The paper napkin trigonometry trick with a smattering of Pythagorean triples.

Take a piece of paper and do the following:

1. Make it into a square (interesting discussion on best way to do this).
2. Fold in half then unfold so you have created crease along a vertical line of symmetry
3. Then take any corner and fold to the midpoint of the opposite edge. Press down to make a crease along the fold line
4. Unfold and now investigate all the triangles you have created, i.e. can you work out their lengths?

Here are some pictures, which also give some hints, although not a complete solution.  The result is very satisfying although I would love to find a way to show this that doesn’t require reams of algebra.  Any takers??