I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more comfortable with algebra than geometry. I think that's mainly because my geometry education was sparse through the years, lacking in consistency etc. So I'd like to revise (and learn more) all at once, catching the basic axioms, understanding why such is such, etc. Essentially, a moderately rigorous textbook in plain Euclidean geometry (nothing fancy). Please don't say "The Elements" - I have browsed it at the bookstore, it is quite good, but not really what I'm looking for right now.
I'm currently working through Robin Hartshorne's Geometry: Euclid and Beyond. It starts out by touching on Euclid's Elements, and then explores Hilbert's axiomatization of Euclidean geometry to make it hold up to modern standards.
There are a good number of challenging exercises in it, and it delves into non-Euclidean geometry as well, so it may be worth checking out if you're interested in brushing up on modern Euclidean geometry and other classical geometry.
For the bare bones beginner who either doesn't know or has completely forgotten all of his or her high school geometry,I cannot recommend more highly:
Kiselev's Geometry, 2 volumes,translated by Alexander Givental
I'd also highly recommend Givental's wonderful introduction in the preface of the first volume on the history of this classic book and what motivated him to bring the Russian classic to an English-speaking audience.
After that,there are basically 3 books you can't go wrong with:
Elementary Geometry From An Advanced Viewpoint, 2nd edition, by Edwin Moise
Euclidean And Non-Euclidean Geometries, 3rd or 4th edition (either will do nicely) by Marvin Greenberg
A Survey of Geometry by Howard Eves, 2nd edition(2 volumes)
Moise is the classic text that develops Euclidean geometry using the metric postulates of G.D. Birkoff. There are several other books that try and do this,but none do as good a job with it as Moise. Greenberg is a remarkable historical tour through the various geometries of the plane as axiomatic systems,from geometry pre-Euclid through 19th century developments of non-Euclidean geometries through a careful analysis of the Hilbert axioms. It also has many pictures and many exercises of varying difficulty incorporated into the body of the text,so you really need to read it with pen in hand. Eves is an older,2 volume work attempting to do for elementary geometry what Birkoff/MacLane did for abstract algebra. Some of it is awkward and dated,but it has a lot of cool stuff in it you can't find anywhere else.
Those 3 are how you get started to me. And if you want to go on from there, it's time to read the awesome classics of Coexter. They are THE detailed textbooks on plane geometry-but they are best read in my opinion after mastering the basics.
You might want to look at Coxeter's Introduction to Geometry.
If the thought of slogging through "The Elements" discourages you, I suggest you look at Benno Artmann's Euclid: The Creation of Mathematics. His book is as much historical as mathematical, but it is very pleasant reading.
From the Preface:
The present book takes a clear position: The Elements are read, interpreted, and commented upon from the point of view of modern mathematics.
and that is exactly what he does.
The following three books are, in my meagre estimation, the most rigorous elementary introductions to elementary geometry:
“A Beautiful Journey Through Olympiad Geometry” is a book that presents all the theorems/methods that you need to know in order to solve Olympiad geometry problems. It progresses step-by step, starting from scratch, so you will definitely be able to follow it without any additional materials. It contains solved problems using these theorems, but also related problems that are left unsolved as a practice for the reader. Check out the book preview and download the book (pay what you want) here.
Check out this one, called "Plain Plane Geometry" by Amol Sasane : http://www.worldscientific.com/worldscibooks/10.1142/9907