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 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