My favourite technology-based maths problem

This is my favourite technology-based maths problem, and possibly my favourite maths problem.  It needs to be attempted in software that has linked geometrical and algebraic views (and preferably some others too).  GeoGebra or TI-Nspire are ideal for this.

Start with a square of variable side with one vertex on the origin and one on the positive x-axis:

The problem is to create a rectangle with the same area as the square whose sides are in the ratio 2:1.

Creating the initial square in GeoGebra

1. Add a point A at the origin.
2. Add a point B on the x-axis.
3. Use the Regular Polygon tool selecting vertices at A and B and set the number of vertices to 4.

Creating the initial square in TI-Nspire

1. Add a new Graphs page.
2. Add a point at the origin by finding the Intersection Point of both axes .
3. Add a new Point On the x-axis.
4. Display the Coordinates of this point.
5. Use Measurement Transfer to transfer the value of the x-coordinate to the y-axis.
6. Add a line Perpendicular to the axis through the point on x-axis.
7. Add a line Perpendicular to the axis through the point on y-axis.
8. Find the Intersection Point of the two lines.
9. Add a Polygon through the four points.  NB it is important to use the Polygon tool and not the Rectangle tool.
10. Measure the Area of the square.

Solutions

If you have a correct solution the area of the square and the rectangle should remain equal as you drag the point on the the positive x-axis.  I know 8½ distinct methods of constructing a solution using GeoGebra and 6½ using Nspire.  How many can you find?