This is fucking clever. Thanks!

Concept: http://i.imgur.com/2T11tBg.png
Practice: http://i.imgur.com/ijXsCB5.gifv
In practice, I cannot keep all segments because two segments cannot intersect at an angle less than 45 degrees. Therefore I have to delete the sqrt(2) and sqrt(3) legs when I do the sqrt(3) and 2 hypotenuses, respectively.
Hope this inspires you :)

Thank you for your feedback. Can you elaborate your way of creating an equilateral triangle since working with qsuareroot-2 and 3 sounds also pretty aüürpxo,ate, since these are irrational numbers.

One thing I want to note, is that the Bezier tool doesn't construct a perfect circular arc. The tool is based on a quadratic function, so it can only approximate a circular arc. I've explained this in my article on creating perfect pentagons. If you want it perfect, I suggest you construct the sides using the Pythagorean theorem, from 1 to square-root 2, square-root 3, and finally to 2. This creates an equilateral triangle. Then you can expand with this equilateral triangle to build a perfect hexagon.