### 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)

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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