A nice problem in high school geometry

Let E be the midpoint of the side AD of a square ABCD.

Problem: Determine which has the greater perimeter, the square ABCD or the circle through E, B, C?  (You can assume you know the value of π. Otherwise try to use elementary methods.)

The neat solution is obvious when you spot it, and I find rather satisfying. Metaproblem: Why is this solution aesthetically pleasing?

I’ll post the answer — to the problem, if not the metaproblem —  tomorrow!