Assumptions "Without Loss of Generality" (WLOG)

Without Loss of Generality, frequently denoted WLOG, essentially means that a problem can be divided into many cases. It then asserts that if one thing is true, then the remaining (completely) symmetrical cases follow. First, a simple example to show what I mean:

Question: Prove that in an isosceles triangle ABC the incenter I lies on the Euler line.

Solution: WLOG, assume that BC is the base, so AB=AC. Now, it is clear that the incenter I lies on the angle bisector of angle <BAC, which, because ABC is isosceles, is the axis of symmetry in the triangle, and it follows, as the other centers are also on the axis of symmetry.

Now a harder example, from this years Harvard-MIT Math Tournament:

Question: In a convex hexagon, what is the highest number of diagonals which can be equal? (HMMT Team #7)

Discussion: This was one of only two problems that our team solved (The other being problem 2). This problem seems to be attackable by assuming the existence of two equilateral triangles and finishing via a greedy argument, but while that yields a construction, it doesn't yield a proof. The proof requires an intriguing insight and many, many WLOGs. This is more of a real application as opposed to the first question.

Solution: 7 diagonals is easily constructible; Take two equilateral triangles, and then space them out so it is satisfied. Finish by connecting the vertices.

Now we prove that 8 isn't possible. There are two cases:

1. The diagonal that is skipped over (as a hexagon has 9 diagonals) is a short one, like AC. WLOG, take it to be AC. Then, AD=AE=CD=CE. Hence, D and E are reflective duals with respect to AC, a contradiction because D and E are on the same side of AC.

2. Now take a long one, WLOG AD. So BF=BE=CE=CF, so B and C are again reflective duals, but remember, E and F cannot lie on different sides of BC!

Hence, the answer is 7, because it can be attained but 8 cannot.