I am looking for a book with some of the most famous proofs properly explained.
For instance: I want a book that proves and explains:
The Pythagorean theorem
Distance Formula
Pi
coterminal-sumplementary angles
Radius, Circumference, of polygons/circles/geometrical shapes etc.
Complex numbers
Golden Ratio
Eulers Identity : $e^{iPi}$ + 1 = 0.
Proof of derivatives, partial derivatives.
Basically I want a book that goes from the most basic elements in mathematics up to multi variable calculus.
I'm not sure that this is really what you want but for sure it is in the same spirit and it is a very reach source in that direction, at least to me:
the book (in three volumes) is "A mathematical gift" by Ueno, Shiga, Morita. AMS, Mathematical World 1995