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It uses the infinite geometric series which is how A and C are determined in terms of a, b, and c (as shown in the article). I think the infinite geometric series is definitely the coolest part of their proof!
I'm certainly no expert on proofs of the Pythagorean theorem, but if their construction is novel, then the above may be too.
I think you can sidestep trigonometry (and the Law of Sines) completely. You can decompose any triangle A, B, C using their construction to create smaller triangle a, b, c where A = 2abc/(b² - a²), C = c(b² + a²)/(b² - a²), and B = c. This can be shown with only similar triangles (it seems like they unnecessarily use sines in the article). It is then just algebra to show A² + B² = c²(b² + a²)²/(b² - a²)² = C².
edit: any right triangle A, B, C using their construction to create smaller right triangle a, b, c