Pythagorean Theorem visualization
One way to think about this proof is that there are two ways to break up an square. One way to do it is to put an square at the top left and a square at the bottom right. That leaves two rectangles at the bottom left and top right. Those can be divided along the diagonals into four right triangles with legs of and a hypotenuse of length (which length we are trying to discover).
Sliding the "twirl" slider, the four right triangles move along the edges of the large square, leaving all the hypotenuses of length forming a smaller square inside the large square. Since both arrangements contain those same four triangles, what remains inside the large square must have the same area, whether in the first square or in the second square. Thus .