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:

1. Line up the two segments of length a  and b on a single line.
2. Draw the circle of center in the midpoint of the lined up segments.
3. 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 semi-circle 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):
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