Tears can be good
There are special circumstances where it's a no-brainer to buy a Tear. I'm specifically talking about Ryze, although Singed can also make great use of a Tear as a sort of alternate strategy. I'd still rather go for a Rod of Ages on Singed, but Tear isn't bad and it lets him use Poison Trail a lot more to be annoying. Singed is all about being annoying, so doing it more is always a plus.
Ryze is a special case since he deals more damage from his mana. While he should not go for AA until he's got most of the rest of his build, an early Tear gives him a lot in the long run, and he trades a lot less to buy one. He gets about as much spell damage as two Doran's Rings (on another character with normal AP scalars) when it's uncharged, and he builds damage continuously as he charges it. An early Tear, transitioned into a lategame AA after much of the rest of his build is done, is a solid strategy.
Cost versus effect
A lot of theorycrafters talk about how much stats are worth. Mana is considered to be worth about 2 gold per point, for instance. However, we have to look more closely at what the actual item impact is worth, particularly in the early game where a Tear would be purchased.
A Tear costs 700 gold and provides 7 mana regeneration, 250 mana, and its unique mana-building passive. By comparison, two Doran's Rings cost 800 gold and give 3 mana regen each, 15 AP, and 60 HP.
In terms of raw stat value, the two rings are similarly efficient, but the value of flat mana in lane, even early game, is kind of dubious. Unless you're Ryze or Singed, for whom flat mana is a big benefit, it doesn't make a lot of sense. Tear extends your lasting time in lane, but it does not extend the lasting time of your health bar or improve your destructive power (like Doran's Ring does). If your lane gets jungle support, you won't have the bonus damage to help clean up as you would if you had some bonus AP.
I sort of liken buying an early Tear to a less extreme version of buying Mejai's Soulstealer. You are taking a risk; if your opponent builds more efficient items, you are about 400-500 gold behind in terms of stats until you finish your AA. Unlike Mejai's, it doesn't put you quite as far behind if it doesn't pay off, and it's guaranteed to pay off as long as you don't lose the game in the laning phase. However, it is effectively putting you back about 500 gold. If you're intending to buy a Tear, you should probably verify whether your lane opponent is worse than you before you buy it. If he's about on your level or better, you probably shouldn't buy a Tear unless you manage an early lane kill.
Tear's bonus to mana regeneration is nice, and it's something that is improved by the application of the blue buff. However, the blue buff is generally enough without a Tear unless you're Anivia or some other very mana-hungry champion. If you're Anivia or Orianna or some other very mana-intensive hero, you probably want to just bite the bullet and buy a Chalice of Harmony as you're going to need it anyway later.
As an amusing aside, Chalice also raises magic resistance so it is generally a better option if you want mana regen. It costs 880 gold, but 25 MR stops a pretty sizable chunk of incoming magic damage. As a mid lane hero, your opponent probably deals magic damage, so it ends up adding a lot of effective hit points.
Philosopher's Stone is generally thought of as an "I'm behind" item, but it costs the same as a Tear, gives more mana regen and health regen, and slowly pays itself off by generating gold over time. Again, it's a pretty good alternative. The fact that Philo gives more in-lane sustainability than a Tear for the same price and gives the GP/10 passive is a pretty big slap in the face for a Tear.
That's the bad news; what's the good news?
Of course, there are times you do want to buy an Archangel's Staff. It's just a really situational pick to buy a Tear in the first place; buying it puts you behind so you want to be ahead, and if that doesn't come early enough, you're probably better off converting your advantage into better items that will win the fights coming now. A lot of characters won't even get enough mana built up in the Tear or AA to turn the AA into a Seraph's Embrace if they rush the Tear, so you really need to have a good feel for about how long it takes each character to fill the mana charge.
Archangel's Staff and Seraph's Embrace give some of the largest AP boosts of any items, although they basically only give AP (the Seraph shield active is pretty nice though). They synergize well with Rabadon's Deathcap and Void Staff to give tons of destructive power that blows through enemy defenses. The problem is getting there.
Also, in an endgame item build, while Seraph's Embrace gives tons of AP, it doesn't give any useful lategame perks other than its shield. It doesn't give cooldown reduction or bonus survivability, so if you need those stats you will need to look in other items. If you've got boots, a Deathcap, a Void Staff, and a Banshee or GA, you have only two item slots left to fill. If you need 30% more CDR, you really can't fit a Seraph in your item build. If you want both a Banshee and a GA or you need some other defensive item, that's also going to make your build hard to fill. In almost all circumstances I'd rather have a Grail to cover any mana regen, since it gives a lot of AP plus a kitchen sink list of good attributes.
There is another option, and it's not AP-based at all! Manamune is the other option for a Tear. It basically follows the same rules as AP-based Tear users, though. Buying mana instead of health, health regen, or attack damage puts you behind the power curve relative to your opponent. On top of that, rushing Manamune is pretty bad, but it also won't build charges on attacks unless you upgrade your Tear.
The bottom line for AD-based Tear users is that it's a risk. There are a few viable builds that get Manamune; it's basically the same as an Archangel's Staff, though. It doesn't provide a lot of extra utility, so if you want attack speed or crit chance or on-hit slow or whatever, you'll need to look elsewhere. The fact is that if you are running dry on mana as an AD character, Tear is the only mana sustaining item that builds into an AD item. Your other option -- and I do this a lot when I'm playing top lane Poppy -- is to just buy a Philosopher's Stone and sell it in the midgame. You can also pick up a Spirit Stone; it's not just for junglers and builds into several powerful options.
Overall, Tear of the Goddess is a very situational item. It gives you a shot at having some really powerful lategame burst, but it is honestly not that great. This is not like Deathcap, where I played devil's advocate on an exceptionally good item. Tear is a legitimately mediocre item with a lot of stronger alternatives. It's not awful, and there are reasons to buy it; I just don't think it's worth getting as an element of any reliable strategy that doesn't involve playing Ryze.
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