Animation follows the same general approach as in
Euclid's Elements: Proposition 47
Claim: The area of the triangle in motion never changes.
Notice that we have two kinds of motion: rotating and skewing. Clearly the area of the rotating triangle isn't changing so let's turn our attention to the skewing process: Refresh the page and observe the motion of the green triangle starting from side a, its tip tracing along side b. Now consider the fact that:
area(triangle) = ½ × base × height
even for obtuse triangles. Also, abc is a right triangle(the angle between sides a and b is 90°) so side b is perpendicular to side a and thus parallel to the base of the skewing triangle. Therefore the height of the skewing triangle never changes, and neither does its area. The area of the skewing triangle after the rotation doesn't change either since the height of the triangle remains the same throughout the skewing process.
QED baby
The animation shows that:
½ area(square) = ½ area(rectangle)
which means
area(square) = area(rectangle)
Combining the results of a and b gives:
a2 + b2 = c2
Full disclosure: The idea for this came to me back in the 90's after reading two books: Fractals Everywhere(fig. 2.5.1: path continuously deformed...) and Euclids Elements(Prop. 47). Jim Morey beat me to the internet with an animated Java applet earning him the grand prize for programming at the first Sun/Java conference in 1995. Java applets are no longer supported so here is an SVG version with commentary. © 2019 Joe Hartman