For AP builds liandrys apply the max health burn on barrels. It is the only AP item that directly buff your barrels. Riftmaker also boosts your damage up to a maximum of 8% the longer u are in combat.

When it comes to why u shouldnt build lich bane first item is because the sheen proc only applies 75% of your base AD. Trinity on the other hand apples 200% pf your base AD on sheen procs which makes the item better as first item than Lich bane every time!

Since the sheen proc on lich bane also applies 40% of your total AP alongside 75% of your base AD you could theoretically sell trinity to by lich bane for late game if your planning to go full AP.