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If you run your setup, rushing Giant's Belt into Rylai's, you're prioritising making your stealth burst champions bulky, and there's a massive 1.1k gold gap where you're not increasing your damage at all. On a champ who does sustained damage, I completely understand the choice to buy a damage/survivability item like Rylai's next, and there are plenty of champs with whom I think rushing an item like this is a great idea. Eve, however, doesn't quite play like this. You're stealthing into people and holding them in place while you and one or more partners are gangraping them. It's supposed to be asymmetrical from the start. Your survivability at this point of the game, which is more focused on ganks rather than large teamfights, comes from your stealth, your heals from champion kills (which you need damage to accomplish), and your ability to escape (movement speed). The last two are handily given to you with Lich Bane.
What you need at this point in the game isn't health; it's more killing power to make you a better assassin. And, as I calculated, the highest possible way to get such damage after building Sheen isn't to rush your next AP item: it's to complete Lich Bane, and then start working on an AP item. Whether the next item is Rylai's or Deathcap depends on how much you think you need damage, health and slows vs. pure damage.
@MissLoraine: Sure. As I said in my guide, the early build order is something I don't find incredibly important, and rushing Sheen before Mejai's might even be a better idea (it's possibility of missing out on stacks if you rush a Sheen vs. possibility of not killing targets if you rush a Mejai's).
nice build but i think that you should make sheen and afterwards Rabadon's Deathcap cuz if you dont have ability lich bane is useless.However, its nice build +1
^ That.
I did read the guide but you could never convice me to do it that way. The only thing about Lich Bane that could ever make it worth rushing before you have more AP is 7% move speed, which you might need to do some games but not all.
I understand that it might do a bit more damage than sheen but I would at least get Rylai's after sheen and then finish Lich Bane. Rylai's will give you lots of beneficial stats as well, and the slows will be welcomed and put your opponents in even more danger. (I wouldn't get Mejai but I know you like it, so I will include it in the flow. I would build this order, and simply not buy mejai's)
Perhaps a better flow would be Boot -> Mejai -> Sheen -> Gian'ts belt -> Amp Tome -> Blasting wand - > Rylai's -> Lich Bane -> etc.
I've considered that. However, when you autoattack out of stealth, the coming out of stealth acts as activating an ability, and therefore procs Lich Bane and adds your AP to your AD. So it feels like wasting a Lich Bane proc (which eventually will hit like a truck) by not maximising its potential (remember that the reason you open with Ravage isn't just because it's a powerful nuke, but because it lowers their armour and magic resist).
Again, I don't rule out DFG as a potential item on Eve (just as you say it's not a must); I just don't like it as much as the others (I love it on Veigar and Malzahar though). If you think you should gank someone and activate DFG on them and then decide it isn't going to work and need to escape, you've wasted that ability for 60 seconds. If you're just using your abilities to gank with, all you've used is Ravage which has, what, a 7.65 second cooldown with this build. And then, as I said earlier with the user telling me I should use Rageblade, it's a question of swapping out one of the other items, all of which (apart from the very situational sixth item) give this build its particular strengths (HP, MR, slows, and a ****load of damage).
+1