Thanks! I should certainly have made my question more clear, particularly to eliminate issues (1) and (3) (although 3 doesn't really solve my particular question - I'm looking for an equivalency and not simply a reduction, mea culpa). Can you speak a bit more about the structure in (2)? I know that GI is self-reducible, but I presume that's not what you're talking about there. Regardless, thank you!

Yes and no - in this case, we can (loosely) define a promise version, but it's not clear that we can explicitly define one - more specifically, I don't see how the construction gives a first-order definable $G$. I'll say more about this in a reply to Scott's comment below, and try to edit my question for clarity...

Just a small note: your description is missing the constraint that the two sides of the triangle have to change - otherwise you could just take for the $\Gamma_n$ a Pappus chain converging towards one of the triangle's vertices and tangent to the two sides incident on that vertex. A fuller description of the six-circles theorem (with a couple of references to elementary-looking proofs) is at mathworld.wolfram.com/SixCirclesTheorem.html .

I'm still confused by this myself (and sadly, I don't have a postscript reader handy to check out the reference). Wikipedia explicitly says 'There may be inputs which are neither yes or no. If such an input is given to an algorithm for solving a promise problem, the algorithm is allowed to output anything.' PlanetMath says something similar; it seems like the promise version of a problem can never be more complex than the non-promise version (just ignore the promise!). I'd prefer to say that his problem 2 is surely in NP but not at all guaranteed to be NP-complete.