Textbooks to fill the mathematical "gap"

Question

I got a terrible education before University, I didn't even know what a theorem was, all math was just "learn how to do it, understanding will not have you pass the exams", I'm not having a bad time in college, but some of my colleagues (who got a strong math background) know things that I not even imagine that existed, and I feel offset about this.

Things Like Geometry for example, I love it, I understand what I'm doing, but some basic concepts I just accept as truth and don't know from where this result came from.

I would like to know some textbooks that would fill this gap (not only in Geometry, but in all basics), I'm not a freshman, so maybe something more rigorous will be interesting.

Answer 1

You might take a look at some books about the history of mathematics:

A History of Greek Mathematics and Vol II - Sir Thomas Heath

Mathematical Thought from Ancient to Modern Times - Morris Kline

And then, you can always find a good text book. For example, in Calculus:

Differential & Integral Calculus and Vol II - Richard Courant

Answer 2

"A Primer of Abstract Mathematics" by Robert B. Ash well-served the basics of formal mathematical proof. As of gemoetry, the subject of college geometry is kind of different to that in high school. Maybe more specific example will help others to give suggestion.