Every geometric problem that has a two-dimensional representation is solved by almost every mathematician by first drawing a diagram, then deriving the correct formal description from this diagram, and then continuing to solve the problem in the algebraic description.
These certainly are ubiquitous "situations in which there is a heuristic argument which is valid and can be formalized."
Also the heuristic argument is "separate (or complementary)" to the "rigorous argument, but the heuristic argument is more enlightening and more explanatory."
