My GeoGebra manipulatives:
- Haga’s First Theorem with free-form paper-folding.
- Haga’s First Theorem with constrained point on the square’s edge.
- Haga’s First Theorem with guidance for exploration.
The history behind these is explained in this blog post.
For guidance in exploring the manipulative, you may find this helps when using the version with the constrained point:
If you click on the picture, you will see a larger version which you can download or print. You will still need to do a bit of calculator work, though… 😛
Addendum:
Since writing this, I have added a more ‘user-friendly’ version for exploring the theorem, but you’ll still need the calculator though!
If you prefer, you can also use the dynamic worksheet online, which doesn’t require a download of GeoGebra.
Sine of the Times: Dividing the Universe by Zero 
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Here’s a close description
http://www.cut-the-knot.org/pythagoras/PaperFolding/SegmentDivision.shtml
with links to other constructions
Thanks for your comment, Alexander. I provided links to theory behind the manipulative in my blog post. This page was just to make it easier for me to point people to the end results! Haga has three theorems related to dividing a paper square. An English version of them is at: http://www.origami.gr.jp/Archives/People/CAGE_/divide/02-e.html