Quote by DaleSpam
Now, transforming to the primed coordinates using the above formulas (v=0.5) gives [itex](t'_A,x'_A)=(0,0)[/itex], [itex](t'_B,x'_B)=(.5,1)[/itex], and [itex](t'_C,x'_C)=(1,.5)[/itex]. So we see that [itex]t_A \ne t_B[/itex] meaning that simultaneity is relative, and the time between A and C is still 1 meaning that time does not dilate.

What happened to gamma?
The way I calculate the three transformed events, I get:
A' = (0,0)
B' = (0.577,1.1547)
C' = (1.1547,0.577)
So A and C do not have the same time coordinates so they are not simultaneous.
EDIT: I see that wasn't your point. I should have said, the time between A and C is not the same as before, it's longer in the primed frame. But I wouldn't call that time dilation, it's just different coordinates for a pair of events.