How to explain relativistic mass with 2 moving systems, but not 3?

All the visual explanations I know work in some kind of "If you are moving relative to something A, while inside A something is moving, the stuff in A has to move slower due time dilation and therefore the mass has to increase so the impact(and momentum) stays the same."

Since all moving objects and not only the ones inside of moving objects have their mass increased I have big trouble understanding why these kind of pictures should lead to any understanding.

Could I argue that if A was moving his whole point-of-view would be moving?
As if everything already would be inside of something moving?
This sounds very wrong and just raises more questions in my head.
You could simply determine who was moving if that would be true, so it isn't.
Or is it?

I don't know where my thoughts got messed up. Hopefully you do.

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Mass doesn't change, it's one of the invariants of the theory. However, in older expositions of relativity, it was customary to introduce a concept of "relativistic mass", which would depend on speed. (Formula: $m=\frac{m_0}{\sqrt{1-v^2/c^2}}$ ) This terminology has been largely abandoned I think. Maybe you should post the exact chapter of the Feynman lectures you are refering to in your question. We could then address this. –  Raskolnikov Mar 4 '12 at 14:07