I have small question regarding this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html
To prove that one side is equal to another, Euclid assumes that one side is bigger than the other. Finally, when Euclid arrives at a contradiction, he dismisses the assumption about the inequality of sides and considers them equal. What I was wondering is, if we assume that a side is unequal to another one (A is bigger than B) and arrive at a contradiction, shouldn't we also try the inverse, B being bigger than A and assure ourselves that we arrive also at a contradiction to conclude that finally, A is equal to B ? Thank you!
If you accept the contradiction you get from assuming $A$ is bigger than $B$, you truly are done. For if you were to switch and now assume that $B$ is bigger than $A$, how would the proof differ? It wouldn't. You'd take the same steps to get to the same contradiction. This means it doesn't matter which of the two is bigger, as long as one is bigger. It just so happens that we called the larger one $A$. But that is completely arbitrary as we could have just as easily called it $B$. Mathematicians will often convey this idea in a proof by using the phrase, "without loss of generality." So without loss of generality, we can take $A$ to be larger than $B$.