Construction of a square with area the same
as that of a rectangle with sides a and b.
Given two segments of lengths a and b, to construct
the segment with length equal to the square root of a.b, we proceed
as follows:

Line up the two segments of length a and b on a single
line.

Draw the circle of center in the midpoint of the lined up
segments.

At the common extremity D to the two segments draw a perpendicular
segment to the diameter, meeting the circle at the point C.
CLAIM: The
square on the segment CD has the same area as the rectangle with sides
of length a^{.}b
WHY?

Any triangle inscribed in a semicircle
is a right triangle!

Triangles ABC, ACD and CBD are all similar right triangles.

Using the smaller triangles we have
the ratios between the corresponding
sides are the same
(being the magnification factor relating the two triangles):

so AD/CD
= CD/BD,

or by "cross multiplying"
CD^{.}CD
= AD^{.}BD
Conclusion: The
square on the side CD has the same area as the rectangle with sides a and
b.