Here's a visualization of the Pythagorean theorem, inspired by Steven Strogatz' March 14, 2010 New York Times Piece Square Dancing.
When you first bring up the visualization by pressing the "Open Geogebra" button below, you will see four congruent right triangles arranged in a big square. It will be apparent that the area of the part of the big square not covered by the four triangles is a2+b2. Then slide the "twirl" slider slowly from a2+b2 to c2. The four triangles will rearrange themselves and ultimately you will notice that the area of the part of the big square not covered by the four triangles is c2. Since in both cases those areas consist of the the part of the big square not covered by the four right triangles, it must be that a2+b2 = c2.
If you like, change a and b to verify that it works for different values of a and b.
- Pythagorean Theorem