The One and the Three

[Tirinor]

The Three were made by Celebrimbor, and were hidden before Sauron could pervert them. However, being Rings of Power, they were still tied to the One (as were the Seven and the Nine). Once the Ruling Ring was destroyed, the Three lost their powers. A good analogy would be cutting off a beast's head -- the other body parts die, as well.

I can see how that might be a good analogy for the other rings, but still don't see it for the three elven rings. If they were crafted in secret, why and how do they have any relation to the One ring? Did they secretly steal some of Sauron's power to fuel their own? I don't see that happening.

An analogy along the lines of what I'm thinking is more like - A bunch of McDonalds franchises being supplied by a cental distribution network, and then, over here, you have three Burger Kings (with their own franchise hub). If McDonalds goes down, Burger King is still good. likewise, if Mcdonalds is successful (powerful) due to it's efficient distribution center, Burger King isn't nessesarily successful, and if they are, it has nothing to do with McDonalds' central distribution network. Burger King is independent, good or bad from Mcdonald's power source.

The Three Rings, being crafted independently, would seem to have no relation, good or bad, to Sauron's One ring of power. But the text claims they are related. How?