Comment by zahlman

We can imagine another copy of the trapezoid, rotated 180 degrees and situated on top; the pair of them create a square with side lengths of a + b. This cancels all the 1/2s out of Garfield's equations, and also makes the result more geometrically obvious: the entire square (a + b)^2 = a^2 + 2ab + b^2 is the inscribed square c^2 plus four copies of the original triangle 4 * ab/2 = 2ab.

This then becomes a restatement of another classic proof (the simple algebraic proof given near the top of the main Wikipedia page for the theorem). So we can imagine Garfield discovering this approach by cutting that diagram (https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/Fil...) in half and describing a different way to construct it.