parallel lines and elliptic and hyperbolic geometry.)
That said, there are some gems. In Book 2, Euclid constructs square roots, and in Book 4 he describes how to draw a regular pentagon (which is surely not obvious). And, the constructions in the first few books are all very interesting. (Of course there is lots of fascinating number theory too, but my course is on geometry.)
What other particularly fascinating tidbits do the Elements contain, which it may be easy to overlook?
 A: If I were to ever incorporate Euclid's elements in a class, it would be through a critical exposition of Euclid's work, not necessarily directly with his books.
The best source I've gotten into for that is Hartshorne's Euclid and beyond. It really helps you see the importance and impact of Euclid's work, without really having to trudge through it. 
As I recall, the gems are strewn along the path of good exposition. The book also introduces Hilbert's axioms, which was the next major revolution of the foundations of geometry that students should really hear about. 
It never lingers on Euclid's presentation long enough to become boring. It revisits it throughout the book, of course. Really, there is no reason to linger on Euclid after students have gotten the general feel of the material.
Maybe you can take a look at how Hartshorne presents results, and then devise a plan. 
A: For me, I like Book 1 Prop 35, whichis the first use of "equal" to mean equi-areal rather than congruent.
Book 2 Prop 11, where Pythagoas is used geometrically to prove a construction for the golden ratio (later used for the regular pentagon construction)
3:35 again a geometrical use of Pythagoras to prove the "power of a point"
6:3 showing that the angle bisector cuts the base of a triangle in the same ratio as the other sides.
6:19 showing that "similar triangles are to one another in the duplicate ratio of their corresponding sides
6:31 pythagoras generalized and proved using ratios
