How to Split-Push as any mage

That's literally it.

Echo helps you clear out minions with ease. Nashor's Tooth increases your attack speed whilst giving you more AP, since every auto attack a mage deals to a turret deals an extra 50% of their AP as magic damage. Lich Bane will turn that 50% into 100%, which is pretty OP. And since Magic Pen applies to turrets, Morellonomicon always comes in handy. Or Void staff, if you're into that sort of thing.

Arc Staff and Rabadon's are just for the extra AP, since they give a LOT of extra AP. With the Transcendence rune, you can easily pick up an extra 60 AP from this build, and you're sure to get an extra 70+ AP from the Arc Staff, and Rabadon's was simply buffed to a very powerful state.

But it ain't just Mages that like it. It's anyone who builds AP and can deal the occasional ability. are whom I like using.