First of all this is my first Evelynn build, so please don't be to harsh :)

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Summoner spells

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Exhaust because it is helps chasing, and securing a gank
ignite because to secure kills or for the healing reduction on people like Dr.mundo
This is a good choice if you are going to jungle but other than that useless
I don't see why eve would need to use this there should be a support on your team to use it if not it isn't needed to have on your team tho it is helpful not a required.
I could see this being useful but you have exhaust and you ultimate makes you run faster so i wouldn't take it
Eve isn't all that mana starved so i wouldn't take, this I find your other options more useful.
I don't personally use this because i have the run speed from my ultimate but i guess if you prefer it over exhaust it could be a possible choice.
This isn't a good spell for eve because as long as she gets a kill or assist she gets a heal from her ultimate anyway so why would you actually need a second heal?
Again why not to use this. Hmm let me think eve is not a support get the support to take it, and if not then the team doesn't need it that much.
This is a great spell for lane control, but why do you have to worry about lane control you shouldn't be laneing very much. You are should be ganking.
I use this more for tanks and i wouldn't use it on eve because she doesn't really get in the situation to use she doesn't live long enough
Again eve is not a support why take this when a support can take it. Again it is not required but is nice to have on your team
This is a nice spell but the cool down on it is just to long to be useful, and i find the ignite exhaust combo better.

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Items

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= because the the spell penetration is very nice.
= because even if you don't do amazing assists do stack it, and the ability power is very good.
= if you really need and explanation eve isn't the champion for you
= the slow is nice + survivability + ability power
= the ability power is nice and so is the spell penetration just an over all good item
= the use on this item i don't usually find useful on eve, but the tremendous amount of ability power is useful.

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Early game

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During the early game I usually take middle lane till you have you majai's and sorcerer's shoes by that time you should be around level 6-8 depending on who you are against.
After you have your starting items it is time to GANK GANK GANK!!!
If first blood hasn't happened when you are about level 5 depending on who you are mid against or even if you just want a kill it is usually easy to get.

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Middle game

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Once it starts to approach middle game you should have your Lich bane, and you should be working on Rylai's. At this point ganking is going to be harder and you should stay with your team, but still ganking when possible.

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Late game

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Now that it has entered late game its pretty much about team fighting, and you should have most or all of your build done pretty much just focus the squishes and try your best not to die because at this point you are pretty easy to kill your self.

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REMINDER

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ALWAYS CHECK FOR ORACLE POTIONS