Hmmm. If I look at Peano's axioms, and I look at the techniques used in plane geometry, I see the former as helping to form the fundamental basis for the validity of the latter. For example, Peano's axiom #3 is essentially a statement of the technique of inductive proofs, which would have some utility in plane geometry.
I'll save you some time. You can't even start to do it. The domain of discourse of plane geometry is is the ideal continuous plane. The domain of peano's axioms is discrete sets.
However, there are discrete entities in plane geometry, for which discrete treatments are possible.
This is like looking for recipes to bake an airplane.
You've no doubt heard of aircraft made from composites -- which are, in fact, baked..... ;-)
Your larger point remains, however. When we limit discussion to areas that can be addressed by logic, we exclude real things that the logic does not, and cannot touch: things like beauty, deliciousness, ugly, boring, interesting (all of which can vary even within the perceptions of a single observer!)
It's also difficult to applly it to something like Tolkein's books -- which are pure fantasy on one level (Tolkein essentially defined a logic in that world); and yet still have significant meaning to us here in the real world.
There is no basis in deductive logic for inductive proofs. No deductive proof exists that induction works. It is something we take on faith, just like we take axioms and predicates on faith. Induction, like the "like" operator, (with which it has much in common), does not have a corresponding logical tautology, nor a proved theorem to it's name. So your example will not help you. I asked you to combine lemmas from the two disciplines to produce a valid proof. When you are talking about lemmas, you are usually engaged in formally serious efforts, and therefore talking about deductive, not inductive proof. Inductive proof is handwaving over the notion that what you have learned to expect is what you expect. It does not provide formal deductive security to a theorem.
The elements of the domain of discourse of plane geometry are points, lines, and planes in a CONTINUOUS field. All of plane geometry is theoretically accessable from the predicates and axioms. This is not true of the domain of discourse of discrete entities in sets. This seems like an unbridgeble difference to me, but if you have an example, by all means point to it.