# Good books for relearning high school geometry from the ground up with complete mathematical rigor and heavy focus on proofs

Before posting I read questions that were similar to mine but I didn't find an answer that was quite what I was looking for.

I want to learn all high school math from the ground up, but I want books that focus extremely heavily on proving every little thing rigorously. I made another post which was closed for being too broad because I specifically requested books for these topics: Algebra, Geometry, Trigonometry, Probability and statistics, Pre-calculus and calculus

So I guess I need to create one question for each ofthese topics so it's less broad (I hope this isn't against the rules or still too broad. I'm just looking for 1 or 2 examples, anything that would help me have a little more guidance and make it a bit easier to learn on my own, I'm certainly not looking for definitive answers, let alone debates).

So, let's start with Geometry. Even though this topic is usually taught in high schools, I'm looking for college level books, since I doubt high school books are as rigorous as I want them to be.

For reference, one of the ones I know about and am going to use for this project as a whole is Euclid's Elements, so I am looking for other books with that level of rigor (though additional explanations are always helpful, since Euclid's Elements is very rigorous but doesn't elaborate much on explanations).

Thanks a lot!

Well, this is not an easy question since Geometry itself is a very broad field in Mathematics and the level of the books that you request may vary depending on what you find to be rigorous.

But based on what you said (rigorous and explanations), at some point you will want to learn about differential geometry of curves and surfaces. This is usually the first step before going for general theory of differentiable manifolds. The number of books and notes covering this topics are huge, really huge. Many of them highly recommended. But if you want to start from the very basic and keeping the rigorousness all the way, then I'd suggest you to study with Differential Geometry of Curves and Surfaces: A Concise Guide, by Victor Andreevich Toponogov.

As you may imagine, this is not going to fill all the knowledge that you should get, but surely other answer might help with that.

• Thanks a lot Edu, this will be very helpful! – José Guedes Oct 6 '16 at 15:20

How about the following book : Euclidean Plane and its Relatives A minimalistic introduction - Anton Petrunin

Maybe highschool student can not cover all materials in the book But a third portion is about Euclidean geometry. From five axioms we derive coordinates on $\mathbb{R}^2$

• Thanks HK Lee, I've downloaded it as a .pdf and it seems to be EXACTLY what I'm looking for. The introduction says it all: "[This] book is meant to be rigorous, conservative, elementary and minimalistic." Thanks a lot! – José Guedes Oct 6 '16 at 15:21