I'm trying to figure out how Dumbledore learned that all Deathly Hallows were real objects.
In Deathly Hallows we see rationalist Hermione arguing with Xenophilius ("lover of the strange") Lovegood.
Ironically, Xenophilius is right about the existence of these objects.
The Elder Wand is the most visible object, one can follow its trace through history. Dumbledore and Grindelwald obviously know the story of the three brothers. They know the stories about the wand that is called the Elder Wand.
But why would they think that the other two Hallows are also real? Perhaps they didn't think so and only later Dumbledore learned that they are in fact real.
James Potter gives Dumbledore the invisibility cloak. Perhaps his 'curiosity' was that this object confirmed the existence of the second Hallow? I can imagine that this can be quite overwhelming for Dumbledore for a moment.
And the same of the resurrection stone when he discovers in the house of the Gaunts, the confirmation that all Hallows are real.