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I think is I think there is a connection even depending on the match up. then: in which case do you choose what to build one item over another? I consider the 3rd item to be the situational one based on the game
I think building Luden's is simply a bad idea for all the reasons I gave in the guide.
I would rather get Tear + rush Rabadon's Deathcap than build Luden's companion if I want to go for max damage.
Right now, I would just build Seraph's embrace every single game. I would also go for Cosmic-Warmog a lot of games after getting Rabadon, building Cryptbloom when the enemy team starts to stack MR.
Pro analysts are slowly starting to realize how strong Warmog is, and Veigar is probably the best midlane mage to abuse it.
About Tear, I would only take it as a starter item in really free/afk farm matchups.
Veigar is a burst mage but a poke mage as well who can apply Liandry's passive pretty easily on the frontline, often made of HP stackers.
That being said, the reason why Liandry is good on Veigar is because of the passive:
"UNIQUE – SUFFERING: For each second in combat with enemy champions, deal 2% increased damage, stacking up to 3 times for a total of 6%."
It's obviously really good in team fights for Veigar as you get 6% increased damage, which is more valuable than AP on Veigar (at some point at least, but you don't want to build Liandry early on Veigar)) for reasons I gave in the guide, while also granting some defensive stats.
Also, Liandry is best in the Warmog setup where you need 1000 bonus HP to activate Warmog's passive.
That being said, I updated the guide again and replaced Liandry by Cosmic Drive in the Warmog set up.
I think it is really situational, but I'll try to provide a framework.
In my opinion, Shurelya's>Seraph's is the safest path: lots of ability haste, mobility and shield, it's also a very strong powerspike at only 5100 golds.
That being said, (and it is maybe counterintuitive to some) if your team is getting stomped too fast (very hard games as Veigar anyway), you need Rabadon's second to maximize your damage output and try to play around shutdowns. As people are snowballing they're also more likely to act cocky and make mistakes. Veigar is really good at punishing mistakes, but if you don't have enough damage (and your team is likely not to have it as well as you're losing) you won't be able to punish anyway.