Transformation of a rectangle to a square

To carry through this transformation initially we construct a square with the same area of the rectangle. That is made using straight edge and compass:
* we line up the two segments of lengths a and b, equal the base and height of the rectangle; we trace a circle with center in the midpoint of the lined up segments; at the common endpoint to the two segments we construct a segment perpendicular to the diameter, meeting the circle. This segment has legnth measured as the square root of a*b, that it is the length of the side of the desired square (to see demonstration)

Having made this construction, we now transform the rectangle to the square. If the side of the rectangle is greater that the double of the side of the square, we cut the rectangle in congruent lesser rectangles successively and "we pile up" these, in order to form a new rectangle. This is done so that, later, the clippings will work well:

Having a rectangle with the longer side that is less than two times the side of the square of same area, the figure below indicates the cutting to be done:

The construction is the following one:
* we overlap the square to the rectangle and trace straight lines FC, ED and BG
* the colors in them indicate clippings from which we obtain the transformation.

As before, intuitively we can see that the parts cut in the rectangle fit perfectly in the square. Formally, we must show the congruences between the pairs of triangles GDJ and BHE, and DCH and JFE. To see the demonstration here click

Two squares transformed into a square
It comes back to the Presentation