Proofs, or any mathematical derivation, appearing in any real setting, such as a book or textbook or talk, or even when we're teaching it in class, includes a great deal of surrounding explanation. But when we ask students to regurgitate proofs, we ask for what is merely the skeletal core of the proof in this rib-like two column format. At the very least, it seems to me, proofs ought to have three columns, including a new multi-row lefthand column that describes (at least) the approach being taken in that section of the proof. By "multi-row" I mean that the proposed new lefthand "row" can encompass multiple rows of the basic two-column proof. The resulting format would look like this:
=================|===============|============== What we're | <c=<c | Reflective Property up to in | tABC~=tDEF | SAS this section | etc... | etc... =================|===============|============== What we're | etc... | etc... up to in | etc... | etc... this section | etc... | etc...
The context would, I think, largely reflect the reasoning and planning that went into (goes into) the proof, and would commonly, I think, end up representing lemmas that participate in the larger proof. One could say that these lemmas ought to be rolled off into prior proofs of their own, and I would agree. But we do not provide a way, in geometry, of naming and organizing proofs usefully so that prior short proofs (technically lemmas) can be looked up and referred to easily.
Because we do not have a clear naming scheme for proofs, we cannot call upon them as one would functions in a programming language. Indeed, one might wonder why student of geometry aren't being taught geometry like one would teach a programming language: Here's a bunch of functions (lemmas) you can use, and here's how to use them. We do do this for some things, like the triangle congruence lemmas (SAS etc), and for some logical rationales (CPCTC, etc), but the dozen random theorems (lemmas) regarding parallelograms, mid segments, and so on aren't ready-to-hard functions with clear naming, so we end up re-deriving/proving them in the middle of other proofs, which makes the proofs into these long-winded, un-memorable and ultimately unwieldy things.
The three-column format I'm proposing at least offers a way to internally organize proofs into logical segments so that even without addressing the problem of the previous paragraph, at least there is a a way of making the substructure explicit.